AlgebraicandHermitian K-TheoryList of Lectures Lecture21(21st January,2021)164 The Quillen plus...

Lecture Notes for

Algebraic and Hermitian K-Theory

Lecturer

Fabian Hebestreit

Notes typed by

Ferdinand Wagner

Winter Term 202021University of Bonn

This text consists of unofficial notes on the lecture Advanced Topics in Algebra ndash Algebraicand Hermitian K-Theory taught at the University of Bonn by Fabian Hebestreit in thewinter term (Wintersemester) 202021

Some additions have been made by the author To distinguish them from the lecturersquosactual contents they are labelled with an asterisk So any Proof or Lemma etc that thereader might encounter are wholly the authorrsquos responsibility

Please report errors typos etc through the Issues feature of GitHub or just tell me beforeor after the lecture

ii

ContentsList of Lectures v

0 Introduction 1Organisational Stuff 1A Fairytale 2Outline of the Course 5

I Recollections and Preliminaries 7infin-Categories and Anima 7CocartesianLeft Fibrations and Luriersquos Straightening Equivalence 14Yonedarsquos Lemma Limits and Colimits in infin-Categories 21

Yonedarsquos Lemma and Adjunctions 21First Properties and Examples of (Co)Limits 23Colimits and the Yoneda embedding 31Kan Extensions 35Bousfield Localisations 44

The Rezk Nerve and Complete Segal Spaces 50

II Symmetric Monoidal and Stable infin-Categories 60E1-Monoids and E1-Groups 60Einfin-Monoids and Einfin-Groups 76Spectra and Stable infin-Categories 89An Interlude on infin-Operads 105

infin-Operads and Symmetric Monoidal infin-Categories 105The Cocartesian and Cartesian Symmetric Monoidal Structures 111Algebras over infin-Operads 117Day Convolution 120Stabilisation of infin-Operads and the Tensor Product on Spectra 125

Some Brave New Algebra 129Einfin-Ring Spectra 131infin-Categories of Modules 135Tensor Products over Arbitrary Bases 143Miscellanea 148

III The Group Completion Theorem and the K-Theory of Finite Fields 151Localisations of Einfin-Ring Spectra 151The Group Completion Theorem and K1(R) 156Towards the K-Theory of Finite Fields 160

Rational K-Theory 160The Cyclic Invariance Condition and Quillenrsquos Plus Construction 163Topological K-Theory 171

iii

Contents

A Sketch of Quillenrsquos Computation 178

IV The K-Theory of Stable infin-Categories 194The Modern Definition of K-Theory 194

K0 of a Stable infin-Category 194The S-Construction and the Q-Construction 198The Resolution Theorem 204

K-Theory as the Universal Additive Invariant 206Verdier Sequences 207The Additivity Theorem and a Proof of Theorem IV12 216The Fibration Theorem 229Localisation and Deacutevissage 236

Bibliography 248

iv

List of LecturesLecture 0 (27th October 2020) 2

From RiemannndashRoch to Quillenrsquos definition of K-theory Outline of the course

Lecture 1 (29th October 2020) 7Fabian gives a history lesson A brief recollection of infin-categories nerve functorsanima Joyalrsquos lifting theorem and Hom spaces in infin-categories

Lecture 2 (3rd November 2020) 11Examples ofinfin-categories DoldndashKan correspondence Straighteningunstraighteningequivalence and an invariant definition of cocartesianleft fibrations

Lecture 3 (5th November 2020) 17The Hom functor and the twisted arrow category Yonedarsquos lemma adjunctions(co)limits Some computations of (co)limits

Lecture 4 (10th November 2020) 25Localisations More properties of (co)limits Left Kan extensions for presheafcategories Hom anima in functor categories

Lecture 5 (12th November 2020) 35Kan extensions and a version of Theorem I2 for infin-categories Lots of examples toshow how algebraic topology becomes a triviality after enough infin-category theorymdash ldquoIrsquom so happy he [Peter Scholze] uses lsquoAnrsquo [to denote anima] I forced this onhim He wanted to call them lsquoAnirsquo Thatrsquos the name of Darth Vaderrdquo

Lecture 6 (17th November 2020) 44Bousfield localisations limits and colimits inside them Applications to derivedcategories of ringsmdash ldquoLook up a paper on Bousfield colocalisations And then cry yourself to sleeprdquo

Lecture 7 (19th November 2020) 50The Rezk nerve and (complete) Segal spacesanimamdash ldquoToday is brainfck time Therersquos no deep content except for the things Irsquom notsayingrdquo

Lecture 8 (24th November 2020) 57A sleep-deprived Fabian proves that Q(C) is a Segal infin-category and introducescartesian monoids

v

List of Lectures

Lecture 9 (26th November 2020) 64More on E1-monoids and E1-groups Group completionmdash ldquoThe fact that you can write down a group or a monoid in semester 2 is sort ofan accidentrdquo

Lecture 10 (1st December 2020) 69A pushout computation Monoidal 1-categories give monoidal infin-categories Defini-tion of algebraic and hermitian K-theory

Lecture 11 (3rd December 2020) 73Topological K-theory and its connection to deep theorems in geometric topologyCartesian commutative monoids and groups

Lecture 12 (8th December 2020) 78Semi-additive and additive infin-categories monoids and groups in these

Lecture 13 (10th December 2020) 88Spectra The recognition principle for infinite loop spaces Stable infin-categoriesmdash ldquoOld-timey topologists hate this proof I love itrdquo

Lecture 14 (15th December 2020) 95Spectra and prespectra spectrification Σinfin and the BarattndashPriddy-Quillen theoremAnother model for spectra Towards a symmetric monoidal structure on Sp

Lecture 15 (17th December 2020) 105infin-operads Symmetric monoidal infin-category as infin-operads strict vs lax monoidalfunctors Some examples A technical lemma about presheaves

Lecture 16 (22nd December 2020) 113Cocartesian and cartesian monoidal structures Algebras over infin-operads Dayconvolution Operadic limits and colimits The tensor product on spectramdash ldquoI didnrsquot feel like presenting this as a gift to you Itrsquos more like a punishment forChristmasrdquo

Lecture 17 (7th January 2021) 129Some tensor product calculations Einfin-ring spectra The infin-operads Assoc andLMod categories of modules over Einfin-ring spectra

Lecture 18 (12th January 2021) 137A userrsquos guide to Luriersquos treatment of brave new algebra Localisations of Einfin-ringspectra

Lecture 19 (14th January 2021) 155The group completion theorem Computation of K1(R) K1(R) as a generaliseddeterminantmdash ldquoAaahhh wersquore not gonna discuss more about this thingrdquo

Lecture 20 (19th January 2021) 159Primitive elements in coalgebra and Hopf algebra structures on rational homologyRational K-theory The cyclic invariance criterion

vi

List of Lectures

Lecture 21 (21st January 2021) 166The Quillen plus construction Topological K-theory Bott periodicity Adamsoperationsmdash ldquoMy mum is calling me [Rejects call] Now my phone has decided to tell methat my mum has called merdquo

Lecture 22 (26th January 2021) 176More on Adams operations A sketch of Quillenrsquos computation of the K-theory offinite fields

Lecture 23 (28th January 2021) 185Completions of spectra Suslin rigidity and further results in the K-theory of fieldsK0 of stable infin-categories

Lecture 24 (2nd February 2021) 198The Waldhausen S-construction and K-theory of stable infin-categories The S-construction vs the Q-construction The resolution theorem

Lecture 25 (4th February 2021) 206Lots of terminology named after Verdier Recollements The additivity theoremmdash ldquoToday there will be a definition thatrsquos longer than a page and then a theoremthatrsquos longer than a page and the theorem will essentially be trivialrdquo

Lecture 26 (9th February 2021) 218End of the proof of the additivity theorem Group completion for additive functorsRezkrsquos equifibrancy lemma

Lecture 27 (11th February 2021) 228The relative Q-construction Verdier quotients Waldhausenrsquos fibration theoremSome final remarks on Karoubi sequences inductive completions Quillenrsquos deacutevissageand non-connective K-theorymdash ldquoWhat is the original formulation of Waldhausenrsquos fibration theorem andwherersquos the fibrationrdquo ldquoI never understood thatrdquo

vii

Chapter 0

Introduction 0Organisational StuffUnder no circumstances you should call Fabian by his last name Also there will be oralexams in the weeks from 15thndash19th February and 22ndndash26th March

Disclaimer mdash These are not official lecture notes Instead Fabian uploads his ownhandwritten notes [AampHK] (please suggest a better shorthand) to the lecturersquos websiteFabianrsquos notes are an excellent resource and they please the eye with their purple and pinkcolour scheme So why should I bother typing my own notes This is because of two mainreasons(a) I like having my notes as one single document with clickable hyperlinks and a somewhat

reasonable file size (I once attempted to open Fabianrsquos also excellent but also handwrittenstraighteningunstraightening notes on my phone with slow mobile internetmdashdonrsquot trythis at home)

(b) Typing notes and polishing them after the lecture is a way to force myself to spendtime thinking about the lecture and in particular its technical details This can be alot of work but itrsquos the best way for me to learn stuff

I also took the opportunity to try and work out some skipped details or omitted proofsmyself (given Fabianrsquos ambitious goals for this lecture some cuts were unavoidable) and Imade a few additions in the hope of clarifying things or sometimes just to unconfuse myselfMy own additions are marked with an asterisk So whenever you encounter a Proof or aLemma or a list item (clowast) be extra careful for mistakes If you happen to spot some donot hesitate to tell me via GitHub or in person

Differences in Numbering mdash Unfortunately my numbers in Chapter I and likewiseat some points in the later chapters tend to diverge from Fabianrsquos notes mainly becausesome examples discussed there havenrsquot been part of the lecture (thatrsquos another reason tocheck out Fabianrsquos notes) Irsquom trying to keep the numbers aligned where possible and tominimize the offset where not

On a related note I sometimes rearrange the material a bit for example by upgrading aside note to the status of a lemma to be citable later Nevertheless I hope that these notesstayed mostly faithful to Fabianrsquos lecture

Acknowledgements mdash Massive thanks to Fabian Bastiaan Branko and Thiago forspotting countless errors and suggesting various improvements

1

A Fairytale

A FairytaleLecture 0

27th Oct 2020We start with an overview of the mathematical developments that eventually led to theinvention of K-theory Donrsquot worry if yoursquore not familiar with the stuff on the next fewpages it is neither a prerequisite for the lecture nor will it play any prominent role in it

Letrsquos begin in the 1850rsquos Consider a compact Riemann surface Σ and let D isin Z[Σ] bea divisor on Σ That is D =

sumDss is a formal sum of points s isin Σ with coefficients

Ds isin Z all but finitely many of which are zero For example if f Σ C is a meromorphicfunction one could consider the principal divisor D(f) given by

D(f)s =

a if f has a zero of order a at sminusa if f has a pole of order a at s0 else

For some divisor D =sumDss put degD =

sumDs and consider the C-vector space

M(D) = f Σ C meromorphic | D(f)s gt minusDs for all s isin Σ

An important problem in the theory of Riemann surfaces is to determine the dimensiondimM(D) Riemann proved the inequality dimM(D) gt degD + 1minus g(Σ) which was soonimproved upon by his student Roch who obtained what is famously known today as theRiemannndashRoch theorem

01 Theorem (RiemannndashRoch) mdash Let Σ be a compact Riemann surface of genus g(Σ)and D be a divisor on Σ Put Dor = KΣ minusD where KΣ is the divisor of any 1-form on ΣThen

dimM(D)minus dimM(Dor) = degD + 1minus g(Σ)

Note that KΣ and thus Dor are only defined up to a principal divisormdashin other wordsonly their divisor class is well-definedmdashbut thatrsquos all we need since both dimM(D) anddegD donrsquot change if D is replaced by D +D(f) for some meromorphic f Σ C

02 Example mdash Consider Σ = CP1 = C cup infin and choose D = ninfin Then M(D) isthe space of all meromorphic functions f CP1 C whose pole at infin has order at most nIn other words

M(D) =f C C holomorphic

∣∣∣∣ |f(z)| is bounded by C|z|n for somesuitable constant C gt 0 as |z|infin

To get KCP1 we can choose the meromorphic 1-form dz which is holomorphic on C and hasa pole of order 2 at infin (because d(zminus1) = minuszminus2dz has a pole of order 2 at 0) Thus thedivisor class of KCP1 is that of minus2infin Plugging in Theorem 01 gives

dimM(ninfin)minus dimM(minus(2 + n)infin) = n+ 1

For n = 0 we obtain M(0) sim= C since all bounded holomorphic functions on C are constantby Liouvillersquos theorem By the same reason M(ninfin) = 0 for n lt 0 Hence

dimM(ninfin) = n+ 1 for all n gt 0

which makes a lot of sense since we would expect (and have just proved) that M(ninfin) isprecisely the space of polynomials of degree 6 n in that case

2

A Fairytale

Now letrsquos try to restate the RiemannndashRoch theorem in more modern terms The firststep is to replace divisors by line bundles which can be done by means of the bijection

divisors on Σ principal divisors sim minus isomorphism classes of line bundles on Σ

which is in fact an isomorphism of abelian groups between the divisor class group Cl(Σ)whose group structure is inherited from Z[Σ] and the Picard group Pic(Σ) whose groupstructure is given by the tensor product of line bundles If a line bundle L corresponds to adivisor D under this isomorphism then the space M(D) corresponds to the C-vector spaceΓ(Σ L) of holomorphic sections of L Thus Theorem 01 can be restated as

dim Γ(Σ L)minus dim Γ(Σ T lowastΣotimes Lminus1) = degL+ 1minus g(Σ)

Observe that Γ(Σ L) = H0sheaf(Σ L) and Γ(Σ T lowastΣotimes Lminus1) = H1

sheaf(Σ L) by Serre dualitySo the term on the left-hand side can be interpreted as the ldquoEuler characteristicrdquo χ(Σ L)This was the starting point for a generalisation to arbitrary dimensions found by Hirzebruchwho was not only the founding father of all mathematics in Bonn after the war but alsoincredibly good at guessing the correct generalisations

03 Theorem (HirzebruchndashRiemannndashRoch) mdash Let E X be a holomorphic vectorbundle over a d-dimensional compact complex manifold X Then

χ(XE) =dsumi=0

(ci(E) cup Tddminusi(TX)

)

Here ci(E) is the ith Chern class of E and Tddminusi(TX) is the (dminus i)th Todd class of TX sothat the right-hand side lives in the cohomology group H2d(XZ) = Z and the above equationmakes sense

Still no sign of K-theory though This is when Grothendieck the master of them allentered the stage Recall that Hi(XE) = RiΓ(XE) Consider the canonical map f X lowastso that the global sections functor is canonically isomorphic to the pushforward along f Informulas flowast = Γ(Xminus) Grothendieckrsquos idea was to generalize Theorem 03 to arbitraryproper morphisms f X Y of complex manifolds by replacing the Hi(XE) (appearing inthe definition of χ(XE)) by RiflowastE This raises an immediate question though What issum

(minus1)iRiflowastE supposed to be The summands are coherent sheaves on Y after all which wecan add (using the direct sum) but surely not subtract one from another And that bringsus straight to K-theory

04 Definition mdash Let X be a complex manifold We define the 0th K-groups K0(X)and K0(X) as follows(a) K0(X) is the group completion of the monoid of isomorphism classes vector bundles

on X (the monoid structure is given by taking direct sums) modulo the relation[E] = [Eprime] + [Eprimeprime] for every short exact sequence 0 Eprime E Eprimeprime 0 (thatrsquos thecondition that was missing in the lecture)

(b) K0(X) is defined in the same way with vector bundles replaced by arbitrary coherentsheaves on X

3

A Fairytale

05 Theorem (GrothendieckndashRiemannndashRoch) mdash Let f X Y be a proper morphism ofcomplex manifolds It induces a morphism f =

sum(minus1)iRiflowast K0(X) K0(Y ) on K-groups

which fits into the following hellish commutative diagram

K0(X) K0(Y )

Hlowast(XZ) Hlowast(YZ)

f

Td(X) ch(minus) ch(minus) Td(Y )

flowast

The bottom line is the pushforward in cohomology (use Poincareacute duality on X then the usualpushforward flowast Hlowast(XZ) Hlowast(YZ) on homology and finally use Poincareacute duality on Yto get back to cohomology)

The theory of K0(X) (and its higher versions Ki(X)) is called topological K-theory andwas developed by Atiyah and Hirzebruch soon after Grothendieck had presented is result atthe Arbeitstagung in Bonn But K0(X) also has an algebraic analogue

06 Definition mdash Let R be a ring The 0th K-group K0(R) is the group completion ofthe monoid of finite projective R-modules (the monoid structure is given by taking directsums as usual)

Since every short exact sequence of projective R-modules splits we donrsquot need to divideout the relation from Definition 04 The group K0(R) is an interesting invariant of rings Itis the universal recipient for a ldquodimension functionrdquo for finite projective R-modules

07 Example mdash (a) If R = k is a field (or more generally a PID) then the usualdimension induces an isomorphism K0(k) simminus Z

(b) In general the map Z K0(R) induced by n 7 [Roplusn] for n gt 0 is injective iff R hasthe invariant basis number property

(c) If R = Q[G] for some finite group G then K0(R) =oplus

V isinIrr(G) Z where the indexingset Irr(G) is the set of isomorphism classes of irreducible G-representations Indeedin this case Q[G] =

prodV isinIrr(G) MatnV (End(V )) for some integers nV gt 0 holds by the

ArtinndashWedderburn theorem which easily gives the above characterisation

We can go one step further and give an ad hoc definition of K1(R)

08 Definition mdash The 1st K-group K1(R) = GLinfin(R)ab is the abelianisation of theinfinite general linear group GLinfin(R) = colimngt0 GLn(R)

Moreover if I sube R is an ideal and S sube R is a multiplicative subset then one can defineK-groups K0(I) and K0(RS) fitting into exact sequences

K1(R) minus K1(RI) partminus K0(I) minus K0(R) minus K0(RI)

K1(R) minus K1(R[Sminus1]

) partminus K0(RS) minus K0(R) minus K0

(R[Sminus1]

)

which look a bit too much like long exact cohomology sequences to be a coincidence Peopleactually managed to produce an ad hoc definition of K2(R) fitting into the sequences abovebut that was where consensus ended People suggested several non-equivalent definitions ofhigher K-groups and it was not at all clear how to continue

4

Outline of the Course

until Quillen came He realized that K0(R) and K1(R) could be written as homotopygroups of a certain simplicial group Let Proj(R) denote the symmetric monoidal groupoidof finite projective R-modules Now consider the inclusion

Picard groupoids sube symmetric monoidal groupoids

Here a symmetric monoidal groupoid (Gotimes) is a Picard groupoid if for all x isin G there existsa y isin G such that xotimes y 1 The above inclusion has a left adjoint (minus)grp and using thiswe can write

Ki(R) = πi N(

Proj(R)grp) for i = 1 2

Here N denotes the nerve constructionQuillenrsquos suggestion although he didnrsquot have the words to say that yet was to do the

same but in the setting of infin-categories That is we consider

Picard groupoids symmetric monoidal groupoids

Picard infin-groupoids symmetric monoidal infin-groupoids

sube

sube sube

(minus)grp

sube

(minus)infin-grp

(the objects on the bottom line are also known as grouplike Einfin-spaces and Einfin-spacesrespectively) In this framework we can finally define the higher K-groups

09 Definition mdash The ith K-group of a ring R is defined as Ki(R) = πi(Proj(R)infin-grp)(these are abelian groups) More importantly we define K(R) = Proj(R)infin-grp which is anobject of the infin-category of infin-groupoids (or anima as we will call them)

010 Warning mdash The functor

(minus)infin-grp symmetric monoidal infin-groupoids minus Picard infin-groupoids

is bloody complicated For example one has Finite setstinfin-grp = ΩinfinS = colimngt0 ΩnSnso even in the simplest casemdashfinite sets and disjoint unionmdashthe homotopy groups of whatcomes out are terrifying they are the stable homotopy groups of spheres

Outline of the CourseThe course will consist of three parts

Part 1 mdash wherein we laid the required infin-categorical foundations After a review ofFabianrsquos previous lectures on infin-categories we will discuss symmetric monoidal structureson infin-categories and anima In particular we will analyse Einfin-monoidsgroups and developthe theory of spectra and stable infin-categories Important examples of stable infin-categoriesare the infin-category Sp of of spectra and the derived infin-category D(R) of R-modules Wewill also prove that Picard infin-groupoids connective spectra which is a result due toBoardmanndashVogt and May

5


Part 2 mdash wherein we finally define K-theory Apart from that we will do some basiccomputations including Quillens computation of the K-theory of finite fields

Part 3 mdash wherein we do some ldquomodernrdquo K-theory In particular we will discuss K-theoryas a functor K Catst

infin Sp from the infin-category of stable infin-categories to the infin-categoryof spectra and sketch the proof of the basic results of ldquolocalisation resolution and deacutevissagerdquoFor this we will follow [LT19] as well as the series of papers [Fab+20I Fab+20II Fab+20III]that Fabian co-authored These papers are actually concerned with hermitian K-theorywhich arises if we replace Proj(R) by

Unimod(R) = (P q) | P is finite projective q is a unimodular form on R

Fabianrsquos original plan was to develop the algebraic and the hermitian theory simultaneouslyAs it turned out the hermitian side of things fell a bit short but still played a role We willalso talk about K-theory being the ldquouniversal additive invariantrdquo in the sense of [BGT13]

6

Chapter I

Recollections and Preliminaries Iinfin-Categories and Anima

Lecture 129th Oct 2020

The plan for today is to give a rapid review of Fabians lectures from the last two semestersNaturally we wonrsquot really prove anything but give references to Fabians notes [HCI HCII]instead

I1 Definition mdash The simplex category ∆∆ is the category of finite totally ordered setsand order preserving maps For C a category we put sC = Fun(∆∆op C) and cC = Fun(∆∆ C)and call this the categories of simplicial and cosimplicial objects in C

I2 Theorem mdash Let A be a small category and let C be a cocomplete category Then theYoneda embedding Y A Fun(AopSet) induces an equivalence of categories

Y lowast FunL(

Fun(AopSet) C) simminus Fun(A C)

Here FunL denotes the full subcategory of functors who are left adjoints (and thus admitright adjoints)

Proof In [HCI Theorem I41] we proved this statement with FunL replaced by Funcolimthe full subcategory of colimit-preserving functors But every such functor automaticallyadmits a right adjoint by [HCI Proposition II18] so Funcolim sube FunL As all left adjointspreserve colimits this inclusion is actually an equality

In particular we can apply Theorem I2 to A = ∆∆ and obtain FunL(sSet C) Fun(∆∆ C)(and of course the right-hand side is the category cC of cosimplicial objects of C) Given afunctor F ∆∆ C we denote the corresponding adjoint pair by

| |F sSet C SingF

In concrete terms | |F can be described as the left Kan extension

∆∆ C

sSet

F

Y LanY F=| |F

of F along the Yoneda embedding Y ∆∆ sSet Moreover as SingF is supposed to beright-adjoint to | |F it is necessarily given by

SingF (X)n = HomC(|∆n|F X

)= HomC

(F ([n]) X

)

7

infin-Categories and Anima

I3 Example mdash The following adjunctions arise in the way explained above(a) π0 sSet Set const where as usual π0 denotes the set of connected components(b) π sSet Cat N Here πX denotes the homotopy category of a simplicial set X and

N(C) the nerve of a category C(c) | | sSet Top Sing the adjunction that motivates the above notation(d) X times minus sSet sSet F(Xminus) for any simplicial set X In particular F(XY ) is

our notation for the simplicial set of maps between X and Y explicitly given byF(XY )n = HomsSet(X times∆n Y )

(e) C[minus] sSet CatsSet Nc taking a simplicial set X to the simplicially enriched categoryC[X] The right adjoint Nc is called the coherent nerve functor

I4 Definition (ldquoHorn filling conditionsrdquo) mdash A quasicategory or infin-category C is asimplicial set such that for all 0 lt i lt n and all solid diagrams

Λni C

∆n

there exists a dashed arrow as indicated rendering it commutative A simplicial set X is aKan complex if it is an infin-category and the above condition holds for i = 0 n as well Thefull subcategories of sSet spanned by quasicategories and Kan complexes are denoted qCatand Kan

If C and D areinfin-categories a functor F C D is a map of simplicial sets or equivalentlya 0-simplex in the simplicial set F(CD) which we usually denote Fun(CD) in this caseSimilarly a natural transformation η F rArr G between functors FG C D is a 1-simplexη in Fun(CD) connecting the 0-simplices F and G

The simplicial set Λni in Definition I4 is called the ith n-horn and is given by removingthe interior of ∆n as well as the face opposing its ith vertex The 2-horns look as follows

0

1

2

Λ20

0

1

2

Λ21

0

1

2

Λ22

The connection between infin-categories as defined in Definition I4 and ordinary categories(ldquo1-categoriesrdquo) is given by the following theorem

I5 Theorem (ldquoUnique horn fillingrdquo) mdash The nerve functor N Cat sSet is fullyfaithful with essential image given by those simplicial sets X such that the dashed arrow inDefinition I4 exists uniquely In fact for a simplicial set X the following are equivalent(a) X sim= N(C) for some category C(b) For all 0 lt i lt n the restriction F(∆n X) simminus F(Λni X) is an isomorphism

8


(c) For all n the restriction F(∆n X) F(In X) is an isomorphism Here In denotesthe spine of ∆n ie the union of all edges connecting consecutive vertices

Moreover an ordinary category C is a groupoid iff N(C) is a Kan complex

Proof In [HCI Theorem II25] we proved the equivalence of (a) (b) and (c) but withHomsSet(minus X) instead of F(minus X) However the weaker version already implies the strongerone We have

F(minus X)n = HomsSet(minustimes∆n X) = HomsSet(minusF(∆n X)

)

and if X sim= N(C) is the nerve of a category then

F(∆n X) = F(

N([n])N(C))

= N Fun([n] C)

so we are mapping into the nerve of a category again The right equality follows froma straightforward calculation using that N Cat sSet is fully faithful The additionalassertion about groupoids and Kan complexes is addressed in Theorem I13

I6 Example mdash ldquoMy first Kan complexesrdquo(a) For all topological spaces X isin Top the simplicial set SingX is a Kan complex which

is pretty easy to check(b) Any simplicial group is a Kan complex Thatrsquos not entirely obvious but not hard See

[Stacks Tag 08NZ] for example(c) If C and D are 1-categories then N(Fun(CD)) sim= Fun(N(C)N(D))

I7 Theorem (KanJoyal) mdash Let C be a simplicial set(a) If C is a Kan complex or aninfin-category then the same holds for F(X C) for all simplicial

sets X(b) C is an infin-category iff F(∆n C) F(Λni C) is a trivial fibration for all 0 lt i lt n which

is again equivalent to F(∆n C) F(In C) being a trivial fibration for all n(c) C is a Kan complex iff the conditions from (b) hold and in addition F(∆n C) F(Λni C)

are trivial fibrations for i = 0 nActually in (b) and (c) it suffices to have the respective conditions only for n = 2

Proof The ldquoifrdquo parts of (b) and (c) follow from the fact that trivial fibrations are surjectiveFor the rest of (a) (b) and (c) see [HCI Corollary V223 and Corollary VI24] The factthat C is already an infin-category if only F(∆2 C) F(Λ2

1 C) is a trivial fibration was provedin [HCI Corollary VI25] This easily implies the corresponding assertion for Kan complexesIf F(∆2 C) F(Λ2

i C) is surjective for i = 0 2 then every morphism in C has a left- andright-inverse hence C is a Kan complex by Theorem I13

I8 Warning mdash In view of Theorem I7(b) it seems tempting to define infin-categoriesas simplicial sets that have lifting against In sube ∆n But thatrsquos wrong The class of simplicialsets obtained in that way called composers is larger than the class of infin-categories If youwant to replace Λni by In you really need the stronger condition that F(∆n C) F(In C)is a trivial fibration rather than just surjective on 0-simplices

9


I9 Definition mdash Let C be an infin-category For 0-simplices a b isin C0 define their homspacemapping animaany combination of these as the pullback

HomC(a b) Fun(∆1 C)

∆0 C times C

(st)(ab)

Here s t Fun(∆1 C) send a 1-simplex in C to its source and target respectively Theinfin-category Fun(∆1 C) is also called the arrow category of C and denoted Ar(C)

I10 Example mdash (a) For any 1-category D one has HomN(D)(a b) sim= const HomD(a b)In other words HomN(D)(a b) is a discrete simplicial sets and it corresponds to theHom set in the original category

(b) If C is an infin-category then the unit adjunction C N(πC) of the (πN) adjunctioninduces isomorphisms

HomN(πC)(a b) sim= const HomπC(a b) sim= constπ0 HomC(a b)

In other words πC is the 1-category having C0 as objects and homotopy classes of C1 asmorphisms

I11 Definition mdash A infin-category C is called an infin-groupoidmdashor in Scholzersquos fancy newterminology an animamdashif πC is a groupoid

I12 Example mdash (a) For a 1-category D let coreD denote the (usually non-full) sub-category spanned by the isomorphisms in D in particular coreD is a groupoid If C isan infin-category we define core C by the pullback

core C C

N(coreπC) N(πC)

Then core C is an anima In fact one can check that it is the largest anima contained inthe infin-category C

(b) If C is an infin-category then HomC(a b) is an anima for all a b isin C (thatrsquos not entirelyobvious though)

Itrsquos pretty easy to check that every Kan complex is an anima Surprisingly and that wasone of the first hard theorems in infin-category theory the converse holds as well

I13 Theorem (Joyal) mdash Every infin-groupoid is in fact a Kan complex

Proof This follows from Joyalrsquos lifting theorem see [HCI Theorem VI320]

By now we havenrsquot been able to produce any examples of infin-categories yet safe fornerves of 1-categories (these guys donrsquot count) The first real source of interesting examplesis given by coherent nerves of Kan enriched categories

10


I14 Theorem (CordierndashPorter) mdash If C isin CatKan is a category enriched in Kan com-plexes then its coherent nerve Nc(C) is an infin-category Moreover for all a b isin C we havecanonical homotopy equivalences

HomNc(C)(a b) FC(a b)

Here FC denotes Kan complex of morphisms in the enriched category C

Proof See [HCI Theorem VII19] (but be warned that stuff gets technical)

I15 Example mdash ldquoMy first infin-categoriesrdquo(a) The full subcategory Kan sube sSet is enriched over itself via FKan(XY ) = F(XY ) Its

coherent nerve Nc(Kan) = An is called the infin-category of anima(b) The full subcategory qCat sube sSet is enriched over Kan via FqCat(CD) = core F(CD)

Its coherent nerve Nc(qCat) = Catinfin is the infin-category of infin-categories Note thatFabian will usually write Cat instead of Catinfin to not drag the index infin around all thetime

(c)Lecture 23rd Nov 2020

The category Cat1 of all small 1-categories is canonically enriched in groupoids viaFCat1(CD) = core Fun(CD) This enrichment can be upgraded to a Kan enrichmentNamely taking the nerves of all hom groupoids (which produces Kan complexes byTheorem I13) induces a functor Nlowast CatGrpd CatKan We thus obtain aninfin-categoryCat(2)

1 = Nc(NlowastCat1) called the 2-category of 1-categories1 Similarly one can define a2-category Grpd(2)

1 We end up with a chain of inclusions

An

N(Set) Grpd(2)1 Catinfin

Cat(2)1

subesube

sube

sube sube

All of these induce homotopy equivalences on mapping anima In other words they arefully faithful

(d) The category Top is Kan enriched via FTop(XY )n = HomTop(X times |∆n| Y ) whichprovides the infin-category of topological spaces after taking coherent nerves However bewarned that Nc(Top) 6 An

(e) Let R be a ring and Ch(R) the category of chain complexes over R It is enriched oversimplicial R-modules and thus Kan-enriched by Example I6 via

FCh(R)(CD)n = HomCh(R)(C otimesZ C

simpbull (∆n) D

)

Here Csimpbull (∆n) is a chain complex of abelian groups given as follows In degree m

we put Z[non-degenerate m-simplices of ∆n] and the differentials are given by partm =summi=0(minus1)idi where di is induced by the corresponding face map di (∆n)m (∆n)mminus1

which preserves non-degenerate simplices (that doesnrsquot hold for arbitrary simplicial sets1Here ldquo2-categoryrdquo means that πi HomCat1 (CD) = 0 for i gt 2 To prove this use Theorem I14 and checkthat nerves of groupoids have vanishing higher homotopy groups

11


though) You probably recognize Csimpbull (X) as the complex that computes the simplicial

homology groups Hsimpi (X)

We let K(R) = Nc(Ch(R)) be the infin-category of chain complexes (this ldquoKrdquo hasnothing to do with ldquoK-theoryrdquo) and let Kgt0(R)K60(R) sube K(R) the full sub-infin-categories spanned by chain complexes concentrated in nonnegativenonpositive degreesInside Kgt0(R) there is the full sub-infin-category Dgt0(R) sube Kgt0(R) spanned by thedegreewise projective chain complexes in concentrated nonnegative degrees This iscalled the bounded below derived category of R Similarly there is the bounded abovederived category D60(R) sube K60(R) spanned by degreewise injective chain complexesconcentrated in nonpositive degrees

Finally there is Dperf(R) sube K(R) which is the full sub-infin-category spanned bybounded complexes of degreewise finite projective modules Once wersquove defined K-theory wersquoll eventually see that K(R) = Proj(R)infin-grp K(Dperf(R)) (and yes thistime ldquoKrdquo is really the ldquoKrdquo from ldquoK-theoryrdquo)

I15a Exercise mdash For complexes CD isin Ch(R) let HomR(CD) denote the internalHom in Ch(R) so that HomR(Cminus) is right-adjoint to C otimesR minus (see the nLab article for anexplicit construction) Show that the Kan enrichment in Example I15(e) satisfies

πi FCh(R)(CD) sim= Hi

(Hom(CD)

)for all i gt 0

In particular π0 FCh(R)(CD) is the set of chain homotopy classes of maps C D

I16 Theorem (DoldndashKan) mdash By a variant of Theorem I2 the cosimplicial object Cbull inChgt0(Z) given by [n] 7 Csimp

bull (∆n) gives rise to mutually inverse equivalences of (ordinary)categories

| |Cbull sAb simsim Chgt0(Z) SingCbull

These equivalences translates homotopy groups into chain homology groups Moreover wehave FCh(Z)(CD) = SingCbull(HomZ(CD)) for all CD isin Chgt0(Z)

Proof See [HA Subsection 123] it takes a bit of fiddling though to check that ourconstructions coincide with the standard ones (ie the ones Lurie uses)

I17 Philosophical Nonsense I mdash We obtain a chain of adjunctions

Top sSet sAb Chgt0(R)Sing

| | Z[minus]

forget

| |Cbull

SingCbull

(all bottom arrows are right-adjoints thatrsquos why the direction of the arrows on the leftis swapped) Going from Top to Chgt0(Z) sends a topological space X to its singularchain complex Csing

bull (X) Going in the reverse direction sends a chain complex C to thecorresponding generalized EilenbergndashMacLane space In particular if C = A[n] is given byan single abelian group A sitting in degree n then it is sent to K(An)

I18 Philosophical Nonsense II mdash To describe what an unbounded chain complexis it suffices to understand bounded below chain complexes Namely a chain complexC isin Ch(R) can be described by the sequence of truncations τgtiC isin Chgti(R) These aregiven by

τgtiC =(

parti+3minusminusminus Ci+2


parti+1minusminusminus ker parti minus 0 minus 0 minus

)

12


(this is chosen in such a way that the canonical map τgtiC C induces isomorphisms onhomology in degrees gt i) One easily checks the following relation with way to many brackets

(τgtiC)[minusi] sim= τgt0

(((τgtiminus1C)[minus(iminus 1)]

)[minus1]

)

Note that under the DoldndashKan correspondence (Theorem I16) the operation C 7 τgt0(C[minus1])on Chgt0(Z) corresponds to X 7 Ω0X on sAb Here Ω0 denotes the simplicial loop space (orat least some model for it) with base point 0 isin X0 ie the neutral element of the simplicialabelian group X So if we write Xi = |(τgtiC)[minusi]|Cbull then we obtain canonical isomorphismsXisim= Ω0Ximinus1 I wonder where Irsquove seen that before so letrsquos define an abelian spectrum to be a sequence of simplicial abelian groups

Xi together with isomorphisms Xisim= Ω0Ximinus1 In that way Theorem I16 extends to an

equivalence abelian spectra Ch(Z) But thatrsquos only the beginning of a long story

I19 Theorem (Joyal) mdash Let C and D be infin-categories(a) A functor F C D of infin-categories is an equivalence (ie there exists G D C

and natural equivalences F G idD and G F idC) if and only if it is essentiallysurjective and fully faithful (ie the induced map Flowast π0 core C π0 coreD is surjectiveand Flowast HomC(x y) HomD(F (x) F (y)) is a homotopy equivalence of anima for allx y isin C)

(b) A natural transformation η F rArr G of functors FG C D is a natural equivalence(ie η is an isomorphism in the homotopy category π Fun(CD)) if and only if inducesequivalences on 0-simplices ie ηx F (x) G(x) is an equivalence for all x isin C

Proof See [HCI Theorem VII1 Theorem VII8]

In 1-category theory Theorem I19 is a triviality (up to axiom of choice business) But ininfin-land it is hard An infin-category contains infinitely more data than just a bunch of objectsand morphism spaces All the more surprising that these actually suffice to characterizeequivalences

As an application we get a precise formulation of Grothendieckrsquos conjectural connectionbetween (infin-)groupoids and spaces that became famously known as the homotopy hypothesis(at that time Grothendieck was already way beyond scribbling devils around his diagramsliving somewhere deep in the Pyrenees)

I20 Corollary (Grothendieckrsquos homotopy hypothesis) mdash Let CW denote the categoryof CW-complexes with its Kan enrichment given by FCW(XY ) = Sing Map(XY ) Thefunctors

| | An Nc(CW) Sing

are inverse equivalences

Proof See [HCI Corollary VII6]

This finishes our recollection of Fabianrsquos Higher Categories I lecture and we start reviewingthe material from Higher Categories II

13

CocartesianLeft Fibrations and Luriersquos Straightening Equivalence

CocartesianLeft Fibrations and Luriersquos StraighteningEquivalenceThe main goal of Fabianrsquos Higher Categories II lecture last semester was to construct afunctor HomC Cop timesC An and prove the infin-analogue of Yonedarsquos lemma We know whatHomC ought to do on 0-simplices of C by Definition I9 but making it into a functor isincredibly complicatedmdashas is any attempt at defining functors into An or Catinfin The key todefine such functors is Luriersquos straighteningunstraightening equivalence

I21 Theorem (Lurie) mdash Let lowastAn denote (any model for) the slice category of animaunder a point To each functor F C An of infin-categories associate an infin-category Un(F )via the pullback

Un(F ) lowastAn

C An

F

This association upgrades to a fully faithful functor Un Fun(CAn) CatinfinC (the ldquoun-straighteningrdquo functor) whose essential image is the full subcategory Left(C) spanned by theleft fibrations over C (to be defined in Definition I24) We let St Left(C) Fun(CAn)(ldquostraighteningrdquo) denote an inverse

Proof This follows from the combined effort of all we did in Higher Categories II so letme randomly pick [HCII Remark X57(iii)] as a reference

I22 Example mdash For any x isin C the straightening of the left fibration xC C is definedto be HomC(xminus) C An That this makes sense can be seen as follows Observe that thefibre of Un(F ) C over some y isin C is just F (y) since the fibre of lowastAn An over someanima X is just X itself Conversely St(p E C)(y) (fibre of G over y) holds for all leftfibrations p E C In particular evaluating HomC(xminus) at y gives the fibre of xC over ywhich is indeed HomC(x y)

Theorem I21 generalizes as follows

I23 Theorem (Lurie again of course) mdash For any infin-category C the equivalences fromTheorem I21 extend to inverse equivalences

St Cocart(C) simsim Fun(CCatinfin) Un

where Cocart(C) is the (non-full) subcategory of CatinfinC spanned by cocartesian fibrationsand maps preserving cocartesian edges (to be defined in Definition I24)

Proof Again I should really cite the entirety of Fabianrsquos previous lecture The proof ofthe statement in question finishes in [HCII Chapter X p 113]

There is also a dual version of Theorem I23 which asserts that there is an equivalenceCart(C) Fun(CopCatinfin) induced by the cartesian straightening and unstraighteningfunctors Whenever there is room for confusion wersquoll use Stcocart Uncocart for the cocartesianversions and Stcart Uncart for the cartesian versions

With that out of the way letrsquos define what this ldquococartesianrdquo thing actually is

14


I24 Definition mdash Let p E C be a functor of infin categories(a) Informally an edge f x y in E is called p-cocartesian if the following condition holds

For all edges g x z in E and all fillers σ ∆2 C as depicted in the right diagrambelow

x z

y

g

f

exist()

p7minus

p(x) p(z)

p(y)

p(g)

p(f)σ

there exists a unique (up to contractible choice) filler τ ∆2 E satisfying p(τ) = σIn other words the space of lifts of σ is contractible

Somewhat more formally f x y is p-cocartesian if for all z isin E the diagram

HomE(y z) HomE(x z)

HomC(p(y) p(z)

)HomC

(p(x) p(z)

)flowast

plowast plowast

p(f)lowast

is a homotopy pullback diagram of anima However that doesnrsquot mean the above is apullback diagram taken inside the 1-category Kan (or sSet)mdashin fact it likely doesnrsquoteven commute on the nose but only up to homotopy And worse The ldquoprecompositionrdquomaps flowast and p(f)lowast are not even canonically defined To construct flowast one has to choosea ldquocomposition lawrdquo in E ie a section of the trivial fibration Fun(∆2 E) Fun(Λ2

1 E)but there is no canonical one

Despite all these difficulties there is a way to give a well-defined definition ofhomotopy pullbacks using model categories (see [HCII Example VIII49(vi)]) Howeveronce we have the infin-categorical machinery available there is a better definition Adiagram of anima is a homotopy pullback if it is a pullback diagram taken inside theinfin-category An The connection to the model-theoretic definition will be made precisein Theorem I34 below

(b) Let p-Cocart sube Ar(E) denote the full subcategory spanned by the cocartesian edgesThen p is called a cocartesian fibration if the dashed arrow q in the diagram below is anequivalence of infin-categories

p-Cocart

P Ar(C)

E C

simq

s

p

s

p

Informally the way to think about this is that all edges of C admit a lift with givenstarting point Also p-Cocart P is automatically fully faithful as wersquoll prove below

(c) We call p a left fibration if it is a cocartesian fibration and satisfies the followingequivalent conditions(i) St(p) C Catinfin factors through An

15


(ii) The derived fibres of p are anima By that we mean the fibres of E prime C whereE simminus E prime C is any factorisation of p into an Joyal equivalence followed byfibration in the Joyal model structure Or just take the fibres as usual but withpullbacks inside the infin-category Catinfin

(iii) p is conservative ie for all edges f in E we have that f is an equivalence iff p(f)is an equivalence in C

(iv) All edges in E are p-cocartesianThere is also a dual notion of cartesian edgesfibrations and right fibrations An edge ofE is p-cartesian if its opposite is a cocartesian edge with respect to pop Eop Cop (inother words we have to reverse all the arrows in the informal part of (a)) Similarly p is acartesianright fibration if pop is a cocartesianleft fibration

In the lecture we had a brief discussion about how Definition I24 relates to the definitionthat the participants of Fabianrsquos previous lectures would have expected Irsquoll give an expandedversion of that discussion below To be safe everything is labelled with an asterisk to indicateit is an addition to the lecture

I24a Warning mdash Definition I24 does not coincide with the way we definedcocartesian and left fibrations in Fabianrsquos previous lectures The main difference is that wedonrsquot require p to be an inner fibration resulting in the advantage that Definition I24 isinvariant under Joyal equivalence (Fabian promised that this will save us some headache) Upto replacing p E C by an isofibration both definitions agree as the following Lemma I24bshows

Also note that it doesnrsquot suffice to replace p by an inner fibration In fact the inclusion0 J of a point into the free-living isomorphism is an inner fibration (as the nerve of amap of 1-categories) and a Joyal equivalence hence it satisfies Definition I24(b) but it isnot an isofibration and thus not cocartesian in the old sense by [HCII Proposition IX2]

I24b Lemma mdash The dashed arrow q in Definition I24(b) is automatically fully faithfulIn particular for a functor p E C the following conditions are equivalent(a) p is a cocartesianleft fibration in the new sense(b) For all factorisations E simminus E prime C into a Joyal equivalence followed by a fibration in

the Joyal model structure E prime C is a cocartesianleft fibration in the old sense(c) The above condition holds for some factorisation E simminus E prime C as aboveMoreover the four conditions from Definition I24(c) are indeed equivalent

Proof Let timesR denote derived pullbacks (or homotopy pullbacks thatrsquos the same) Letf x y and g xprime yprime be cocartesian edges Since p-Cocart sube Ar(E) is a full subcategorywe can use the calculation of Hom anima in arrow categories from [HCII Proposition VIII5]to obtain

Homp-Cocart(f g) HomE(x xprime)timesRHomE(xyprime) HomE(y yprime) Our definition of Hom anima in Definition I9 commutes with pullbacks of simplicial sets (aslong as the pullback still gives an infin-category) so we get

HomP

(q(f) q(g)

)= HomE(x xprime)timesHomC(p(x)p(xprime)) HomAr(C)

(p(f) p(g)

)

This is a pullback on the nose but also a homotopy pullback because s Ar(C) C is aninner fibration (even a cartesian fibration by the dual of Example I25 this also implies

16


that P is an infin-category) hence slowast HomAr(C)(p(f) p(g)) HomC(p(x) p(xprime)) is a Kanfibration Now replacing HomE(y yprime) by the homotopy pullback from Definition I24(a)HomAr(C)(p(f) p(g)) by its description from [HCII Proposition VIII5] and simplifyingthe resulting homotopy pullbacks shows that qlowast Homp-Cocart(f g) HomP (q(f) q(g)) isindeed an equivalence hence q is fully faithful

To prove equivalence of (a) (b) (c) note that Definition I24(b) is invariant under Joyalequivalences so we may assume p E C is a fibration in the Joyal model structure orequivalently an isofibration since both are infin-categories If p is cocartesian in the old sensethen q p-Cocart P is surjective on the nose hence a Joyal equivalence by what we justproved This is enough to prove (b) rArr (c) rArr (a) For (a) rArr (b) we may again assume p isan isofibration and we must show that q is surjective on the nose rather than just essentiallysurjective We will even show that q is a trivial fibration Since q is a Joyal equivalenceit suffices to show that it is an isofibration Observe that Ar(E) P is an isofibration by[HCI Corollary VII11] Hence its restriction q to the full subcategory p-Cocart sube Ar(E) isan inner fibration But p-Cocart is closed under equivalences in Ar(E) (which is evident fromthe homotopy pullback condition in Definition I24(a)) so an easy argument shows that qalso inherits lifting agains 0 J from the isofibration Ar(E) P

Now that we know equivalence of (a) (b) (c) (except for the assertions about left fibrationsbut these will follow immediately) itrsquos easy to show that the conditions in Definition I24are indeed equivalent Note that the values of St(p) are given by the (derived) fibres of phence (i) hArr (ii) and (ii) hArr (iii) hArr (iv) follows from [HCII Proposition IX3]

I25 Example mdash ldquoMy first cocartesian edgesfibrationsrdquo(a) Any equivalence in E is p-cocartesian by Joyalrsquos lifting theorem(b) The ldquotarget morphismrdquo t Ar(C) C (given by restriction along 1 ∆1) is a

cocartesian fibration (in both the old and new sense since it is an isofibration) Letf x xprime and g y yprime be objects in Ar(C) Then a morphism σ f g in Ar(C) iea commutative square

x y

xprime yprime

f σ g

in C is a t-cocartesian edge if and only if the induced morphism σ0 x y is anequivalence in C We will prove this in Lemma I25a below (Fabian gave the idea inthe lecture but some more details canrsquot hurt)

This immediately implies that t is a cocartesian fibration because for every objectf x xprime of Ar(C) and all morphisms xprime yprime in C we can take

x x

xprime yprime

f

as a t-cocartesian lift However t is usually not a left fibration The straightening St(t)is the slice category functor Cminus C Catinfin which not necessarily factors over An

Lecture 35th Nov 2020

As a fun fact let us mention that t Ar(C) C is also a cartesian fibration providedC has pullbacks (wersquoll talk more about limits in infin-categories soon) which we leave asExercise I25b Spoiler warning Irsquove also included my attempt at a solution there

17


(c) For all x isin C the slice category projection Cx C is a cartesian fibration and infact even a right fibration since its (derived or non-derived) fibre over y is the animaHomC(y x)

I25a Lemma mdash With notation as in Example I25(b) σ is t-cocartesian iff the inducedmorphism σ0 x y is an equivalence in C

Proof Indeed if x y is an equivalence then an easy calculation involving the characteri-sation of Hom anima in arrow categories ([HCII Proposition VIII5]) shows that σ satisfiesthe homotopy pullback condition from Definition I24(a) Conversely if σ is cocartesianthen any diagram

z

x y

zprime

xprime yprime

h

f

g

in which the front face is σ and only the top and right back face are missing admits afiller (at least up to equivalence of such diagrams) Choose the left back face λ as followsPut z = x and zprime = yprime also choose λ0 x x as the identity on x and λ1 xprime yprime asσ1 xprime yprime and h x yprime as a composition of f and σ1 in C Then a filler of the abovediagram provides a left inverse of x y Moreover the morphism ρ g h in Ar(C) inducedby the right back face ρ is again t-cocartesian This is because λ is a composition of ρ and σand both σ and λ are t-cocartesian (for λ this follows from x z being the identity on xhence an equivalence) so [HCII Proposition IX5] can be applied Therefore we may repeatthe all the arguments for ρ instead of σ which shows that the left inverse of σ0 x y has aleft inverse itself whence σ0 must indeed be an equivalence

I25b Exercise mdash Show that C admits pullbacks (in the infin-categorical sense defined inExample I33(b) below) iff the cocartesian fibration t Ar(C) C is also a cartesian fibrationMore generally the t-cartesian edges are given precisely by the pullback squares

Proof sketch If the claimed characterisation of cartesian edges is true then the liftingcondition for t to be a cartesian fibration is satisfied if and only if C has all pullbacks So itsuffices to show the second assertion

Wersquoll freely use the yoga of limits ininfin-categories which is developed from Example I33(b)onward By Corollary I50 HomC(zminus) C An preserves all limits for every z isin C Alsolimits in An can be computed by homotopy limits thanks to Theorem I34 below

Now let f x xprime and g y yprime be objects of Ar(C) and let σ f g be an edge inAr(C) Then σ is cartesian iff for all h z zprime we can compute HomAr(C)(h f) by a certainhomotopy pullback dual to that in Definition I24(a) Writing that condition out explicitlyand simplifying using the description of Hom anima in arrow categories given in [HCIIProposition VIII5] we obtain the condition

HomC(z x)timesRHomC(zxprime) HomC(zprime xprime) HomC(z y)timesRHomC(zyprime) HomC(zprime xprime)

If z = zprime and h = idz then this condition precisely says that x is a pullback of y yprime xprime

by our remark on HomC(zminus) in the previous paragraph We should perhaps explain why

18


it suffices to check that on objects z isin C This is because one has in fact a naturaltransformation HomC(minus x) rArr HomC(minus y)timesRHomC(minusyprime) HomC(minus xprime) of functors Cop An(to construct this one has to dive into the straightening construction) and whether this isan equivalence can indeed be checked object-wise by Theorem I19 Conversely if x is apullback of y yprime xprime one easily checks that the above equivalence holds

I26 Constructing the functor HomC Cop times C An mdash There are at least fourequivalent ways to do so(a) Recall An = Nc(Kan) so Cop times C An corresponds to a simplicially enriched functor

C[Cop times C] Kan This is provided by the commutative square

C[Cop times C] Kan

C[C]op times C[C] sSetFC[C]

Exinfin

where as usual FC[C] C[C]optimesC[C] sSet denotes the simplicial set of morphisms in thesimplicially enriched category C[C] Instead of Exinfin we could use any functorial weakequivalence into a Kan complexes for example Lurie uses the functor X 7 Sing |X| in[HTT sect513]

(b) We construct the corresponding left fibration to CoptimesC explicitly It is the twisted arrowcategory TwAr(C) defined by TwAr(C)n = HomsSet((∆n)op ∆n C) The functorialinclusions (∆n)op t∆n sube (∆n)op ∆n induce a projection

TwAr(C) minus Cop times C

which turns out to be a left fibration with the correct fibres See [HA sect521] for proofsThe twisted arrow category construction of HomC is the one that is most commonlyused today

(c) Fabianrsquos favourite construction considers the commutative diagram

Ar(C) C times C

C

(st)

t pr2

which is a morphism in Cocart(C) (in particular (s t) preserves cocartesian edges)by Example I25(b) and the following simple fact For all infin-categories C and D theprojection pr2 CtimesD D is cartesian and cocartesian and the pr2-cartesiancocartesianedges are those of the form (equivalence in C arbitrary morphism in D)

Applying the cocartesian straightening construction gives a natural transformationStcocart(t)rArr const C of functors C Catinfin by Theorem I23 This natural transforma-tion defines a map ∆1 Fun(CCatinfin) hence a functor S C Ar(Catinfin) by adjoiningInformally S takes x isin C to the object Cx C in the arrow category of Catinfin Inparticular these are right fibrations by Example I25(c) and the target is always Cwhence S factors through Right(C) Ar(Catinfin) We denote the resulting functorS C Right(C) as well Composing S with Stcart Right(C) simminus Fun(CopAn) thecartesian straightening equivalence finally provides a functor

Y C C minus Fun(CopAn)

19


(the Yoneda embedding) which gives the desired HomC Cop times C An after ldquocurryingrdquo(which is how Fabian sometimes calls the adjunction Fun(CFun(D E)) Fun(CtimesD E))

(d) You can also dualize all steps in (c) and start with

Ar(C) C times C

C

(st)

s pr1

I27 Theorem mdash All constructions in I26 give the same functor HomC Cop times C Anup to equivalence

We wonrsquot prove Theorem I27 Lurie shows that I26(a) and (b) are equivalent in [HAProposition 52111] and for equivalence of (c) and (d) you probably shouldnrsquot go for a directproof but show that either one is equivalent to (b) Fabian believes that if one works withany of these constructions long enough it will become obvious that the others are equivalent

I28 More on the twisted arrow category mdash By I26(b) and Theorem I21 thetwisted arrow category of C sits inside a pullback (up to equivalence or take the pullback inCatinfin)

TwAr(C) lowastAn

Cop times C An

HomC

(if you donrsquot want to use I26(b) to construct HomC this pullback square can also be takenas a definition of TwAr(C))

Unwinding this shows that the objects of TwAr(C) are given by pairs (x y) isin Cop times Ctogether with a map f ∆0 HomC(x y) In other words objects are arrows f x y inC Informally an edge f g in TwAr(C) between f x y and g xprime yprime is given by aldquotwisted squarerdquo (∆1)op ∆1 C and a 2-simplex ∆2 An as follows

x xprime

y yprime

f

ms

g

mt

andHomC(x y) HomC(xprime yprime)

∆0

mlowastsmtlowast

f g

More formally since TwAr(C) sits in the pullback above HomTwAr(C)(f g) can be computedas the pullback of the corresponding Hom anima in Cop times C An and lowastAn This pullback isalso a homotopy pullback as lowastAn An is an inner fibration (even a left fibration in theold sense) hence induces Kan fibrations on Hom anima Plugging in the calculation of Homanima in slice categories from [HCII Corollary VIII6] thus provides a homotopy pullback

HomTwAr(C)(f g) g

HomC(xprime x)timesHomC(y yprime) HomC(xprime yprime)

h

where the bottom arrow sends (msmt) 7 mt f ms for some choice of composition

20

Yonedarsquos Lemma Limits and Colimits in infin-Categories

The twisted arrow category will be important to us since it has to do with spans inC This becomes apparent when we look at TwAr(∆n) Its objects are arrows in ∆nWe organize these arrows in an upwards pointing triangle with rows numbered 0 1 nfrom the bottom to the top In the ith row we put the morphisms that ldquogo up by irdquo ie(0 i) (1 i+ 1) (nminus i n) In particular the bottom row contains the identitiesand the top row contains only the morphism (0 n) If we draw in some of the edges inTwAr(C) (every other edge can be obtained as a composite of these) we obtain a picture asfollows

And voilagrave plenty of spans The corresponding picture for Ar(∆n) is instead the following

As a bit of foreshadowing Fabian mentioned that the first picture is related to the QuillenQ-construction (see I70) whereas the second is related to the Segal S-construction (seeConstruction IV5) which people keep calling Waldhausen S-construction even thoughWaldhausen himself wrote Segal S-constructionI28a Exercise mdash Show that core TwAr(C) core Ar(C) holds for all infin-categories CProof sketch See [AampHK Proposition I31] for a rough sketch

Yonedarsquos Lemma Limits and Colimits in infin-CategoriesYonedarsquos Lemma and AdjunctionsNow that we have a sufficient supply of mutually equivalent definitions of the Hom functorthe next step is to prove Yonedarsquos lemmaI29 Theorem (Yonedarsquos lemma) mdash Let C be aninfin-category Given a functor F C Anand an object x isin C the evaluation map

evidx Nat(

HomC(xminus) F) simminus F (x)

is an equivalence (we use Nat as a shortcut for HomFun(CAn)) Morover adjoining thefunctor HomC Cop times C Fun(CAn) gives fully faithful functors (ldquoYoneda embeddingsrdquo)

Y C C minus Fun(CopAn) and YC Cop minus Fun(CAn)

21


Proof of Theorem I29 See [HCII Corollaries XI2 and XI4]

I30 Definition mdash For an infin-category C we denote by P(C) = Fun(CopAn) the infin-category of presheaves on C

Onwards to adjunctions

I31 Definition mdash Let R D C be a functor of infin-categories(a) Given objects y isin C x isin D and a morphism η y Rx in C we say η witnesses x as

a left-adjoint object to y under R if the composite

HomD(xminus) Rminus HomC(RxRminus) ηlowast

minus HomC(yRminus)

is an equivalence of functors D An (which may be tested on objects by Theo-rem I19(b))

(b) An adjunction between R and some functor L C D is an equivalence

HomD(Lminusminus) HomC(minus Rminus)

as functors Cop times C An

A simple but still somewhat surprising and incredibly useful consequence of Yonedarsquoslemma is that to define left-adjoint functors it suffices to do so on objects (something that iswildly false for arbitrary functors)

I32 Corollary mdash A functor R D C of infin-categories admits a left adjoint if and onlyif every y isin C admits a left-adjoint object under R More generally if CR sube C is the fullsubcategory spanned by those objects y isin C which admit a left-adjoint object extracting theseleft-adjoint objects defines a functor

L CR minus D

Proof We repeat the proof given in [HCII Lemma XI6] Via currying the functorHomC(minus Rminus) Cop times D An corresponds to a functor H Cop Fun(DAn) Whenrestricted to Cop

R it lands in the representables ie in the essential image of the Yonedaembedding YD Dop Fun(DAn) By Theorem I29 YD is fully faithful hence an equiva-lence onto its essential image by Theorem I19(a) Composing H|Cop

Rwith an inverse of this

equivalence gives a functor CopR Dop Taking (minus)op we get L CR D as required

By construction YD Lop is equivalent to H|CopR

in Fun(CopR Fun(DAn)) But this means

that HomD(Lminusminus) and HomC(minus Rminus) are equivalent in Fun(CopR timesDAn) If CR = C this

proves that L is indeed a left adjoint of R

I33 Example mdash ldquoMy first adjoint pairs of functorsrdquo(a) The inclusion An sube Catinfin has both a left adjoint the functor | | Catinfin An (which

sends an infin-category to its localisation at all its morphisms) and a right adjoint givenby core Catinfin An We proved this in [HCII Example XI8] Moreover the inclusionSet An has π0 An Set as a left adjoint In pictures

Set An Catinfinsube sube

π0 | |

core

22


Of course we should actually write N(Set) instead of Set but from now on wersquoll frequentlyabuse notation that way since writing N all the time will inevitably get on our nerves (sorry)

(b) Let I be any simplicial set (which we think of as a ldquodiagram shaperdquo) and consider thefunctor const C Fun(I C) Given a map F I C a left-adjoint object to F underconst is called a colimit of F and denoted colimI F Similarly a right-adjoint object iscalled a limit and denoted limI F In particular one has

HomC(

colimI

F x) Nat(F constx)

for all objects x isin C and similarly for limI F We will discuss limits and colimits indetail in the next subsection

First Properties and Examples of (Co)LimitsFirst wersquoll discuss the main existence theorem for limits and colimits in infin-categories Forthat matter recall the notion of Kan simplicial model categories from [HCII Digression IVDefinition G2] For a (not necessarily Kan simplicial) model category A we denote by Acf itsfull subcategory of bifibrant objects (Lurie uses the notation A instead) Also (minus)c A Aand (minus)f A A denote any choice of cofibrantfibrant replacement functors

I34 Theorem mdash If A is a Kan simplicial model category then Nc(Acf) has all limitsand colimits For a diagram F I Nc(Acf) these are given by

colimI

F (

hocolimC[I]

F)f

and limIF

(holimC[I]

F)c

where F C[I] Acf is adjoint to F I Nc(Acf)

Proof See [HCII Theorem XI21]

I34a Remark mdash Wersquoll soon talk about cofinal and final maps (see Definition I44or [HTT sect41]) but let me already mention that the theory of these allows us to assumethat all diagram shapes I are infin-categories This hardly ever matters but we formulatedTheorem I23 for infin-categories only so I sleep better knowing that I can be chosen that way

I35 Example mdash Applying Theorem I34 to A = sSet with its KanndashQuillen modelstructure or A = sSet+ (the category of marked simplicial sets) with its marked Joyal modelstructure which are Kan simplicial model categories by [HCII Digression IV Examples G3]shows that An and Catinfin are complete and cocomplete (ie have all limits and colimits)

Another example of a complete and cocomplete infin-category is K(R) for any ring R asdefined in Example I15(e) In this case though we canrsquot apply Theorem I34 since K(R)is (probably) not given by a Kan simplicial model structure on Ch(R) at least not for theprojective or injective model structure (these have quasi-isomorphisms as weak equivalenceswhereas K(R) wants homotopy equivalences) Instead one has to use some more propertiesof limits and colimits which we will develop until Exercise I50b

I35a Example mdash Per Bastiaanrsquos suggestion we give two more examples Recall thata pullback on the nose in a model category is also a homotopy pullback if all objects arefibrant and one of its legs is a fibration (this follows from Reedyrsquos lemma for example see[HCII Corollary VIII52])

23


(a) Combining this observation with Theorem I34 shows that the pullback definingHomC(a b) from Definition I9 is also a pullback in Catinfin because its vertical legFun(∆1 C) Fun(part∆1 C) CtimesC is an isofibration (by [HCI Corollary VII11]) hencea fibration in the Joyal model structure on sSet

(b) Likewise the pullback defining Un(F ) in Theorem I21 is a pullback in Catinfin becauselowastAn An is a left fibration hence an isofibration

For a cocartesian fibration p E S we denote by Γ(p) its infin-category of sectionsdefined by the pullback

Γ(p) Fun(SE)

idS Fun(S S)

plowast

If we use the new notion from Definition I24(b) and p is not necessarily an isofibration weneed to take this pullback in Catinfin otherwise it can also be taken in sSet by the argumentsfrom Example I35a above We let Γcocart(p) sube Γ(p) denote the full sub-infin-category spannedby sections that take all edges in S to p-cocartesian edges

I36 Proposition (Lurie) mdash Given a diagram F I Catinfin we have

colimI

F Uncocart(F )[cocartesian edgesminus1]

limIF Γcocart

(Uncocart(F )

)

In particular if F I An takes values in anima then

colimI

F ∣∣Un(F )

∣∣ and limIF Γ

(Un(F )

)

Proof sketch In [HCII Theorem XI23] Fabian gave a proof thatrsquos basically the same aswhat we are going to do now but more on the model category side (but also less sketchy)We only do the case of limits Consider the diagram

Fun(ICatinfin)

Catinfin

Cocart(I)

sim

St

const

in which the bottom arrow sends C isin Catinfin to the cocartesian fibration pr2 C times I Iover I Let p E I denote the cocartesian unstraightening of F If X limI F then wemust have equivalences Nat(const C F ) HomCatinfin(C X) for all infin-categories C ApplyingTheorem I23 this translates into

HomCocart(I)((pr2 C times I I) (p E I)

) HomCatinfin(C X)

which wersquoll prove now is fulfilled for X Γcocart(p) (or rather we give an idea why itholds) A map ϕ C times I E corresponds to a map ϕ C Fun(I E) via currying Byinspection ϕ commutes with the projection down to I iff ϕ factors over Γ(p) E Since pr2-cocartesian edges are precisely those of the form (equivalence in C arbitrary morphism in I)

24


we see that ϕ preserves cocartesian edges iff ϕ sends equivalences f c cprime in C to naturaltransformations η ϕ(c) rArr ϕ(cprime) isin Fun(I E)1 such that for all edges g i iprime in I thecomposition (or some choice of it)

ϕ(c)(i) ηiminus ϕ(cprime)(i) glowastminus ϕ(cprime)(iprime)

is p-cocartesian in E Since f c cprime is an equivalence in C we get that ηi is an equivalencein the fibre EtimesI i (even in E) hence automatically p-cocartesian by [HCII Corollary IX8]Thus the composition is p-cocartesian iff glowast is p-cocartesian in other words if ϕ(cprime) sends allmorphisms in I to p-cocartesian edges Since this holds for all cprime isin C (we can always takef = idcprime) we finally obtain that ϕ preserves pr2-cocartesian edges iff ϕ C Γ(p) factorsover Γcocart(p) ldquoThusrdquo X Γcocart(p)

So far this is no proof at all since we only explained why 0-simplices of Nat(const C F )correspond to 0-simplices of HomCatinfin(CΓcocart(p)) but itrsquos not clear how to proceed forhigher simplices nor how to make this an infin-natural transformation To fix this first notethat we have a map of cocartesian fibrations

Γcocart(p)times I E

I I

pr2 p

induced by the evaluation map Fun(I E)times I E After straightening this gives a naturaltransformation η const Γcocart(p)rArr F hence we get

HomCatinfin(minusΓcocart(p)

) const lowast====rArr Nat(

constminus const Γcocart(p)) ηlowast=rArr Nat(constminus F )

which is the desired natural transformation Whether this is an equivalence can be checkedpointwise Since instead of Nat(constminus F ) we can take the corresponding Hom anima inCocart(I) we arrive at the equivalence from the beginning of the proof Since we know howto compute Hom spaces in Catinfin and in CatinfinI (the latter by [HCII Corollary VIII6])one easily checks HomCatinfin(CΓ(p)) HomCatinfinI((pr2 C times I I) (p E I)) Now theHom anima we are interested in are full subanima of these so it really suffices to check thattheir 0-simplices correspond which we did above

I37 Example mdashLecture 410th Nov 2020

Since core Catinfin An is right-adjoint to the inclusion An sube Catinfin(Example I33(a)) we get a natural counit transformation corerArr idCatinfin which is given bya map ∆1 Fun(CatinfinCatinfin) By currying this yields a functor

Catinfin minus Ar(Catinfin)C 7minus (core C C)

If F W C is an object in Ar(Catinfin) a left-adjoint object to F under the above functor iscalled a localisation of C at F (or rather at the image of π0 core Ar(W ) π0 core Ar(C)) ofF and denoted C[Wminus1] Unwinding this means that

HomCatinfin(C[Wminus1]D

)sube HomCatinfin(CD)

25


is a collection of path components given by those G C D such that the compositeG F W D lands in coreD The localisation C[Wminus1] always exists as one can take

W C

|W | C[Wminus1]

F

p

(the pushout is taken in Catinfin of course) Futhermore it turns out that

plowast Fun(C[Wminus1]D

)minus Fun(CD)

is fully faithful with essential image spanned by the same functors G C D as above (see[HCII Proposition VIII7] for a proof) This has the funny consequence that the Yonedaembedding induces fully faithful inclusions C[Wminus1] sube P

(C[Wminus1]

)sube P(C) In other words

all localisations of C occur as full subcategories of P(C) which puts a size bound on howlarge C[Wminus1] can be

Warning mdash Localisations are difficult Localisations of locally small infin-categoriesmay not be locally small any more Moreover the localisation of a 1-category may not be a1-category any more

I38 Example mdash Letrsquos compute the pushout

∆0 ∆1

∆1

1

0

in Catinfin One way to do this would be to notice that 0 ∆0 ∆1 is a cofibration in theJoyal model structure on sSet (and so is 1 ∆0 ∆1 but we donrsquot even need this) andall objects are cofibrant hence the pushout on the nose which is the 2-spine I2 is also ahomotopy pushout Now I2 sube ∆2 is inner anodyne hence ∆2 is a fibrant replacement of I2

in the Joyal model structure which shows that ∆2 is really the pushout in question byTheorem I34

For educational purposes Fabian presented a less efficient way to show ∆2 usingProposition I36 Consider the pushout diagram above as a map F Λ2

0 Catinfin Since allmaps in the diagram are nerves of functors of 1-categories the cocartesian unstraighteningD = Uncocart(F ) is simply given by the Grothendieck construction Therefore the cocartesianfibration D Λ2

0 can be pictured as follows

Λ20

D Dprime

The two pink edges are the cocartesian ones and thus sent to equivalences in the localisationD[cocartesian edgesminus1] = D[pink edgesminus1] Let Dprime be the infin-category obtained from D

26


by removing the purple and the pink edge on the left together with the 2-simplex spannedby them We claim that the restriction

Funpink(D E) simminus Funpink(Dprime E)

is an equivalence for any infin-category E where Funpink denotes functors that send pinkedges to equivalences Indeed the purple edge on the left is ldquosuperfluous datardquo ie anymap Dr purple edge E has a contractible space of lifts to D by Joyalrsquos lifting theoremMoreover any map Dprime E has a contractible space of lifts to a map Dr purple edge Ethat sends the left pink edge to an equivalence This proves the above equivalence

A similar argument shows that it doesnrsquot matter whether we send the pink edge on theright to an equivalence or to an actual identity in E Thus

Funpink(Dprime E) Fun(Λ21 E) Fun(∆2 E)

which proves that D[cocartesian edgesminus1] ∆2 as claimed

I39 Lemma mdash If C is complete or cocomplete then so is Fun(D C) for any D Further-more for any f E D the precomposition functor

flowast Fun(D C) minus Fun(E C)

preserves limits and colimits In particular (taking f to be d D for any d isin D) limitsand colimits in functor categories are computed pointwise

For the proof we need the following observation

I40 Observation mdash If L C D R are adjoint functors then so are

Llowast Fun(E C) Fun(E D) Rlowast and Rlowast Fun(D E) Fun(C E) Llowast

for any infin-category E

Proof Being adjoints can be characterized by the existence of a unit and a counit transfor-mation satisfying the triangle identities (see [HCII Proposition XI14] and Llowast Rlowast inheritthese transformations from L R Same for Rlowast Llowast

I41 Observation mdash Left-adjoint functors preserve colimits right-adjoint functorspreserve limits

Proof Straightforward calculation using Observation I40 See [HCII Lemma XI22] fordetails

Proof of Lemma I39 For any diagram shape I consider the commutative diagram

Fun(IFun(D C)

)Fun(D C)

Fun(DFun(I C)

)const

const lowast

colimlowast

limlowast

27


By Observation I40 colimlowast and limlowast are left and right adjoints of const lowast respectively whichproves that const Fun(D C) Fun(IFun(D C)) has a left and right adjoint as well HenceFun(D C) is complete and cocomplete again Moreover the assertion that limits and colimitsare computed pointwise follows by unraveling

It remains to show that flowast Fun(D C) Fun(E C) preserves limits and colimits Forany diagram F I Fun(D C) we get a map

colimI

flowastF minus flowast colimI

F

(unique up to contractible choice) Whether this is an equivalence in Fun(E C) can be checkedpointwise (by Theorem I19(b)) ie after composition with xlowast Fun(E C) Fun(x C) Cfor all x isin E Since we already know that colimits in Fun(E C) and Fun(D C) are computedpointwise we see that both xlowast colimI flowastF and xlowastflowast colimI F are colimits of the diagramI times f(x) C induced by currying of F and restriction along f(x) D

I42 Proposition mdash Let p E C be a cocartesian fibration and F E D a diagramSuppose that the restrictions F|c E|c D of F to the fibres E|c = E timesC c of p have colimitsfor all c isin C Then these assemble into a functor G C D sending c 7 colimE|c F|c and

colimE

F colimC

G

whenever either exists In particular (taking p to be the projection pr2 I times C C for someinfin-category I) this means that ldquocolimits commuterdquo Also a dual assertion holds with p acartesian fibration and all colimits replaced by limits

Proof Fabian claims (I think) this follows from [HTT Proposition 43212] but Luriersquosproof is rather laborious so Irsquoll give my own proof As it turns out the arguments arepretty much the same as for the proof of Theorems I51 and I52 in Fabians notes Irsquoll freelyreference later results (without running into circular arguments I hope)

We wonrsquot prove the assertion as stated above but its dualized version The reason isthat the direct proof of the undualized version is quite confusing (for example neither of theYoneda embeddings preserves colimits so we would have to consider (Y D)op Dop P(D)op)and it seems cleaner to prove the dualized statement and then deduce the undualized one byapplying the dual statement to pop Eop Cop

So let p E C be cartesian instead We want to construct G as the right Kan extensionG plowastF Letrsquos first consider the case where D is replaced by P(D) Then the dual ofTheorem I21 implies Fun(CP(D)) Fun(CtimesDopAn) Right(CoptimesD) and same for E sowe may apply Lemma I42a below to the cocartesian fibration poptimes idD EoptimesD CoptimesDto see that plowast Fun(CP(D)) Fun(E P(D)) has a right adjoint plowast Now consider thediagram

Fun(CP(D)

)Fun

(E P(D)

)Fun

(cP(D)

)Fun

(E|cP(D)

)plowast

clowast ilowast

plowast|c

where i E|c E denotes the inclusion of the fibre The second part of Lemma I42a ensuresthat clowastplowastF p|clowastilowastF By inspection right Kan extension along the functor p|c E|c cis given by taking a functor G E|c P(D) to limE|c G This implies plowastF (c) limE|c F|c so

28


plowastF takes the correct values Moreover limits over C are given by right Kan extension alongthe unique map π C lowast and similar for π p E lowast hence

limCplowastF πlowastplowastF (π p)lowastF lim

EF

since right adjoints compose This finishes the proof for P(D)Now for the case of general D The Yoneda embedding Y D D P(D) is fully faithful

hence an equivalence onto its essential image To obtain a functor plowastF C D it thussuffices to check that plowast(Y D F ) C P(D) has image in the representable functors Byassumption all limits limE|c F|c exist in D and Y D preserves limits by Corollary I50 hencethe description of the values plowast(Y F )(c) for c isin C shows that indeed it takes values inthe representables Thus plowastF C D exists The additional assertion about limits followsas above noting that the limits in question exist in D if and only if the limits taken inP(D) (where they definitely exist) are representable since Y D is fully faithful and preserveslimits

I42a Lemma mdash Let p C D be a cocartesian fibration of infin-categories Consider apullback square (inside Catinfin) as on the left

Cprime Dprime

C D

q

g f

p

Right(Cprime) Right(Dprime)

Right(C) Right(D)

qlowast

qlowast

glowast

plowast

flowast

plowast

inducing a square as on the right Then plowast and qlowast have right adjoints plowast and qlowast as indicatedMoreover therersquos a natural equivalence

flowastplowastsim=rArr qlowastg

lowast

Proof Without loss of generality all cartesian or right fibrations can be taken in the oldsense Recall (eg from the dual of [HTT sect214]) that there exists the contravariant modelstructure on sSetC in which the cofibrations are monomorphisms of simplicial sets and fibrantobjects are given by right fibrations over C (general fibrations are more complicated) Thefunctor plowast sSetD sSetC has a right adjoint plowast for abstract reasons Itrsquos straightforwardto check that plowast preserves colimits so Theorem I2 can be applied to sSetD P((∆D)op)(here ∆S denotes the slice category with respect to the functor ∆ ∆∆ sSet sending[n] 7 ∆n) We claim that

plowast sSetD sSetC plowastis even a Quillen adjunction when p is a cocartesian fibration Indeed itrsquos clear that plowastpreserves cofibrations and cofibrant objects For preservation of trivial cofibrations we referto [HTT Proposition 41215] or the dual of [HCII Proposition X43] for the essential case

As for any Quillen adjunction of model categories we get an induced adjunction

Lplowast (sSetC)infin (sSetD)infin Rplowast

on underlying infin-categories defined as the localisations at all weak equivalences or equiv-alently as Nc((sSetC)cf) and Nc((sSetC)cf) since sSetC and sSetD can be made into aKan simplicial model categories so [HCII Digression III Theorem D] applies The latter

29


description shows that the underlying infin-categories identify with Right(C) and Right(D)Hence Lplowast (which we abbreviate to plowast in the following) has indeed an adjoint Rplowast (whichwe abbreviate to plowast)

So plowast and qlowast exist The counit plowastplowast rArr id induces a transformation qlowastflowastplowast glowastplowastplowast rArr glowastwhose adjoint is the transformation flowastplowast rArr qlowastg

lowast Whether this is an equivalence can bechecked on objects However explicitly computing plowast and qlowast is nasty so we employ a trickBy Yoneda it suffices to check that HomRight(Dprime)(minus flowastplowastminus) rArr HomRight(Dprime)(minus qlowastglowastminus) isa natural equivalence But flowast and glowast have left adjoints f and g by Theorem I47 so wemay as well show that HomRight(C)(plowastfminusminus) rArr HomRight(C)(gq

lowastminusminus) is an equivalenceie that gq

lowast rArr plowastf is an equivalence This is now easily checked on objects Unravellingwe need to prove that if X Dprime is a right fibration and X Y D is a factorisation intoa right anodyne and a right fibration then X timesD C Y timesD C is right anodyne again whichfollows from the references above since p C D is cocartesian

I43 Theorem (Joyalrsquos version of Quillenrsquos Theorem A) mdash For a functor α I J thefollowing statements are equivalent(a) A functor F J C has a colimit iff the composition F α I C has a colimit and

in this casecolimI

F α colimJ

F

(b) For all j isin J we have |jα| lowast ie the slice category jα is weakly contractible Herewe define jα by the pullback

jα Ar(J )

j times I J times J

(st)(inclα)

Also observe that jα = I timesJ jJ which is how people usually denote this in theliterature

Proof Combine [HTT Proposition 4118] (for more equivalent characterisations) and[HTT Theorem 4131] (for the actual proof)

I44 Definition mdash Maps α I J as in Theorem I43 are called cofinal Dually ifprecomposition with α preserves limits and detects their existence then α is called final

Cofinal maps induce homotopy equivalences |α| |I| |J | of anima which can be seenvia the chain of homotopy equivalences

|I| colimI

const lowast colimJ

const lowast |J |

(the colimits in the middle are taken in An) Here we use |I| colimI const lowast which can beseen in several ways One could use Theorem I34 and the BousfieldndashKan formula ([HCIIDigression III]) Alternatively compute HomAn(colimI const lowastK) limI HomAn(lowastK) limI K for any anima K using Corollary I50 below and apply Proposition I36 to getlimI K Γ(pr2 KtimesI I) The space of sections of pr2 KtimesI I clearly identifies withFun(IK) core Fun(IK) HomCatinfin(IK) so colimI const lowast is indeed a left-adjointobject of I under the inclusion An sube Catinfin

30


Examples of cofinal maps are given by right-adjoint functors and localisations Indeed ifα I J is a right adjoint then consider the diagram

C

Fun(J C) Fun(I C)con

st const

αlowast

colimJ colimI

By Observation I40 αlowast Fun(J C) Fun(I C) is a left-adjoint Since left-adjoints composewe obtain colimJ colimI αlowast as required If p C C[Wminus1] is a localisation andF C[Wminus1] D is any diagram then

Nat(F constx) Nat(F p constx)

holds for every x isin C since plowast Fun(C[Wminus1]D) Fun(CD) is fully faithful which againproves cofinality by means of Theorem I43(a)

I45 Corollary mdash Let G I Catinfin be a diagram and put J colimI G Let F J Dbe another diagram and suppose that the restrictions

G(i) minus J Fminus D

have colimits for all i isin I Then these colimits assemble into a functor H I D sendingi 7 colimG(i) F and we have

colimJ

F colimI

H

if either exists

Proof By Proposition I36 we have J colimI G Uncocart(G)[cocartesian edgesminus1]Since localisations are cofinal we obtain

colimJ

F colimUncocart(G)

F

Now recall that G(i) Uncocart(G)timesI i so we are done by Proposition I42

I46 ldquoCorollaryrdquo mdash If C has coproducts and pushouts then it is cocomplete Dually ifC has products and pullbacks then it is complete

ldquoProofrdquo sketch Decompose any test category I into its skeleta and repeatedly apply Corol-lary I45 using colim∆n F = F (n) since n ∆n is cofinalmdashin fact all right anodyne mapsare cofinal For a complete proof check out [HTT Proposition 4426]

Colimits and the Yoneda embeddingWe start with an existence result for ldquoKan extensionsrdquo which was already used in the proofof Proposition I42

I47 Theorem (Joyal) mdash Given an arbitrary functor f C D of infin-categories thepullback functor

flowast Right(D) minus Right(C)

31


admits a left-adjoint Its value on p E C is obtained by factoring E C D as a cofinalmap followed by a right fibration pprime E prime D In diagrams

E E prime

C D

p pprime

f

where the top arrow is cofinal

Proof sketch One way to prove this would be to use the contravariant model structures onsSetC and sSetD as in the proof of Lemma I42a One shows that flowast sSetD sSetChas a left Quillen adjoint f sending E C to E C D Since right anodyne maps arecofinal ([HTT Proposition 4113] or [HCII Theorem XI28]) a quick unraveling shows thatLf can indeed be described as above

However Bastiaan has pointed out a simpler proof that doesnrsquot need any model categoriesWe will proceed in three steps(1) The pullback functor flowast CatinfinD CatinfinC (where we take the pullback along f in

Catinfin) has a left adjoint On objects it is the ldquoforgetful functorrdquo sending p E C tof p E C D

(2) Under the fully faithful inclusions Right(C) sube CatinfinC and Right(D) sube CatinfinD thepullback functor flowast CatinfinD CatinfinC restricts to flowast Right(D) Right(C) fromthe formulation of the theorem

(3) The fully faithful inclusion Right(D) sube CatinfinD has a left adjoint sending an objectq E D to qprime E prime D where E E prime D is any factorisation into a cofinal mapfollowed by a right fibrationSince left adjoints compose combining (1) (2) and (3) shows that flowast Right(D)

Right(C) has a left adjoint f with the desired description on objectsTo show (1) we use Corollary I32 to see that it suffices to show that f p is a left adjoint

object of p for any (p E C) isin CatinfinC If we had the dual of Corollary I50 below alreadythis would be almost trivial but as it is we need to be a bit careful Factor f p into aJoyal equivalence E E prime followed by an isofibration pprime E prime D Then flowastE prime equals the samepullback taken in sSet by Theorem I34 Hence there is a natural map E flowastE prime Togetherwith functoriality of flowast it induces natural transformations

HomCatinfinD(E minus) HomCatinfinD(E primeminus) =rArr HomCatinfinC(flowastE prime flowast(minus)

)=rArr HomCatinfinC

(E flowast(minus)

)

We are to show that the composite is an equivalence By Theorem I19(b) this can be doneon objects Let T D be an bject of CatinfinD which we may choose to be an isofibrationwithout restriction so that flowast(T ) agrees with the corresponding pullback in sSet ConsiderFunD(E T ) = Fun(E T )timesFun(ED) f p and define FunC(E flowast(T )) similarly Since T Dis an isofibration it doesnrsquot matter whether this pullback is taken in sSet or in CatinfinMoreover core(minus) transforms pullbacks in Catinfin into pullbacks in An (it is a right adjoint byExample I33(a) so Observation I41 can be applied) hence our computation of Hom animain slice categories from [HCII Corollary VIII6] shows HomCatinfinD(E T ) core FunD(E T )Likewise HomCatinfinC(E flowast(T )) core FunC(E flowast(T )) Now the universal property of flowast(T )as a pullback in sSet implies FunD(E T ) FunC(E flowast(T )) which proves step (1)

32


For (2) observe that flowast of a right fibration in the old sense can also be taken in sSet byTheorem I34 again and right fibrations (in the old sense) are preserved under pullbacksformed in sSet Hence right fibrations in the new sense are preserved by pullbacks in Catinfin

Finally to show (3) we can argue as (1) to reduce the assertion to FunD(E prime T ) FunD(E T ) for any right fibration T D and any cofinal map E prime E over D But this isprecisely how Lurie defines cofinal maps in [HTT Definition 4111] and his definition isequivalent to our Definition I44

I48 Corollary mdash The functor flowast P(D) P(C) has a left-adjoint f (given by left Kanextension) such that the diagram

C D

P(C) P(D)

f

Y C Y D

f

commutes in the infin-category Catinfin

Proof The first statement immediately follows from Theorem I47 and (the dual of) The-orem I21 which asserts that the cartesian straightening Stcart Right(C) simminus P(C) is anequivalence with inverse the cartesian unstraightening Uncart

For the second statement we have to show that f HomC(minus x) HomD(minus f(x)) is anequivalence for all x isin C After unstraightening this translates to the statement that thediagram

Cx Df(x)

C Df

exhibits Df(x) D as f(Cx C) The right vertical arrow is already a right fibrationso it suffices to check that Cx Df(x) is cofinal This holds because

∆0

Cx Df(x)

x f(x)

f

has both downward arrows cofinal (as x and f(x) are terminal objects in Cx and Df(x)respectively and thus both sloped arrows are right anodyne by [HCII Digression I Corol-lary D7])

I49 Corollary mdash Given functors FG C D of infin-categories we have an equivalenceof anima

Nat(FG) lim(xy)isinTwAr(C)

HomD(F (x) G(y)

)

Explicitly the limit is taken over TwAr(C) Cop times C F optimesGminusminusminusminusminus Dop timesD HomDminusminusminusminus An

Note that Corollary I49 explains why natural transformations are complicated ininfin-landπ0 does not commute with arbitrary limits

33


Proof of Corollary I49 By Proposition I36 the right-hand side can be computed as theanima of sections of p P TwAr(C) where P denotes the correct unstraightening Explicitlywe have a diagram

P P prime TwAr(D) lowastAn

TwAr(C) Cop C Dop timesD An

(st) F optimesG HomD

which shows Γ(p) HomCatinfinCoptimesC(TwAr(C) P prime) But these are both left fibrationsover Cop times C so the Hom anima on the right-hand side can be equivalently computed asNat(HomC HomD (F optimesG)) Now the currying equality Fun(CoptimesCAn) = Fun(CP(C))sends HomC to Y C and HomD (F op times G) to F lowast Y D G hence the Hom anima underconsideration is given by

Nat(Y C F lowast Y D G)

) Nat(F Y C Y D G) Nat(Y D F Y D G) Nat(FG)

as claimed For the left equivalence we use that F minus is an adjoint of F lowast minus by constructionand Observation I40 the middle equivalence follows from Corollary I48 and the right onesince Y D D P(D) is fully faithful by Yonedarsquos lemma (Theorem I29)

I50 Corollary mdash Let F I C be a functor of infin-categories A natural transformationη F rArr const c exhibits c isin C as the colimit of F if and only if the natural map

ηlowast HomC(c x) simminus limiisinIop

HomC(F (i) x

)is an equivalence for all x isin C A similar assertion holds for limits In particular thefunctors HomC(minus d) Cop An and Y C C P(C) preserve limits

Proof We are done if we can show that the right-hand side is just Nat(F constx) FromCorollary I49 we get a map

Nat(F constx) lim(iiprime)isinTwAr(I)

HomC(F (i) x) minus limiisinIop

HomC(F (i) x)

So we are done if we show that s TwAr(I) Iop is final By (the dual of) Theorem I43(b)we must show |si| lowast for all i isin Iop Applying Exercise I50a to the cocartesian fibrations = pr2(s t) TwAr(I) IoptimesI Iop we find that |sminus1i| |si| since adjoints induceinverse homotopy equivalences after applying | | alternatively one can use that left adjointsare final Now sminus1i has an initial object which immediately shows |sminus1i| lowast In fact wehave sminus1i iI (and idi is initial in the right-hand side) Depending on your definitionsthis is either something one has to check (as in the proof of [HA Proposition 52110]) orfollows from the fact that t sminus1i I represents the functor HomI(iminus) and is thus givenby the left fibration iI I

I50a Exercise mdash Let p E C be a cocartesian fibration (in the old sense if you wantthis to work in the new sense as well all fibre products need to be taken inside Catinfin) Showthat the natural map E timesC c E timesC Cc has a left adjoint for all c isin C

Fabian wrote is instead of si in the lecture and in [AampHK Corollar I50] and theexercise was formulated with cC instead of Cc Irsquom pretty sure that we really need theother slicesmdashafter all we need to apply the dual version of Theorem I43(b) also it willbecome apparent during the proof of Exercise I50a that we really need to use Cc

34


Proof of Exercise I50a Since adjoints can be constructed objectwise by Corollary I32 weonly need to show that every object (eprime ϕ cprime c) isin E timesC Cc admits a left-adjoint objectLet η eprime e be a p-cocartesian lift of ϕ This induces a morphism η (eprime ϕ) (e idc) inE timesC Cc We claim that η exhibits e as a left-adjoint object of (eprime ϕ) so we need to showthat the composite

HomEtimesCc(e eprimeprime) minus HomEtimesCCc

((e idc) (eprimeprime idc)

) ηlowast

minus HomEtimesCCc((eprime ϕ) (eprimeprime idc)

)is an equivalence for all eprimeprime isin E timesC c Plugging in the homotopy pullback that describesHomE(e eprimeprime) due to eprime e being p-cocartesian as well as the homotopy pullback thatcomputes HomCc(cprime c) due to [HCII Corollary VIII6] shows that both the left-hand sideand the right-hand side are homotopy equivalent to HomE(eprime e)timesRHomC(cprimec) ϕ and that theabove composite induces the identity on that anima

I50b Exercise mdash Use Corollary I50 to show that K(R) and Nc(Top) are (co)complete

Public Service Announcement mdash From now on all pullbacks of infin-categories oranima will be taken in Catinfin or An unless specified otherwise Still in many cases they willagree with the corresponding pullbacks in sSet by arguments like Example I35a

Kan ExtensionsLecture 5

12th Nov 2020The first goal for today is to discuss two closely related theorems which provide aninfin-analogueof Theorem I2

I51 Theorem mdash For every cocomplete infin-category D and every small infin-category Cthe restriction of Y Clowast Fun(P(C)D) Fun(CD) to colimit-preserving functors in thesource is an equivalence Furthermore any such functor has a right adjoint In view ofObservation I41 we thus get an equivalence

Y Clowast FunL(P(C)D

) simminus Fun(CD)

where FunL denotes the full sub-infin-category spanned by left adjoint functors

I52 Theorem mdash If C is a small infin-category and D cocomplete or complete then forany f C E the functor

flowast Fun(E D) minus Fun(CD)

has a left adjoint Lanf or a right adjoint Ranf (sometimes also denoted f and flowast) whichsatisfy

Lanf F (e) colim(cf(c)e)isinfe

F (c) and Ranf F (e) lim(cef(c))isinef

F (c)

In fact if D is not necessarily cocomplete or complete but these colimits or limits happento exist for all e isin E then they assemble into a functor Lanf F E D or Ranf F E Dwhich is a left- or right-adjoint object of F with respect to flowast

I53 Definition mdash If G E D is a left or right adjoint object of F C D underflowast Fun(E D) Fun(CD) then G is called a left or right Kan extension of F along f

35


A first example is given by f P(C) P(D) from Corollary I48 which sends a presheafF Cop An to its left Kan extension fF Dop An along Cop Dop

In the official lectures notes (starting from [AampHK Observation I56]) Fabian outlinestwo proofs of Theorems I51 and I52 one due to Cisinski and one due to Lurie In thesenotes wersquoll present a third proof that Fabian briefly sketched in the lecture Itrsquos perhaps a bitmore straightforward than the other two in that it really does the same as for 1-categoriesbut itrsquos not really shorter

Proof of Theorem I52 Consider the slice category fE defined by the pullback

fE Ar(E)

C E

s

f

(this is not equivalent to the thin or fat slice Ef or Ef) Informally fE consists ofpairs (c f(c) e) where c isin C e isin E It comes with morphisms s fE C andt fE E sending such a pair to c or e respectively Then t is a cocartesian fibrationby an easy generalisation of Example I25(b) and its fibres tminus1e fe are the slicecategories from the definition of Lanf F (e) We may thus apply Proposition I42 to thecocartesian fibration t and the functor F s fE D to see that the pointwise formulasindeed assemble into a functor Lanf F Going through the proof of Proposition I42 wesee that (Lanf F )op top

lowast (F op sop) is a right-adjoint object of F op sop with respect to(top)lowast Fun(EopDop) Fun((fE)opDop) hence Lanf F is a left-adjoint object of F sunder tlowast Fun(E D) Fun(fE D) Thus we only need to show that there is an equivalence

Nat(Fminus f) sim=rArr Nat(F sminus t)

Observe that s has a section r C fE defined by the universal property of pullbacksand the maps idC C C and flowast const C Ar(E) Then f = t r and there are naturaltransformations η f srArr t and ηprime r srArr idfE in diagrams

C fE C

Ef

r

t

s

flArr=η

To get η we compose fE times∆1 Ar(E)times∆1 with the evaluation map Ar(E)times∆1 E Toget ηprime fE times∆1 fE we must choose homotopies fE times∆1 C and fE times∆1 Ar(E)The first can be chosen to be constant at s the second is equivalently given by a map∆1 times∆1 Fun(fE E) which we choose to be

f s f s

f s t

η η

η

In particular t ηprime = η Now the desired equivalence can be obtained via the compositeηlowast slowast Nat(Fminus f) rArr Nat(F sminus f s) rArr Nat(F sminus t) To show that it isan equivalence we simply check that rlowast Nat(F sminus t) rArr Nat(Fminus f) (here we uses r = idC and t r = f) is an inverse by a straightforward computation

36


To prove Theorem I51 wersquoll check that LanY C Fun(CD) Fun(P(C)D) constitutesan inverse to Y Clowast For this we need to check that LanY C really takes values in FunL(P(C)D)which needs some preparations The first is a lemma that I copied straight from Fabiansnotes [AampHK Proposition I51] (but Fabian proves it differently)

I53a Lemma mdash Every element in P(C) is a colimit of representable presheaves Moreprecisely for every E isin P(C) the canonical map

colim(ccE)isinY CE

HomC(minus c) simminus E

is an equivalence

Proof By Theorem I19(b) it suffices to check that both sides agree after evaluation atx isin Cop Since colimits in functor categories are computed pointwise by Lemma I39 we seethat the left-hand side when evaluated at x is given by the colimit of

Y CEsminus C HomC(xminus)

minusminusminusminusminusminusminus An

We know how to compute colimits in An by Proposition I36 The cocartesian unstraighteningof HomC(xminus) is xC C hence the cocartesian unstraightening U of the functor in questionis given by the pullback

U xC

Y CE C

s

Wersquoll see in the proof of Lemma I53b that Y CE C is cartesian (even a right fibration)hence the dual of Exercise I50a provides an adjunction U timesxC idx U In particularwe get a homotopy equivalence |U timesxC idx| |U | (see [HCII Corollary XI17]) NowProposition I36 shows that |U | is precisely the colimit wersquore interested in and

U timesxC idx Y CE timesC x HomP(C)(Y C(x) E

) E(x)

holds by a quick unravelling of definitions and Yonedarsquos lemma

I53b Lemma mdash Let D be cocomplete For every functor F C D the left Kanextension LanY C F P(C) D commutes with colimits

Proof One is tempted to say ldquoThis follows immediately from the pointwise formula (Theo-rem I52) and the fact that colimits commute (Proposition I42)rdquo but in reality it is way moresubtle than that If you donrsquot believe me have a look at Warning I53c below Anywayletrsquos move onward to the problem at hand

Let Θ I P(C) be a diagram with colimit E colimI Θ Abusively we will also writeΘ(i) = Ei and E colimiisinI Ei Define J as the pullback

J Y CP(C) C D

I P(C)

t

s F

Θ LanY C F

37


Note that t Y CP(C) P(C) is a cocartesian fibration by an easy generalisation of Exam-ple I25(b) hence so is J I The pointwise formula from Theorem I52 yields

LanY C F(

colimiisinI

Ei

) LanY C F (E) colim

(Y CE minus C F

minus D)

Using the pointwise formula for each Ei together with Proposition I42 we find that

colimiisinI

LanY C F (Ei) colim(J minus C F

minus D)

We obtain a canonical map J Y CE as follows Extend Θ I P(C) to its colimitcocone Θ I P(C) Since I = (I times∆1)(I times 1) this can be further extended to amap I times∆1 P(C) which is constE on I times 1 Pulling back t Y CP(C) P(C) alongI times ∆1 P(C) gives a cocartesian fibration over I times ∆1 hence a cocartesian fibrationover ∆1 by composing with pr2 I times∆1 ∆1 After applying Stcocart this gives a mapJ I times Y CE providing the required map J Y CE

Now observe that J C is a cartesian fibration and Y CE C is even a right fibrationIndeed by construction and some abstract nonsense about pullbacks we get pullbackdiagrams

J C

P(C)Θ P(C)

Y C

s

andY CE C

P(C)E P(C)

Y C

s

Now P(C)Θ P(C) is a cartesian fibration by an easy generalisation of the dual ofExample I25(b) and P(C)E P(C) is a right fibration The right-hand side shows thatStcart(Y CE C) HomP(C)(Y C(minus) E) E(minus) as functors Cop An by Yonedarsquos lemmaHence the map J Y CE constructed above induces a natural transformation

Stcart(J C) =rArr E

Since An sube Catinfin has a left-adjoint | | Catinfin An this transformation factors over|Stcart(J C)| In fact we will show that |Stcart(J C)| E This can be done objectwiseso we need to show |J timesC x| E(x) for all x isin C We have E(x) colimiisinI Ei(x) ascolimits of presheaves are computed pointwise by Lemma I39 The colimit on the right-hand is precisely the colimit over evx Θ I P(C) An This can be computed viaProposition I36 so it suffices to show J timesC x Uncocart(evx Θ) By Yonedarsquos lemmathe diagram

I An

P(C)

Θ

evxΘ

HomP(C)

(Y C(x)minus

)commutes hence Uncocart(evx Θ) is the pullback of the left fibration Y C(x)P(C) P(C)along Θ By abstract pullback nonsense again this agrees with J timesC x as required

Now that we know |Stcart(J C)| E we can finally prove that J Y CE iscofinal which ultimately shows what we want Since | | Catinfin An is a left adjointof An sube Catinfin we obtain HomFun(CopCatinfin)(Stcart(J C) G) HomP(C)(EG) for allpresheaves G Cop An hence

HomCart(C)(J X) HomRight(C)(Y CEX

)

38


for right fibrations X C This easily implies FunY CE(J X) FunY CE(Y CEX) forall right fibrations X Y CE hence J Y CE is indeed cofinal in the sense of [HTTDefinition 4111] which is equivalent to our Definition I44

I53c Warning mdash In the situation of Theorem I52 there examples where f C Eis fully faithful and colimit-dense but still Lanf F doesnrsquot commute with colimits (but ofcourse the functor Lanf itself does preserve colimits as it is a left adjoint) Herersquos acounterexample I think Let C = Ntc be the poset consisting of N and a totally unrelatedadditional point c such that HomC(c n) = empty = HomC(n c) for all n isin N Let E = C be thecocone below C with tip e isin E and let f C E be the canonical inclusion Let F C Anbe the functor that is constant with value empty isin An on N and takes F (c) = lowast isin An ThencolimnisinN n e and we have

colimnisinN

Lanf F (n) colimnisinN

empty = empty

(here we use Corollary I54 below) However Lanf F (e) 6= empty because there is a mapF (c) Lanf F (e) by the pointwise formula

The key problem in this counterexample is the existence of a map c e colimnisinN n thatdoesnrsquot factor over any c n and the only reason why this doesnrsquot happen for Y C C P(C)as well is the Yoneda lemma This shows that Lemma I53b critically depends on the Yonedalemma and perhaps justifies the effort we had to put into its proof

Proof of Theorem I51 From Theorem I52 we obtain that Y Clowast admits a left adjointLanY C Fun(CD) Fun(P(C)D) We have verified in Lemma I53b that LanY C takesvalues in the full sub-infin-category Funcolim(P(C)D) of colimit-preserving functors We firstshow that

LanY C Fun(CD) simsim Funcolim (P(C)D

)Y Clowast

is an equivalence It suffices to show that the unit and counit of (LanY C Y Clowast) are equivalencesFor the counit this follows from Corollary I54 below For the unit let G P(C) D bea colimit-preserving functor We need to prove that G LanY C (G Y Clowast) But both sidescoincide on representable presheaves (ie after precomposition with Y C as we have justverified) and every presheaf can be written as a colimit of representable ones by Lemma I53ahence the claim

It remains to check that FunL(P(C)D) sube Funcolim(P(C)D) is an equivalence ie thatevery colimit-preserving functor P(C) D admits a right adjoint This can be written downexplicitly which we do in Corollary I55 below

I54 Corollary mdash If f C E is fully faithful in the situtation of Theorem I52 then

F Lanf F f and F Ranf F f

holds for all functors F C D

Proof The natural transformations in question are induced by the unit of the (Lanf flowast)-adjunction and the counit of the (flowastRanf )-adjunction so it suffices to check both of themon objects c isin C But if f is fully faithful then the slice categories ff(c) have (c idf(c))as terminal objects hence colimits over ff(c) are computed by evaluation at that objectproving Lanf F (f(c)) F (c) Same for Ranf F (f(c))

39


I55 Corollary mdash In the situation of Theorem I51 let F C D be a functor cor-responding to LanY C F P(C) D Then LanY C F admits a right-adjoint R D P(C)which is given by

R(d) HomD(F minus d) Cop minus An

on objects d isin D

Proof To check that LanY C F admits a right adjoint it suffices to check that all d isin Dadmit right-adjoint objects under LanY C F by Corollary I32 We already have a candidatefor R(d) so we need to check that there is an equivalence

HomD(

LanY C F minus d) Nat

(minus R(d)

)of functors P(C)op An But regarding both sides as functors P(C) Anop instead notethat they preserve colimits and they agree on representables since both become HomD(F minus d)after precomposition with Y C C P(C) (for the left-hand side we need to use Corollary I54for the right-hand side use Yonedarsquos lemma) Hence these two functors must be equivalentby Theorem I51 To complete the proof that R(d) is a right-adjoint object of d we needto check that the above equivalence is actually induced by a map LanY C F (R(d)) dbut this we get for free by plugging R(d) into the above equivalence and taking the ofidR(d) isin Nat(R(d) R(d)) in the left-hand side

This finishes the proofs of Theorem I51 and Theorem I52 Time for examples

I56 Very Long Example mdash We will show that algebraic topology is a corollary ofTheorem I51 Let C = lowast Then the theorem shows that the evaluation at the point lowast isin Anis an equivalence

evlowast FunL(AnD) simminus D

for every cocomplete D For example take D = Dgt0(Z) to be the derived category of abeliangroups (which is cocompletemdashafter ldquoCorollaryrdquo I46 and Corollary I50 there isnrsquot much todo) Then the equivalence Dgt0(Z) FunL(AnDgt0(Z)) from above takes Z[0] isin Dgt0(Z) tothe ldquonormalized chain complexrdquo functor

Cbull An minus Dgt0(Z)

that is the homology of Cbull(X) gives the unreduced homology of X (as a simplicial set orequivalently the unreduced cellularsingular homology of the realisation |X| isin CW) The factthat Cbull preserves coproducts and pushouts precisely gives that simplicialcellularsingularhomology takes disjoint unions to direct sums and satisfies a MayerndashVietoris sequence

The right adjoint of Cbull isK Dgt0(Z) minus An

(this is another K that has nothing to do with K-theory) which is now our official definitionof EilenbergndashMacLane spaces anima On objects C isin Dgt0(Z) it can be explicitly describedas K(C) HomDgt0(Z)(Z[0] C) by Corollary I55 The (CbullK) adjunction can be upgradedto an adjunction

Cbull lowastAn Dgt0(Z) K

Here Cbull(Xx) is the complex computing the reduced homology of a pointed anima (Xx)For example if Si is your favourite model for the i-sphere in anima then Cbull(Si lowast) Z[i]

40


This new adjunction needs a bit of justification but to not interrupt the flow wersquoll discussthat later in Remark I56c For now letrsquos compute

πiK(C) = π0 HomlowastAn((Si lowast)K(C)

)= π0 HomDgt0(Z)

(Cbull(Si lowast) C

)= π0 HomDgt0(Z)

(Z[i] C

)= Hi(C)

In fact we can say more Let A be an abelian group and A[n] isin Dgt0(Z) the complexconsisting of A placed in degree n (technically we have to replace A[n] by a projectiveresolution to be consistent with Example I15(e)) Then K(An) = K(A[n]) satisfies

πi HomAn(XK(An)

)= πi HomDgt0(Z)

(Cbull(X) A[n]

)= HiminusnC

bull(XA)= Hnminusi(XA)

where Cbull(XA) isin D60(Z) is the complex of simplicialcellularsingular cochains withcoefficients in A If X is pointed and connected the same calculation works with reducedcohomology (this only matters for n = i) We thus obtain

I56a Theorem (EilenbergndashMacLane) mdash The functors Hn(minus A) πAn Set (unre-duced cohomology) and Hn(minus A) π(lowastAn) Set (reduced cohomology) are represented byK(An)

In particular if X isin lowastAn satisfies πiX = 0 for i 6= n then H = HomlowastAn(XK(An)) isgiven by the discrete anima Hn(XA) Indeed we have πiH = Hnminusi(XA) which vanishesfor 1 6 i 6 n by Hurewicz and the universal coefficient theorem and for i gt n by the factthat cohomology is zero in negative degrees So H π0H is a homotopy equivalence byWhiteheadrsquos theorem Using universal coefficients again we can summarize this by

HomlowastAn(XK(An)

) Hn(XA) HomAb

(Hn(X) A

)

For n gt 1 an isomorphism on the right-hand side gives an equivalence on the left-hand sideby a combination of Hurewicz and Whitehead (for n = 1 we really need that A is abelian forthis argument to work) This shows that K(minus n) Ab lowastAn is fully faithful for n gt 1

Now consider the full sub-infin-category lowastAn6n sube lowastAn spanned by n-truncated animaie those with vanishing πi for i gt n The inclusion has a left-adjoint τ6n lowastAn lowastAn6nOne can construct τ6nX by iteratively attaching cells to X to kill higher and higher homotopygroups and then use that adjunctions can be constructed objectwise by Corollary I32 Infact one can take τ6nX = coskn+1X since the right-hand side is a really brutal way ofkilling homotopy groups in degrees n and higher In particular the n-truncation τ6nXsatisfies

πi(τ6nX) =

0 if i gt n

πi(X) if i 6 n

The unit map X τ6nX factors over τ6n+1X τ6nX since τ6nX is already (n + 1)-truncated We claim that the homotopy fibre F (ie the fibre taken inside theinfin-category An)of this map is equivalent to K(πn+1Xn+ 1) Indeed the long exact sequence of homotopy

41


groups shows that πiF = 0 for i 6= n+ 1 and that πn+1F πn+1X Hence by the discussionafter Theorem I56a

HomlowastAn(FK(πn+1Xn+ 1)

) HomAb

(Hn+1(F ) πn+1X

)

and an isomorphism on the right-hand side induces an isomorphism on the left-hand sideBut Hn+1(F ) πn+1X by Hurewicz hence we do obtain an equivalence as desired

The diagram

X minus( minus τ6n+1 minus τ6nX minus minus τ61X

)is called the Postnikov tower of X Itrsquos not hard to show that the Postnikov tower induces anequivalence X simminus limnisinNop τ6nX in lowastAn when X is connected (see [Hat02 Corollary 468]for example) The Postnikov tower is an important organisational and computationaltool since it ldquostratifiesrdquo lowastAn into pieces whose ldquodifference is controlled by Abrdquo via theEilenbergndashMacLane functor K(minus n) Ab lowastAn

I56b Exercise mdash Show that every chain complex over Z (in fact over any PID) isquasi-isomorphic via a zigzag to its homology considered as a chain complex all of whosedifferentials are 0

In particular Exercise I56b implies that for C isin Dgt0(Z) there is a (non-canonical)equivalence

C infinoplusi=0

Hi(C)[i] infinprodi=0

Hi(C)[i]

Since the right-adjoint functor K preserves limits (Observation I41) this implies K(C) prodinfini=0K(Hi(C) i) In general anima X with the property

X infinprodi=0

K(πiX i)

are called gems (a pun on ldquogeneralized EilenbergndashMacLane spacesrdquo) It is also easy to checkthat the following diagram is a pullback in the infin-category Dgt0(Z)

τgt0(C[minus1]

)0

0 C

Again since K preserves limits we get K(τgt0C[minus1]) ΩK(C) (indeed if you take the samepullback in An it will give you the loop space) In particular K(Anminus 1) ΩK(An) forall n gt 1 And thatrsquos our first spectrum

I56c Remark mdash To construct a lift K Dgt0(Z) lowastAn Fabian explained that0 isin Dgt0(Z) is an initial object and K(0) HomDgt0(Z)(Z[0] 0) lowast so K lifts to a functorK Dgt0(Z) 0Dgt0(Z) lowastAn on slice categories I claimed that K(0) empty in a previousversion of these notes but thatrsquos nonsense and Fabian was completely right

Yet the fact that K has a left adjoint is not entirely automatic more precisely itrsquosnot an instance where an adjunction automatically descends to slice categories (this will

42


happen occasionally in the future) The reason is Cbull(lowast) Z[0] so Cbull doesnrsquot lift to afunctor lowastAn 0Dgt0(Z) Dgt0(Z) on slice categories Instead we have to define a functorCbull lowast An Dgt0(Z) taking a pointed anima (Xx) to the pushout

Z[0] Cbull(x) Cbull(X)

0 Cbull(Xx)

or in other words to the cofibre cofib(Cbull(x) Cbull(X)) Itrsquos straightforward to check thatCbull is indeed left-adjoint to K Indeed from Corollary I50 and the computation of Homanima in slice categories ([HCII Proposition VIII6]) we get

HomDgt0(Z)(Cbull(Xx) D

) HomDgt0(Z)

(Cbull(X) D

)timesHomDgt0(Z)(Cbull(x)D) 0

HomAn(XK(D)

)timesHomAn(xK(D)) K(0) lowast K(D)

HomlowastAn((Xx)K(D)

)(all pullbacks are taken in An as announced on Page 35) The complex Cbull(Xx) computesreduced homology For example if Si denotes the i-sphere then Cbull(Si lowast) cofib(Z[0] Z[0]oplus Z[i]) Z[i] as one would expect

I57 Example mdash Two more applications of Theorem I51(a) If D is an infin-category we will again write sD and cD for the functor categories

Fun(∆∆opD) and Fun(∆∆D) respectively If D is cocomplete and F isin cD a cosimplicialobject in D we obtain an adjunction

| |F sAn D SingF

In the case where D is a 1-category we also get an adjunction | |1F sSet D Sing1F

from Theorem I2 To see how these two adjunctions are related recall that sSet sube sAnhas a left adjoint π0 sAn sSet by Example I33(a) and Observation I40 which yieldsa factorisation

| |F sAn sSet D SingFπ0

incl

| |1F

Sing1F

of the adjunction above

Warning mdash Right now there is another possible definition of π0 Recall thatπ0 sSet Set can also be written as colim∆∆op = | |1const lowast where the cosimplicialset const lowast ∆∆ Set sends everything to a point Analogously we have a functorcolim∆∆op = | |const lowast sAn An which arises via Theorem I51 from the cosimplicialanima const lowast ∆∆ An We will never call this functor π0 but colim∆∆op instead whichis what it really is

(b) Consider ∆∆ Cat1 sube Cat(2)1 sube Catinfin sending [n] to itself considered as a poset By

(a) this cosimplicial infin-category induces an adjunction

asscat sAn Catinfin Nr

43


(Fabian remarks that ldquoasscatrdquo is short for ldquoassociated categoryrdquo and has nothing todo with ldquoArschkatzenrdquo) The right adjoint Nr is called the Rezk nerve and is a veryimportant construction If C is an infin-category Corollary I55 provides the explicitdescription

Nrn(C) HomCatinfin

([n] C

) core Fun

([n] C

)

Beware that if C is a 1-category then Nr(C) is different from the ordinary nerveN(C) = ([n] 7 HomCat1([n] C)) since Cat1 sube Cat(2)

1 is not fully faithful In fact N(C)is a discrete simplicial anima but Nr(C) is usually not After a brief detour to Bousfieldlocalisations we will come back to Rezk nerves in TheoremDefinition I64

Bousfield LocalisationsLecture 6

17th Nov 2020I58 PropositionDefinition mdash Let L C D R be an adjoint pair(a) The functor R is fully faithful iff the counit c LRrArr idD is an equivalence(b) A morphism f x y in C is taken to an equivalence by L iff it is a left R-local

equivalence ie iff the induced map

flowast HomC(yRd) minus HomC(xRd)

is an equivalence for all d isin D We will often drop the ldquoleftrdquo part and just write ldquoR-localequivalencerdquo

Moreover if the equivalent conditions from (a) hold then(c) The unit c RLc is always a left R-local equivalence(d) L C D is a localisation at the left R-local equivalencesOne then says that L is a left Bousfield localisation In particular C[R-local equivminus1] isagain locally small Of course therersquos also a dual notion of right Bousfield localisations

Proof Fabian decided to prove (b) first By Yonedarsquos lemma for Lf Lx Ly to bean equivalence it suffices to show that it induces an equivalence after HomD(minus d) for alld isin D Using the natural adjunction equivalence HomD(Lminusminus) HomC(minus Rminus) we obtaina homotopy commutative diagram

HomC(yRd) HomC(xRd)

HomD(Ly d) HomD(Lx d)

flowast

sim sim

Lflowast

in anima which immediately proves (b)Next up we show (a) Bastiaan found a proof which is considerably simpler than the one

from the lecture so we shall include his proof here Observe that the diagram

HomD(d x) HomD(LRd x)

HomC(RdRx)

clowastd

Rsim

44


commutes (in πAn) for all d x isin D Indeed since Yonedarsquos lemma tells us that

evidd Nat(

HomD(dminus)HomD(LRdminus)) simminus HomD(LRd d)

is an equivalence it suffices to check commutativity in the special case x = d on the singleelement idd isin HomD(d d) But idd is clearly mapped to cd isin HomD(LRd d) in both casesproving that the diagram does indeed commute By Theorem I19(b) and Yonedarsquos lemmaagain c LRrArr idD is an equivalence iff clowastd HomD(d x) simminus HomD(LRd x) is an equivalencefor all x isin D By the diagram above this holds iff R HomD(d x) simminus HomC(RdRx) is anequivalence for all d x isin D ie iff R is fully faithful

Now for (c) By one of the triangle identities the composition

LxLuxminusminus LRLx

cLxminusminus Lx

is equivalent to idLx for all x isin C Since we assume that the conditions from (a) hold cLx isan equivalence hence Lux is one too By (b) this means that ux is an R-local equivalence

For (d) let E be any test category and consider the adjunction

Rlowast Fun(C E) Fun(D E) Llowast

obtained from Observation I40 Since the counit of (LR) is an equivalence the counitof (Rlowast Llowast) is an equivalence too (it is induced by the former counit) But the direction ofthe adjunction has switched so now Llowast is fully faithful by (a) We are left to show thatthe essential image of Llowast is FunR-local equiv(C E) ie those functors that invert R-localequivalences But any LlowastF F L inverts the R-local equivalences by (b) hence the essentialimage of Llowast is contained in FunR-local equiv(C E) Conversely for F C E inverting theR-local equivalences we have F FRL Llowast(FR) since the unit ux x RLx is an R-localequivalence for all x isin C by (c) Hence F is contained in the essential image of Llowast

Now that we know when adjunctions are Bousfield localisations we would like to studythe ldquoconverserdquo question ie when localisations are Bousfield localisations

I59 Proposition mdash Let W sube π0 core Ar(C) be some collection of morphisms withassociated localisation p C C[Wminus1] Then a map τ px y in C[Wminus1] exhibits x as aright-adjoint object to y under p iff τ is an equivalence and HomC(minus x) Cop An invertsall morphisms from W

In particular a right-adjoint functor R to p is automatically fully faithful Note that thisalso characterizes the essential image of R as the (left) R-local objects ie those c isin C suchthat HomC(minus c) Cop An inverts (left) R-local equivalences

We first prove the following lemma

I60 Lemma mdash Let p C C[Wminus1] be a localisation(a) For all c isin C the functor HomC[Wminus1](minus pc) is the left Kan extension of HomC(minus c)

along pop In diagramsCop An

C[Wminus1]op

pop

HomC(minusc)

HomC[Wminus1](minuspc)

45


(b) Any functor F C[Wminus1] D is left Kan extended from F p along p In diagrams

C D

C[Wminus1]

p

Fp

F

The same is true for functors F C[Wminus1]op D and left Kan extension along pop Infact pop Cop C[Wminus1]op is a localisation of Cop at W op

Warning mdash Donrsquot get too excited and think ldquoYeah now Theorem I52 allows meto compute Hom anima in localisationsrdquo since the pointwise formula is useless (or rathertautological) here

Proof of Lemma I60 For (a) we apply Yonedarsquos lemma twice to obtain

Nat(HomC[Wminus1](minus pc) G

) G(pc) Nat

(HomC(minus c) G pop)

for all G C[Wminus1]op D This is precisely what we want from a Kan extension along popFor (b) recall that plowast Fun(C[Wminus1] E) Fun(C E) is fully faithful ie

plowast Nat(FG) simminus Nat(F pG p)

is an equivalence for all F and G This again exhibits F as a left Kan extension along p thistime For the additional assertion use the universal property or the explicit constructionfrom [HCII Theorem VIII8]

Proof of Proposition I59 If HomC(minus x) inverts all arrows from W then it descends to afunctor F C[Wminus1]op An which is left Kan extended from HomC(minus x) by Lemma I60(b)But so is HomC[Wminus1](minus px) by Lemma I60(a) Hence

HomC(minus x) HomC[Wminus1](pminus px) τlowastminus HomC[Wminus1](pminus y)

is an equivalence since τ is one This proves that x is indeed a right-adjoint object of yFor the converse just read the argument backwards As Sil and Bastiaan pointed out wehave to use that p C C[Wminus1] is essentially surjective (use its universal property or theconstruction from [HCII Theorem VIII8]) so for τ to be an equivalence it does suffice thatτlowast HomC[Wminus1](pminus px) HomC[Wminus1](pminus y) is an equivalence

For the additional assertions assume that R is a right adjoint of p Then the counitpRrArr id consists of morphisms τ as above hence it is an equivalence hence R is fully faithfulby PropositionDefinition I58(a) A morphism in C is an R-local equivalence iff it becomesan equivalence in C[Wminus1] by PropositionDefinition I58(b) so it is clear that the essentialimage of R is contained in the R-local objects by what we showed above But conversely anR-local object x isin C is a right-adjoint object of px via the witness idpx px px hence x isin the essential image of R

I61 Corollary mdash If L C D is a left Bousfield localisation and C is complete orcocomplete then so is D More precisely we have

colimI

F L(

colimI

RF)

and limIF L

(limIRF)

for any diagram F I D

46


Proof We start with the formula for colimits Compute

HomD(L(

colimI

RF)minus) HomC

(colimI

RFRminus) Nat

(RF const(Rminus)

)Now R is fully faithful by PropositionDefinition I58(a) hence so is the postcompositionfunctor Rlowast Fun(ID) Fun(I C) which shows

Nat(RF const(Rminus)

) Nat(F constminus)

Hence L(colimI RF ) satisfies the desired universal property for colimI F For the limit part first note that R-local objects are closed under limits Indeed if

x limjisinJ xj with each xj an R-local object then

HomC(minus x) limjisinJ

HomC(minus xj)

by the dual of Corollary I50 hence HomC(minus x) inverts R-local equivalences because eachHomC(minus xj) does Moreover R-local equivalences between R-local objects are actualequivalences by Yonedarsquos lemma applied to the full sub-infin-category of C spanned by R-localobjects Now PropositionDefinition I58 shows that the unit

u limIRF simminus RL

(limIRF)

is an R-local equivalence hence an actual equivalence as RF takes values in the R-localobjects by the addendum to Proposition I59 Using this and the fact that R is fully faithfulwe can now compute

HomD(minus L

(limIRF)) HomC

(Rminus RL

(limIRF)) HomC

(Rminus lim

IRF)

Nat(

const(Rminus) RF)

Nat(constminus F )

so L(limI RF ) satisfies the desired universal property for limI F

Before we discuss our first examples of Bousfield localisations we need to discuss aproposition which Fabian originally forgot to put here but which came up later in thecourse

I61a Proposition (see [HTT Proposition 5274]) mdash Let L C C be a functortogether with a natural transformation η idrArr L such that both maps

ηLx Lx simminus LLx and Lηx Lx simminus LLx

are equivalences for all x isin C Then L C im(L) is left-ajoint to the inclusion im(L) sube Cwith unit η In particular L is a Bousfield localisation and Lη ηL as natural transformations(by the triangle identities)

Proof We will prove the assertion by reduction to the case of 1-categories for which wealready know this statement We have to show that ηlowastc HomC(Lc Ld) simminus HomC(c Ld) isan equivalence for all c d isin C For this it suffices to show that for all K isin An the inducedmap

π0 HomAn(KHomC(Lc Ld)

) simminus π0 HomAn(KHomC(c Ld)

)

47


is a bijection But now consider the functor of 1-categories

πLlowast π Fun(K C) minus π Fun(K C)

together with the transformation πηlowast idrArr πLlowast satisfying that πηlowastπLlowast πLlowastsim=rArr πLlowastπLlowast

and πLlowastπηlowast πLlowastsim=rArr πLlowastπLlowast are natural equivalences So we get the same situation as

before but now π Fun(K C) is a 1-category For 1-categories we already know the statement istrue For example wersquove seen this on exercise sheet 2 of Higher Categories I and also Luriegives another proof Plugging in the functors const c constLc constLd isin Fun(K C) thenshows that π0 HomAn(KHomC(Lc Ld)) simminus π0 HomAn(KHomC(c Ld)) is an equivalenceas desired Thus we have proved that L C im(L) is a Bousfield localisation

To obtain the additional assertion that Lη ηL write down the triangle identities andnote that both sides have the same left inverse namely the counit evaluated at Lx

I62 Example mdash ldquoMy first Bousfield localisationsrdquo(a) We now recognize all the left pointing arrows in the diagram from Example I33(a)


π0

sube

| |

core

as Bousfield localisations More precisely π0 and | | are left Bousfield localisationswhereas core is a right one

(b) Recall from Example I15(d) that Top has a Kan enrichment giving rise to the infin-category Nc(Top) of topological spaces Consider the localisation

p Nc(Top) minus Nc(Top)[weak homotopy equivminus1

]

The functor HomNc(Top)(Xminus) inverts weak homotopy equivalences if X is a CWcomplex (thatrsquos a well-known extended version of Whiteheadrsquos theorem) and everytopological space admits a weak equivalence from a CW complex (ldquoCW approximationrdquo)Hence the dual of Proposition I59 implies that p is a right Bousfield localisation

This also shows that An Nc(CW) Nc(Top)[weak homotopy equivminus1] is anequivalence In particular we get for free that CW approximation is infin-functorial

(c) Let R be a ring Recall from Example I15(e) that there is an infin-category K(R) =Nc(Ch(R)) of chain complexes over R We define the derived infin-category of R

D(R) = K(R)[quasi-isosminus1]

as the localisation of K(R) at the quasi-isomorphisms Then the fundamental lemma ofhomological algebra2 says that for a bounded below C isin K(R) any projective resolution

2Herersquos the complete argument We need to show that HomK(R)(Pminus) inverts quasi-isomorphismsSo let ϕ C D be one The fundamental lemma or at least a strong form of it says thatϕlowast homotopy classes of maps P C simminus homotopy classes of maps P D is bijective In otherwords ϕlowast H0 HomR(PC) sim

minus H0 HomR(PD) is an isomorphism where HomR is the internal Homin Ch(R) Replacing P by its shifts P [minusi] shows the same for ϕlowast Hi HomR(PC) simminus Hi HomR(PD)Now Exercise I15a shows

πi HomK(R)(minusminus) Hi HomR(minusminus)and we are done

48


P C is a left-adjoint object to C Similarly for a bounded above C an injectiveresolution C I is a right-adjoint object In particular

Dprojgt0 (R) sube K(R) minus D(R)

is fully faithful with essential image the non-negative chain complexes (and similarly forDinj60(R)) But in fact K(R) D(R) admits both adjoints so its both a left and a right

Bousfield localisation This boils down to the classical result of Spaltenstein that thereare enough K-projective and K-injective complexes (again a K that has nothing to dowith K-theory) see [Spa88] (thanks to Sil for the reference) In particular combiningExercise I50b and Corollary I61 shows that D(R) is both complete and cocomplete

Also note that the homotopy category πD(R) is equivalent to the usual derived1-category D(R) by [HCII Proposition VIII12]

(d) Given a ring morphism ϕ R S we can look at the functor

S otimesR minus Ch(R) minus Ch(S)

This is simplicially enriched (basically since the enrichment of Ch(R) is defined via tensorproducts and these are associative) hence it defines a functor S otimesR minus K(R) K(S)Now define the derived tensor product as the composition

ϕ = S otimesLR minus D(R) K-projminusminusminusminusminus K(R) SotimesRminusminusminusminusminus K(S) minus D(S)

The map K-proj D(R) K(R) is a left adjoint of the right Bousfield localisationp K(R) D(R) and given by taking K-projective resolutions We have seen in (c) thatp also admits a right adjoint but that wouldnrsquot give the correct functor Also note thatHi(S otimesLRM) = TorRi (SM) holds for all R-modules M by unravelling of constructions

It follows from general homological nonsense that ϕ has a right-adjoint ϕlowast (theldquoforgetful functorrdquo) which has another right adjoint ϕlowast = RHomR(Sminus) In diagrams

D(R)ϕ

ϕlowastD(S)

ϕlowast

ϕlowastD(R)

I63 PropositionDefinition mdash For a ring morphism ϕ R S the following condi-tions are equivalent(a) ϕ = S otimesLR minus D(R) D(S) is a left Bousfield localisation(b) ϕlowast = forget D(S) D(R) is fully faithful(c) ϕlowast = RHomR(Sminus) D(R) D(S) is a right Bousfield localisation(d) The multiplication map S otimesLR S S is an equivalence in D(S) As Sil points out we

should actually write S otimesLR ϕlowastS but letrsquos not do that here(e) The cofibre cofib(ϕ) = SLR ie the pushout

R S

0 SLR

ϕ

in D(R) satisfies S otimesLR (SLR) 0If these conditions are satisfied ϕ is called a derived localisation (or sometimes ldquoTor-unitalrdquo)

49

The Rezk Nerve and Complete Segal Spaces

I63a Examples mdash ldquoMy first derived localisationsrdquo(a) If W sube R is a multiplicative subset in the commutative ring R then R R[Wminus1]

is a derived localisation This easily follows from PropositionDefinition I63(d) sinceR[Wminus1] is flat over R

(b) For non-commutative rings one needs the (left or right who knows) Ore condition forR R[Wminus1] to be a derived localisation

(c) If U sube V is an open embedding of affine schemes then Γ(VOV ) Γ(UOU ) is aderived localisation (but not necessarily an ordinary localisation) Indeed the conditionfrom PropositionDefinition I63(d) easily implies that a ring morphism ϕ R S is aderived localisation iff Rp Sq is a derived localisation for all prime ideals q isin SpecSand p isin SpecR such that p = ϕminus1(q) In our case Rp and Sq correspond to the localrings OVv and OUv for v isin V and these are isomorphic since U is an open subscheme ofV Hence the morphisms OVv simminus OUv for v isin V are derived localisations for obviousreasons

We will later see that K-theory has exact sequences for derived localisations

Proof of PropositionDefinition I63 We already know (a) hArr (b) hArr (c) from Proposi-tionDefinition I58 and its dual For (d)hArr (e) first note that SotimesLRR

simminus S is an equivalenceIndeed R can be taken as its own projective resolution hence S otimesLR R S otimesR R S

Now S otimesLR minus being a left adjoint commutes with colimits which implies that

S S otimesLR S

0 S otimesLR (SLR)

SotimesLRϕ

is a pushout square This immediately implies S otimesLR S S iff S otimesLR (SLR) 0 as desiredWe finish the proof by showing (b) hArr (d) We must show that the counit S otimesLR C C is

an equivalence for all C isin D(S) iff it is an equivalence for S The ldquoonly ifrdquo direction beingobvious letrsquos assume it is an equivalence for S Then it is also an equivalence for all shiftsS[i] But the S[i]iisinZ generate D(S) under colimits and S otimesLR minus commutes with colimitsso we are done

The Rezk Nerve and Complete Segal SpacesLecture 7

19th Nov 2020From now on a good portion of the lecture will take place in the land of simplicial animaUnfortunately this leads to ldquo∆nrdquo having two different meanings It could mean the infin-category ∆n = N([n]) or the simplicial set ∆n which we consider as a discrete simplicialanima via sSet sube sAn To avoid confusion we adopt the following convention The infin-category will always be denoted [n] whereas the simplicial anima will be denoted ∆n

Under the Rezk nerve functor Nr Catinfin sAn from Example I57(b) these two arerelated via Nr([n]) ∆n Indeed we have

Nrm

([n]) HomCatinfin

([m] [n]

) core Fun

([m] [n]

)

and the right-hand side is the discrete set (∆n)m because every isomorphism of functors inFun([m] [n]) must be an identity (as [n] itself has no non-identity isomorphisms)

50


I64 TheoremDefinition (Rezk JoyalndashTierney Lurie) mdash The Rezk nerve functorNr Catinfin sAn is fully faithful and its essential image consists of the complete Segalspacesanima Here a Segal anima is a simplicial anima X ∆∆op An such that

Xn HomsAn(∆n X) simminus HomsAn(In X) X timesX0 X1 timesX0 middot middot middot timesX0 X1

is an equivalence for all n (here In sube ∆n denotes the nth spine ie the union of all edgesbetween consecutive 0-simplices of ∆n) We call X complete if the following equivalentconditions hold(a) Also the restriction X0 HomsAn(∆0 X) simminus HomsAn(JX) along J ∆0 is an

equivalence where J is the (ordinary nerve of the) free-living isomorphism consideredas a simplicial setdiscrete simplicial anima

(b) The diagramX0 X0 timesX0

X3 X1 timesX1

s

∆

(ss)(d02d13)

is cartesian (note that ∆ denotes a good old honest diagonal and not some simplicialstuff )

(c) The degeneracy map s X0simminus Xtimes1 is an equivalence where Xtimes1 sube X1 is the collection

of path components of those g isin X1 for which the following holds Let x = d1(g) andy = d0(g) and for arbitrary w z isin X0 put

Pwz X1

lowast X0 timesX0

(d1d0)(wz)

Moreover there is a ldquocomposition maprdquo

X1 timesd1X0d0 X1 X2d1minus X1

(d0d2)sim

which after restriction to Pwx Pwxtimesx g and Pyz gtimesyPyz induces ldquopost-and precomposition mapsrdquo

glowast Pwx minus Pwy and glowast Pyz minus Pxz

Then we put g isin Xtimes1 iff glowast and glowast are equivalences for all w z isin X0(d) The simplicial subanima Xtimes given by

[n] 7minus Xtimesn =

collection of path components inXn such that all edges lie in Xtimes1

is constant Note that Xtimes0 = X0 so it is actually constant on X0

51


I65 Some Explanations mdash Before we (not so much) talk about the proof someexplanations are in order

First of the Segal condition Just like Segal sets Segal anima should be thought ofsimplicial anima with ldquounique spine liftingrdquo ie unique (up to contractible choice) liftingagainst In sube ∆n To see why HomsAn(In X) X1 timesX0 middot middot middot timesX0 X1 we must show thatIn is an iterated pushout of copies of ∆1 and then invoke Corollary I50 The pushoutcondition is certainly true in sSet In particular the discrete set (In)m is an iterated pushoutof copies of (∆1)m and the maps along the pushouts are taken are injective Now pushoutsin sAn are computed degreewise by Lemma I39 the simplicial anima In ∆1 and ∆0 arediscrete sets in every degree and Set sube An preserves pushouts along injective maps so inthis particular special case the pushout in sSet agrees with the one in sAn Wersquoll also linkthe Segal condition to monoidal structures soon see Chapter II

Now about completeness As Bastiaan explained to me the completeness conditionis probably best understood in the words of Emily Riehl ldquo[The completeness condition]says that isomorphisms are equivalent to identitiesrdquo Although these words were spokenin a slightly different context they fit here just wonderfully For conditions (a) and (c)it is intuitive that ldquoisomorphisms are equivalent to identitiesrdquo is what theyrsquore saying andcondition (d) is basically just the statement that all higher simplices having equivalences asedges are degenerate To make it more intuitive why also condition (b) is saying the samething Fabian drew the following picture of a 3-simplex in X

0

1

2

3

id

id

What (b) is trying to say is that whenever the ∆02- and ∆13-edges are degenerate asindicated then all edges are degenerate and the 3-simplex itself is equivalent to a degenerateone Notice that this is just the higher categorical way of phrasing the requirement thatidentity morphisms are closed under the 2-out-of-6-property This is also equivalent toall equivalences being identities since the equivalences in an infin-category are the smallestcollection of arrows containing the identities and closed under the 2-out-of-6-property

Not a proof of TheoremDefinition I64 Itrsquos relatively easy to see that Nr(C) for an infin-category C is a complete Segal anima The Segal condition reduces to the claim that[n] [1]t[0] middot middot middott[0] [1] in Catinfin which can be shown as in Example I38 For completeness oneunwinds that Nr

1(C)times sube core Ar(C) consists of the equivalences hence s Nr0(C) simminus Nr

1(C)timesbeing an equivalence reduces to [1] [0] being a localisation Fabian also recommends toprove the equivalence of the four conditions as a (not easy but doable) exercise

However the proof that Nr is fully faithful is difficult The original proof of JoyalndashTierneycan be found in [JT07] and Luriersquos poof is in [Lur09] The key step is to give a descriptionof asscat(X) for X a (not necessarily complete) Segal anima First there is a canonicalequivalence

core asscat(X) |Xtimes| (I651)

where | | colim∆∆op sAn An denotes the colimit functor over ∆∆op (Fabian points outthat this notation is very much consistent with the other kinds of realisation wersquove seen so

52


far see Remark I66 below) Second one can show that for every (not necessarily complete)Segal anima X there are pullback diagrams

Homasscat(X)(x y) X1

lowast X0 timesX0

(d1d0)

(xy)

(I652)

for all x y isin X0 In particular Homasscat(X)(x y) Pxy in the notation of (b)With (I651) and (I652) the proof of TheoremDefinition I64 becomes quite easy To

prove that Nr is fully faithful we must show that the counit asscat(Nr(C)) simminus C is anequivalence for all C isin Catinfin It is an equivalence on cores because

core asscat(

Nr(C)) |Nr(C)times| and core C Nr

0(C)

since Nr(C) is complete Nr(C)times is constant on Nr0(C) hence the colimit over the weakly

contractible category ∆∆op is just |Nr(C)times| Nr0(C) by Proposition I36 as claimed This

shows that the counit is essentially surjectiveTo prove that it is fully faithful ie induces equivalences on Hom anima observe that

using (I652) it induces the following equivalence of pullback squares

Homasscat(Nr(C))(x y) HomC(x y)

Nr1(C) core Ar(C)

lowast lowast

Nr0(C)timesNr

0(C) core C times core C

sim

(xy)(xy)

sim

For the right pullback square recall the usual pullback diagram for HomC(x y) can also betaken in Catinfin (Example I35a(a)) and core(minus) preserves pullbacks because it is a rightadjoint Up to verifying the actual hard stuff ((I651) and (I652)) this finishes the proofthat Nr is fully faithful To show that itrsquos image are the complete Segal anima one canargue similarly for the unit X Nr(asscat(X))

I66 Remark mdash Note that the notation | | colim∆∆op is consistent with our notationfor the geometric realisation of simplicial sets If T ∆∆op Set sube An is a simplicialsetdiscrete simplicial anima then the homotopy colimit of T (considered as a functorT ∆∆op Set sube sSet where sSet is equipped with the KanndashQuillen model structure) isweakly homotopy equivalent to T so

colim∆∆op

T (

hocolimC[∆∆op]

T)f T f Sing |T |

using Theorem I34 With the technique that we will introduce in the proof of Lemma IV7one can also show directly that colim∆∆op T is the anima corresponding to the CW complex|T | The notation is moreover consistent with | | Catinfin An in that generally for asimplicial anima T ∆∆op An we have

| asscat(T )| |T |

53


Indeed this follows from Theorem I51 as both sides are colimit-preserving functors thatsend ∆n 7 lowast isin An for all n isin N

I67 LemmaDefinition mdash Let CSAn sube sAn denote the full subminusinfin-category of com-plete Segal anima The functor

comp sAn asscatminusminusminusminus Catinfin

Nrminus CSAn

is called completion and it is a left adjoint to the inclusion CSAn sube sAn

Proof Indeed if X isin sAn and Y isin CSAn so Y Nr(C) for some infin-category C thenHomsAn(compXY ) HomCatinfin(asscatX C) HomsAn(XNr(C)) since Nr is fully faithfuland right-adjoint to asscat sAn Catinfin

Fabian remarks that comp sAn CSAn is impossible to control outside of (not neces-sarily complete) Segal spaces

I68 Example mdash (a) The ordinary nerve N(C) isin sSet sube sAn of a 1-category is a Segalset and thus also a Segal anima since Set sube An preserves limits But it is usually notcomplete In fact we have

N0(C) set of objects in C

N1(C) set of isomorphisms in C

s

where the right vertical arrow takes x isin C to idx But usually there are more isomor-phisms in a category than just the identities hence N1(C)times is usually larger than N(C)0This is also another instance of Emily Riehlrsquos wise words from I65 ldquoThe completenesscondition says that isomorphisms are equivalent to identitiesrdquo

Note however that the composition asscat(N(C)) asscat(Nr(C)) C is an equiva-lence so Nr(C) is the completion of N(C) in the sense of LemmaDefinition I67 Indeeditrsquos clear from (I652) that asscat(N(C)) C induces equivalences on Hom anima soit remains to see whether it is essentially surjective By (I651) it suffices to show|N(C)times| core C But Nn(C)times is given by the set of composable sequences (f1 fn)of isomorphisms in C hence it is the ordinary N(core C) Since this is already a Kancomplex Remark I66 shows |N(core C)| core(C) as required

(b) The two functorssAn asscatminusminusminusminus Catinfin

πminus Cat(2)

1

sAn sπ0minusminus sSet hminus Cat1

are different even though both are colimit-preserving and take ∆n 7 [n] The reason is ofcourse that Cat1 sube Cat(2)

1 doesnrsquot preserve colimits To obtain explicit counterexamplesyou can take constS1 or Nr(finite-dimensional vector spaces) for example Bothfunctors do however agree on ordinary nerves N(C) of 1-categories NeverthelessFabian warns you to never look at the lower functor

(c) Let sAnconst sube sAn denote the full subcategory of constant simplicial anima Theadjunction asscat sAn Catinfin Nr restricts to an equivalence

ev0 sAnconstsimsim An const

54


Indeed Nrn(K) core Fun([n]K) Fun(|[n]|K) K as |[n]| lowast

I69 Lemma mdash For any infin-category C we can describe Nr(TwAr(C)) as follows

Nrn

(TwAr(C)

) HomCatinfin

([n]op [n] C

) HomsAn

((∆n)op ∆nNr(C)

)

Here [n]op [n] denotes the usual join of categories and (∆n)op ∆n is formed as a join insSet and then imported into sAn

Proof Recall from I28 that there is a pullback square

TwAr(C) lowastAn

Cop times C An

HomC

(depending on your definitions this is a pullback both in simplicial sets and in Catinfin or onlyin Catinfin) Now we do a computation in several steps each of which will be justified below

Nrn

(TwAr(C)

) HomCatinfin

([n]TwAr(C)

)(1) HomCatinfin

([n] Cop times C

)timesHomCatinfin ([n]An) HomCatinfin

([n] lowastAn

)(2) HomCatinfin

([n] Cop times C

)timescore(An) core(lowastAn)

(3) HomCatinfin

([n] Cop times C

)timescore(CoptimesC) core

((Cop times C)timesAn lowastAn

)(4)(

HomCatinfin([n]op C

)timesHomCatinfin

([n] C

))timescore(CoptimesC) core TwAr(C)

(5)(

HomCatinfin([n]op C

)timesHomCatinfin

([n] C

))timescore(CtimesC) core Ar(C)

(6) core Fun

([n]op t0 [1] t0 [n] C

)(7) HomCatinfin

([n]op [n] C

)(8) HomsAn

((∆n)op ∆nNr(C)

)

For step (1) we plug in TwAr(C) (Cop times C)timesAn lowastAn and use that HomCatinfin([n]minus)preserves pullbacks by the dual of Corollary I50

For step (2) we observe that restriction along 0 [n] ie ldquoevaluation at 0rdquo in-duces morphisms ev0 HomCatinfin([n] lowastAn) HomCatinfin(0 lowastAn) core(lowastAn) andev0 HomCatinfin([n]An) HomCatinfin(0An) core(lowastAn) We claim that these fit into apullback diagram

HomCatinfin([n] lowastAn

)core(lowastAn)

HomCatinfin([n]An

)core(An)

ev0

ev0

in An To see this note that Fun([n] lowastAn) Funlowast([n + 1]An) holds by the join-sliceadjunction (or by inspection) where Funlowast denotes the full sub-infin-categories of functorstaking 0 7 lowast isin An Similarly lowastAn Funlowast([1]An) Now [n+1] can be written as a pushout

55


[1] t1 [1 n + 1] in Catinfin (where [1 n + 1] denotes the poset 1 n + 1) Indeed byTheorem I34 we may compute this pushout as a fibrant replacement of the correspondinghomotopy pushout in sSet Since 1 ∆1 is a cofibration the homotopy pushout is justthe ordinary pushout and ∆1 t1∆1n+1 ∆n+1 is inner anodyne so [n+ 1] is indeedthe correct pushout in Catinfin Since HomCatinfin(minusAn) core Fun(minusAn) takes pushouts topullbacks by Corollary I50 we get

core Fun([n+ 1]An

) core Fun

([1]An

)timescore Fun(1An) core Fun

([1 n+ 1]An

)

and this restricts to an equivalence

core Funlowast([n+ 1]An

) core Funlowast

([1]An


([1 n+ 1]An

)

This shows that the pullback diagram above is correct Plugging it into (1) gives (2)Step (3) is a formal pullback manipulation together with the fact that core(minus) commutes

with pullbacks since it is a right adjointFor (4) note that HomCatinfin([n] CoptimesC) HomCatinfin([n] Cop)timesHomCatinfin([n] C) and we

have Fun([n] Cop) core Fun([n]op C)op But the (minus)op vanishes upon taking cores hence

HomCatinfin([n] Cop) HomCatinfin

([n]op C

)

This explains the left factor in the pullback For the right factor we simply plug inTwAr(C) (Cop times C)timesAn lowastAn

Step (5) follows straight from Exercise I28a Step (6) is immediate from Corollary I50and for (7) we need to check that [n]op t0 [1]t0 [n] when taken as a pushout in Catinfin isgiven by [n]op [n] This can be done as before Take the pushout in sSet and verify thatthe canonical map to [n]op [n] is inner anodyne

Finally (8) is immediate once we convinced ourselves that

Nr([n]op [n]) (∆n)op ∆n

since Nr is fully faithful by TheoremDefinition I64 To obtain the equivalence abovenote that HomCatinfin([m] [n]op [n]) HomCat1([m] [n]op [n]) even though Cat1 sube Catinfinis not fully faithful in general Here it works since every isomorphism of functors inFun([m] [n]op[n]) must be an identity (as [n]op[n] has no non-identity isomorphisms) ThusNrm([n]op [n]) is given by the discrete set HomsSet(∆m (∆n)op ∆n) since N Cat1 sSet

is fully faithful and compatible with minus minus and (minus)op This shows that Nr([n]op [n]) indeedagrees with (∆n)op ∆n

I70 The Quillen Q-Construction mdash Let C be an infin-category and let

Qn(C) sube Fun(

TwAr([n])op C)

be the full sub-infin-category consisting of those functors that take all ldquosquaresrdquo in TwAr([n])op

to pullbacks To make sense of this recall our description of TwAr([n]) from I28 In thatway a functor TwAr([n])op C can be pictured as a diagram

56


in C (shown here for n = 3) and to be in Qn(C) we require that all squares are pullbacks asindicated Note that then in fact all rectangles you can find in this picture must be pullbacks

The Qn(C) assemble into a simplicial infin-category Q(C) ∆∆op Catinfin To see thisone first shows that Fun(TwAr([minus])op C) ∆∆op Catinfin gives a functor Itrsquos clear that allits ingredients are functorial except perhaps for TwAr(minus) But TwAr(minus) sSet sSet isright-adjoint to (minus)op (minus) sSet sSet (indeed this guy has a right adjoint by Theorem I2now unravel that TwAr(minus) as defined in I26(b) fits the general description of such rightadjoints) which one can show is a left Quillen functor for the Joyal model structure (by[HCII Proposition D5] for example)

Now that we know Fun(TwAr([minus])op C) ∆∆op Catinfin is a functor we only need to checkthat all the boundary and degeneracy maps preserve the full subcategories Qn(C) describedabove We leave this as an exercise As you might have guessed assigning to C the simplicialinfin-category Q(C) is called the Quillen Q-construction

I71 PropositionDefinition mdashLecture 824th Nov 2020

Let C be an infin-category with pullbacks The simplicialanima

coreQ(C) ∆∆op minus An

is a complete Segal space Its associated category Span(C) asscat(coreQ(C)) is called theinfin-category of spans in C

Proof We will actually prove that Q(C) itself satisfies the Segal and completeness conditionsfor simplicial infin-categories rather than anima (really what this says is that therersquos an infin-double category Span(2)(C) but never mind that) Then all assertions for coreQ(C) willfollow from the fact that core Catinfin An preserves limits as it is a right-adjoint

Recall that the 0-simplices in TwAr([n])op are given by morphisms in the poset [n] ieby relations (i 6 j) Now let Jn sube TwAr([n])op be the subposet spanned by all 0-simplices(i 6 j) with j 6 i+ 1 In pictures Jn (for n = 3) looks as follows

0 6 0 1 6 1 2 6 2 3 6 3

0 6 1 1 6 2 2 6 3

0 6 2 1 6 3

0 6 3

J3

Then one checks from the pointwise formula for Kan extensions (Theorem I52) that a functorF TwAr([n])op C is in Qn(C) iff it is right-Kan extended from F |Jn To see this in theexample above note that Kan extensions can be computed in steps (because right adjointscompose) so we may first Kan extend to the 0-simplex (0 6 2) then to (1 6 3) and thenfinally to (0 6 3) In each step the Kan extension is given by pullback as one can see byrestricting the limit from Theorem I52 to a suitable cofinal sub-infin-category of the respectiveslice categories The case for general n works precisely the same

In particular since Jn sube TwAr([n])op is fully faithful we see that restriction to Jn inducesan equivalence

Qn(C) simminus Fun(Jn C)

57


by Corollary I54 The reason we donrsquot work with the Jn right away is that they do not forma cosimplicial infin-category So we have to include some redundant junk However the Jn areclosed under the maps induced by the Segal maps ei [1] [n] (sending [1] to i i+1 sube [n])These maps induce an equivalence Jn J1 tJ0 middot middot middot tJ0 J1 in Catinfin hence

Qn(C) Fun(J1 tJ0 middot middot middot tJ0 J1 C) Fun(J1 C)timesFun(J0C) middot middot middot timesFun(J0C) Fun(J1 C) Q1(C)timesQ0(C) middot middot middot timesQ0(C) Q1(C)

by Corollary I50 which proves that Q(C) is indeed a Segal object in CatinfinFor completeness we have to check the condition from TheoremDefinition I64(b) ie

we need to show thatQ0(C) Q0(C)timesQ0(C)

Q3(C) Q1(C)timesQ1(C)

s

∆

(ss)(d02d13)

is cartesian So let P be the pullback We first check that the induced map Q0(C) Pis fully faithful To see this first note that the canonical map TwAr([n])op TwAr([0])op

induces a fully faithful map s Fun(TwAr([0])op C) Fun(TwAr([n])op C) for all n IndeedTwAr([0])op lowast is an anima hence s factors over the full subcategory Fun(|TwAr([n])op| C)But |TwAr([n])op| lowast because TwAr([n])op contains (0 6 n) as an initial element and nowitrsquos clear that s is indeed fully faithful Hence so is s Q0(C) Qn(C) for all n

In particular this is true for n = 3 Moreover P Q3(C) is fully faithful too becauseit is a pullback of the fully faithful map (s s) Q0(C)timesQ0(C) Q1(C)timesQ1(C) Now thediagram

Q0(C) P

Q3(C)s

and the corresponding two-out-of-three property for fully faithful maps show that Q0(C) Pis fully faithful too

Now for essential surjectivity The actual image of Q0(C) P consists of the constantdiagrams F TwAr([3])op C hence (by an easy argument) the essential image contains alldiagrams that map all edges to equivalences To see that every 0-simplex in P is a diagramF TwAr([3])op C of that form we have to show the following

F (0 6 0) F (1 6 1) F (2 6 2) F (3 6 3)

F (0 6 1) F (1 6 2) F (2 6 3)

F (0 6 2) F (1 6 3)

F (0 6 3)

(1) (2)(4) (4)

(2) (1)

(1) (3) (3) (1)

(1) (1)

58


Suppose we are given a diagram F TwAr([3])op C in which all squares are pullbacks asindicated and such that the four dashed purple arrows are equivalences Then all arrows areequivalences

Indeed then the two dashed pink arrows F (0 6 3) F (0 6 1) and F (0 6 3) F (2 6 3)are equivalences as well since they are pullbacks the purple arrows By two-out-of-six wesee that all arrows labelled ldquo(1)rdquo must be equivalences

Now the commutative square

F (0 6 3) F (1 6 3)

F (0 6 1) F (1 6 1)

sim

sim

simshows that F (0 6 1) F (1 6 1) is an equivalence An analogous argument applies toF (2 6 3) F (2 6 2) In other words the two arrows labelled ldquo(2)rdquo are equivalencesThis implies that the two arrows labelled ldquo(3)rdquo are equivalences since they are pullbacksof the former Finally the two remaining arrows labelled ldquo(4)rdquo must be equivalences bytwo-out-of-three We are done

59

Chapter II

Symmetric Monoidal andStable infin-Categories

IIE1-Monoids and E1-GroupsWe start things off slowly by only considering (not necessarily symmetric) monoidal infin-categories and E1-spaces

II0 Stasheffrsquos Definition of Coherently Associative Monoids mdash The naive wayto define a monoid in An would be to have an object M isin An together with a multiplicationmap

micro M timesM minusM

Now micro induces two different maps M3 M (ldquotwo ways of bracketingrdquo) so there ought to bea homotopy H micro (microtimes idM ) micro (idM timesmicro) between them But then there are five mapsM4 M (ldquofive ways of bracketingrdquo) and we see that H induces a loop in HomAn(M4M)Since M should be associative up to coherent homotopy this loop has to be filled The storygoes on There are a number of maps M5 M (too lazy to count them) and now there aresome 3-simplices to be filled and so on

Although this looks horrible it is possible to turn these considerations into a precisedefinition and one obtains Stasheffrsquos Ainfin-spaces But we can take a simpler route Indeedthanks to our efforts so far we now have the luxury of saying ldquoWell a coherently associativemonoid is just an infin-category with only one objectrdquo This leads to the following definition

II1 Definition mdash Let C be an infin-category with finite products (in particular C has afinal object lowast isin C) A cartesian monoid in C is a functor X ∆∆op C such that(a) X0 lowast(b) It satisfies the Segal condition ie the Segal maps ei [1] [n] define an equivalence

Xnsimminus

nprodi=1

X1

Let Mon(C) sube sC be the full sub-infin-category spanned by cartesian monoids If C = Anthese are also called E1-monoids Ainfin-spaces coherent monoids special ∆-spaces ForC = Catinfin we simply call them monoidal infin-categories (note that these can also be encodedas cocartesian fibrations over ∆∆op)

To make sense of the condition from Definition II1(b) note that since X0 lowast is terminalwe have X1 timesX0 times middot middot middot timesX0 X1

prodni=1X1 so this condition really is the Segal condition

Somewhat weird though is that we donrsquot impose any completeness conditions and in fact

60

E1-Monoids and E1-Groups

cartesian monoids in An are usually not complete Segal spaces This is in some sense fixedby the following proposition

II2 Proposition mdash The completion functor comp sAn CSAn from LemmaDefi-nition I67 restricts to a fully faithful functor

comp Mon(An) minus lowastCSAn

with essential image those pointed complete segal spaces (Xx) with π0X0 = lowast

In other words monoids really are categories with one object (up to equivalence but notup to contractible choice as Bastiaan pointed out in the lecture) since the target is alsopointed categories with connected base

Proof of Proposition II2 The proof is not hard but quite lengthy so we break it down intofour major steps Be aware that Step (3) wasnrsquot done in the lecture so any errors in it aremy fault(1) There exists a reasonable candidate decomp lowastCSAn Mon(An) of an inverse functor

(ldquodecompletionrdquo)If (Xx) is a pointed complete Segal space and if Y isin Mon(An) is the ldquouniversal wayrdquo

to make (Xx) connected and still satisfy the Segal condition then itrsquos reasonable to expectthat Yn sits inside a pullback

Yn Xn

lowast Xn+10

(xx)

for all [n] isin ∆∆op We will see that this gives indeed the correct functorLetrsquos first address the elephant in the room ie how to make the pointwise-defined Yn

into a functor Y ∆∆op An which in turn should be functorial in X Since taking pullbacksis functorial wersquoll have solved both problems at once if we show that the cospan diagramlowast Xn+1

0 Xn is functorial in (Xx) and in [n] To this end consider X as a functor∆∆ Anop and colimit-extend it to a functor

| |X sAn minus Anop

This can be done functorially in X as Theorem I51 shows In more abstract terms what wedo here is to identify sAn Fun(∆∆Anop)op FunL(sAnAnop)op We define two functors∆∆0 ∆∆ sSet sube sAn via

∆([n])

= ∆n and ∆0([n])

=n∐i=0

∆0

Since ∆ and ∆0 are functors between 1-categories (before we compose them with the inclusionsSet sube sAn) we donrsquot need to check any higher coherences and immediately obtain that ∆and ∆0 are indeed functors Moreover the canonical map

∐ni=0 ∆0 ∆n sending the ith

component to the ith vertex of ∆n is clearly functorial in [n] isin ∆∆ hence it induces a naturaltransformation ∆0 rArr ∆ (which exhibits ∆0 as the ldquo0-skeletonrdquo of ∆ hence the notation)Now consider the two composites

sAn Fun (∆∆Anop)op Fun (sAnAnop)op ∆lowast

∆lowast0Fun (∆∆Anop)op sAn

61


which we denote |∆|(minus) and |∆0|(minus) The transformation ∆0 rArr ∆ induces a transformation|∆|(minus) rArr |∆0|(minus) going in the other direction since there has been an (minus)op in betweenUpon closer inspection we find that ∆ ∆∆ sAn is actually nothing else but the Yonedaembedding hence Theorem I51 shows that |∆|(minus) id Moreover since | |X preservescolimits we easily obtain

Xn+10

∣∣∣∣∣n∐i=0

∆0

∣∣∣∣∣X

hence the Xn+10 can be organized into a simplicial anima |∆0|X The transformation

id |∆|(minus) rArr |∆0|(minus) gives a natural map X |∆0|X This shows that the maps Xn+1

0 Xn can indeed be made functorial in X and [n] Asimilar argument applies shows that the maps lowast Xn+1

0 which assemble into a functorialmap ∆0 |∆0|X We conclude that Y X times|∆0|X ∆0 as above is indeed a simplicialanima and functorial in X We have Y0 lowast by construction and also Yn Y1 timesY0 middot middot middot timesY0 Y1follows from the fact that X itself satisfies the Segal conditions together with some abstractpullback nonsense (basically that ldquolimits commuterdquo see the dual of Proposition I42) SoY isin Mon(An) and we can finally write down the functor

decomp lowastCSAn minus Mon(An)

This finishes Step (1)(2) The completion functor comp Mon(An) CSAn from LemmaDefinition I67 factors

over lowastCSAn CSAn and its essential image is contained in those pointed completeSegal spaces (Xx) with π0X0 = lowast

To get the factorisation observe that Mon(An) has an initial object namely ∆0 isin sSet subesAn (in the lecture we called it ldquoconst lowastrdquo but itrsquos really just ∆0) Indeed if T isin Mon(An)then

HomMon(An)(∆0 T ) HomsAn(∆0 T ) T0 lowast

by Definition II1(a) Also comp(∆0) lowast so the completion functor really lifts to a functorMon(An) ∆0Mon(An) lowastCSAn as required

To check that π0(compT )0 = lowast recall that (compT )0 core asscatT |Ttimes| by (I651)Now we have to use the following general fact(lowast) For all X isin sAn the map π0X0 π0|X| is surjectiveThis shows what we want since (Ttimes)0 T0 lowast holds by assumption To see where (lowast)comes from observe that π0 being a left adjoint (Example I33(a)) commutes with colimitsThis shows π0|X| π0 colim∆∆op X colim∆∆op π0X where the colimit on the right-handside is a good old colimit taken in sSet Now d0 d1 [1] [0] sube ∆∆op is 1-cofinal (but notinfin-cofinal) hence

colim∆∆op

π0X = Coeq(π0X1

d1

d0π0X0

)

This immediately implies (lowast) and thus Step (2) is done(3) The functors from Step (1) and Step (2) fit into an adjunction

comp Mon(An) lowastCSAn decomp

62


Consider T isin Mon(An) which is naturally pointed via the unique (up to contractiblechoice) map 1T ∆0 T and let (Xx) be a pointed complete Segal space Then

HomlowastCSAn(

compT (Xx)) HomlowastsAn

((T 1T ) (Xx)

) HomsAn(TX)timesHomsAn(const 1T X) constx HomsAn(TX)timesHomAn(1T X0) x

The first equivalence follows from LemmaDefinition I67 (and the fact that adjunctionsextend to slice categories) the second follows from [HCII Corollary VIII6] The last onefollows by inspection All of them are functorial in (Xx) Moreover we have

HomMon(An)(T decomp(Xx)

) HomsAn

(TX times|∆0|X ∆0)

HomsAn(TX)timesHomsAn(T|∆0|X) HomsAn(T∆0)

Since id rArr |∆0|(minus) is a natural transformation one quickly checks that the morphismHomsAn(TX) HomsAn(T |∆0|X) factors over HomsAn(|∆0|T |∆0|X) But T0 lowast hence|∆0|T ∆0 Also HomsAn(T∆0) lowast Putting everything together we can rewrite


) HomsAn(TX)timesHomAn(T0X0) x

and after another inspection we see that our calculations of HomlowastCSAn(compT (Xx)) andHomMon(An)(T decomp(Xx)) agree as they are supposed to This finishes Step (3)(4) The restriction decomp (lowastCSAn)π0=lowast Mon(An) is indeed an inverse to comp

where (lowastCSAn)π0=lowast sube lowastCSAn denotes the full subcategory spanned by those (Xx)with π0X0 = lowast

Thanks to Step (3) we already have unit and counit transformations idrArr decomp compand comp decomp rArr id so we only need to show that they are equivalences Letrsquos startwith the counit We must show

comp(

decomp(Xx)) (Xx)

for all (Xx) isin (lowastCSAn)π0=lowast Observe that compX X because X is already completeHence it suffices to show asscat(decomp(Xx)) asscatX (and check that the chosenbase points on both sides correspond but thatrsquos easy) because comp Nr asscat byLemmaDefinition I67 and Nr is fully faithful by TheoremDefinition I64 Using (I651)and the fact that Xtimes is a constant simplicial anima with value X0 as X is complete we seethat

π0 core(asscatX) π0|Xtimes| π0X0 lowast

so asscat(decomp(Xx)) asscatX is essentially surjective for trivial reasons Moreoverwe have decomp(Xx)0 x by construction and (I652) (which is applicable here becausedecomp(Xx) is Segal by construction) easily shows

Homasscat(decomp(Xx))(x x) Pxx HomasscatX(x x)

so asscat(decomp(Xx)) asscat(X) is fully faithful too This shows that the counit is anequivalence as claimed The argument for the unit is similar We have to show

T decomp(compT )

63


for all T isin Mon(An) Since both sides are functors ∆∆op An and a natural transformationbetween them is already given we can do this degree-wise by Theorem I19(b) Since bothsides are cartesian monoids is suffices to check that the map in degree 1 is an equivalenceie that T1 decomp(compT )1 Write (Xx) compT By construction we have

decomp(Xx)1 Pxx and T1 P1T 1T

where as usual 1T ∆0 T is the natural pointing of T As observed at the very beginningof this proof the spaces Pyz arenrsquot affected by completion hence Pxx P1T 1T and wersquoredone


Under the equivalence Catinfin CSAn from TheoremDefinition I64 the constructionof decomp from the proof of Proposition II2 corresponds to a functor lowastCatinfin Mon(An)which becomes an equivalence when restricted to the full subcategory (lowastCatinfin)gt1 sube lowastCatinfinspanned by those C with π0 core C lowast In diagrams

lowastCatinfin Mon(An)

(lowastCatinfin)gt1

sube sim

Explicitly this functor sends a pointed infin-category (x C) to HomC(x x) in degree 1(and then all other degrees are determined by the conditions from Definition II1) We thusobtain

II3 Corollary mdash If C is an infin-category and x isin C then HomC(x x) carries a canonicalstructure of an E1-monoid

II4 Definition mdash Let C be an infin-category with finite products A cartesian monoid Xin C is called a cartesian group (or E1-group in the case C = An) if the map

(pr1 ) X1 timesX1simminus X1 timesX1

(f g) 7minus (f f g)

is an equivalence Here ldquordquo is the composition as defined in TheoremDefinition I64(c) Wedenote by Grp(C) sube Mon(C) the full sub-infin-category of cartesian groups

II5 Proposition mdash The equivalences Mon(An) (lowastCatinfin) (lowastCSAn)π0=lowast restrictto equivalences

Mon(An) Grp(An)

(lowastCatinfin)gt1 (lowastAn)gt1

(lowastCSAn)π0=lowast (lowastsAnconst)π0=lowast

sim

supe

sim

sim

supe

sim

supe

In the lower right corner we use the notation from Example I68(c)

64


Proof sketch In the lecture we noted only that X isin Mon(An) is an E1-group iff the setπ0X1 with its induced ordinary monoid structure is an ordinary group

To elaborate on this a bit more first note that we may assume X HomC(x x) forsome infin-category C with π0 core C lowast as was argued above If X is an E1-group thenπ0 HomC(x x) must be an ordinary group which implies that all endomorphisms of x areequivalences hence all morphisms in C are equivalences as π0 core C lowast This proves that Cis an anima by Theorem I13 Conversely if C is an anima either inclusion [1] J into thefree-living isomorphism induces a trivial fibration Fun(J C) simminus Ar(C) Choosing a sectionand composing it with the the other map Fun(J C) Ar(C) (induced by the other inclusion[1] J) yields an ldquoinversionrdquo map (minus)minus1 HomC(x x) HomC(x x) From this we canconstruct an inverse of the map in Definition II4

II6 Some Explicit Calculations mdash The equivalence Grp(An) (lowastAn)gt1 is explic-itly given as follows

B = | | Grp(An) simsim (lowastAn)gt1 Ω

The use B to denote the functor | | = colim∆∆op is common in the literature as it is short forldquobar constructionrdquo Fabian doesnrsquot like it since he finds it stupid to name something afterthe way it was typeset in the pre-LATEX age (where people had to resort to bars to denotetheir tensor products) Amusingly though his preferred notation consists of actual bars In these notes however Irsquoll show my bad taste and consistently use B

Anyway back to the matters at hand To make sense of why the above equivalence isindeed given by B = | | and Ω recall that the composition

Mon(An) compminusminusminus CSAn ev0minus An

sends X 7 |Xtimes| as follows from (I651) and some unravelling If X is an E1-group oneeasily verifies Xtimes = X and thus X isin Grp(An) is sent to BX = |X| as claimed Converselythe composition

(lowastAn)gt1decompminusminusminusminusminus Grp(An) ev1minus An

sends a pointed connected anima (K k) to HomK(k k) as argued previously So to justifywhy (lowastAn)gt1

simminus Grp(An) should be thought of as a ldquoloop space functorrdquo we must explainwhy ΩkK HomK(k k) This can be done by several arguments and I will present a differentone than in the lecture The loop space is usually defined by the left of the following pullbackdiagrams in An

ΩkK lowast

lowast K

k

k

simminus

HomLK(k k) Kk

lowast K

k

The map lowast K can be factored into the left anodyne map lowast Kk followed by the leftfibration Kk K which is even a Kan fibration since K is an anima Replacing oneof the lowast K accordingly we obtain the pullback diagram on the right which is alreadya pullback in sSet as one leg is a Kan fibration (see Theorem I34) Its pullback is thusHomL

K(k k) HomK(k k) as left-Hom anima coincide with the regular Hom anima by[HCII Digression I Corollary D2]

65


In particular we have proved another result from algebraic topology (or rather deducedit from the much more general statement of TheoremDefinition I64 which we didnrsquot provebut never mind that)II7 Corollary (Recognition Principle for Loop Spaces Stasheff) mdash The loop functorΩ lowastAn An lifts to an equivalence

Ω (lowastAn)gt1simminus Grp(An)

Note that ev1 Mon(An) An preserves limits since Mon(An) is closed under limits insAn (the Segal condition is given by a limit and limits commute by the dual of Proposition I42)Colimits however are bloody complicated and even though one can show that Mon(An) iscocomplete colimits in it are by far not just colimits of underlying simplicial anima same ascolimits of ordinary monoids are much more complicated than colimits in setsII8 Proposition mdash The inclusion Grp(An) sube Mon(An) has a left adjoint called groupcompletion and denoted

(minus)infin-grp Mon(An) minus Grp(An)

It is given by Xinfin-grp ΩBX where B Mon(An) (lowastAn)gt1 denotes a functor extendingthe B from II6Proof We know that An sube Catinfin has a left-adjoint | | Catinfin An The unit and counitare equivalences on lowast isin An hence the adjunction passes to slices under it Moreover the(minus)gt1-parts on both sides are clearly preserved thus | | descends to a left adjoint

| | (lowastCatinfin)gt1 minus (lowastAn)gt1

of the obvious inclusion in the other direction Now use Proposition II5 to get the desiredleft-adjoint (minus)infin-grp Mon(An) Grp(An) The explicit description follows from ourcalculations in II6 and Remark I66 which ensures that the different meanings of | |(colim∆∆op or localisation of an infin-category at all its morphisms) are actually compatible

A consequence that wasnrsquot explicitly mentioned but used later in the course is thefollowingII8a Corollary mdash Without restricting them as in II6 the functors B and Ω form anadjunction

B Mon(An) lowastAn Ω

Proof For a pointed anima (Xx) let Xx denote the connected component containing xThen ΩxX ΩxXx so the equivalence Ω (lowastAn)gt1

simminus Grp(An) from II6 extends to afunctor Ω lowastAn Grp(An) Now if M isin Mon(An) is an E1-monoid we can compute

HomMon(An)(MΩxX) HomGrp(An)(Minfin-grpΩxXx) HomGrp(An)

(ΩBMΩxXx

) Hom(lowastAn)gt1

(BMXx

) HomlowastAn

(BMX

)

which establishes the desired adjunction The second equivalence follows from Proposition II8the third equivalence uses that Ω (lowastAn)gt1

simminus Grp(An) is an equivalence and the fourthequivalence follows from the fact that BM is connected (by construction) hence any pointedmap BM X only hits the connected component Xx

66


Next we will discuss some examples We begin with free monoids and groups

II9 Proposition mdash The evaluation functor ev1 Mon(An) An has a left adjointFreeMon An Mon(An) It is given explicitly by the ldquoanima of words of arbitrary lengthrdquo

FreeMon(K)1 ∐ngt0

Kn

Proof We didnrsquot prove this in the lecture but herersquos a very simple argument that Fabianexplained to me We call a morphism in ∆∆ inert if it is the inclusion of an interval and activeif it preserves the largest and the smallest element Note that the Segal maps ei [1] [n]are precisely the inert maps with source [1] and that every map can be uniquely factoredinto an inert followed by an active We let ∆∆op

int sube ∆∆op denote the (non-full) subcategoryspanned by the inert maps Letrsquos first verify the following two claims(1) Consider the full sub-infin-category FunSeg(∆∆op

intAn) spanned by the ldquoreduced Segal func-torsrdquo ie those X ∆∆op

int An satisfying X0 lowast and Xn prodni=1X1 via the Segal

maps Then evaluation at [1] induces an equivalence

ev1 FunSeg(∆∆opintAn) simminus An

(2) Consider the left-Kan extension functor Lani Fun(∆∆opintAn) Fun(∆∆opAn) along

i ∆∆opint ∆∆opThen Lani preserves reduced Segal functors

To show (1) we verify that right-Kan extension along the inclusion j [1] ∆∆opint defines

an inverse Using the pointwise formulas from Theorem I52 together with the fact that theinert maps [1] [n] are precisely the Segal maps we see that Ranj indeed takes values inthe reduced Segal functors and that Ranj ev1 id To show ev1 Ranj id observe thatj is fully faithful and use Corollary I54 This proves (1)

To prove (2) let X ∆∆opint An be reduced Segal By Theorem I52

(LaniX)n colim([k][n])isin∆∆op

int[n]Xk colim

([k][n])isin∆∆opint[n]

Xk1

An object [k] [n] of ∆∆opint[n] corresponds to a map [n] [k] in ∆∆ Any such map can be

uniquely factored into an inert [nprime] [k] followed by an active α [n] [nprime] This allows usto decompose ∆∆op

int[n] into a disjoint union∐α Cα indexed by all active maps α [n] [nprime]

with source [n] Moreover each Cα has a terminal object which is given by α itself (or ratherthe corresponding object αop [nprime] [n] in ∆∆op

int[n])For n = 0 this shows (LaniX)n lowast since id[0] [0] [0] is the only active map

with source [0] For n = 1 we get a unique active map [1] [n] for all n which shows(LaniX)1

∐kgt0X

k1 For general n we get

(LaniX)n ∐

k1kngt0Xk1+middotmiddotmiddot+kn

1

Upon inspection this shows that LaniX is again reduced Segal Wersquove thus proved (2)Combining (1) and (2) shows that FreeMon An Mon(An) is given by Lani Ranj Butthen the calculations above show FreeMon(K)1

∐ngt0K

n as claimed

67


By the explicit construction of FreeMon the diagram

Set Mon(Set)

An Mon(An)

Free

FreeMon

commutes But be aware that this is not automatic and in fact wersquoll see it very much breaksonce we impose commutativity (see pages 88 and 101) As some first examples one hasFreeMon(lowast) N and FreeMon(empty) lowast which is initial in Mon(An)

II10 Proposition mdash The evaluation functor ev1 Grp(An) An has a left-adjointFreeGrp An Grp(An) too Explicitly it is given by the composite

An minus lowastAn Σminus lowastAn Ω

minus Grp(An)X 7minus X t lowast

In other words the free E1-group on X is FreeGrp(X) ΩΣX+ where X+ = X t lowast Asusual the suspension functor Σ An An is given by the pushout diagram

X lowast

lowast ΣX

II10a Remark mdash For the proof wersquoll need a functor Σ lowastAn lowastAn instead It isinduced by its unpointed variant of course but we can also define it by taking the abovepushout in lowastAn In fact it doesnrsquot matter whether the pushout is taken in lowastAn or Ansince lowastAn An commutes with weakly contractible colimits (but not with arbitrary oneswhich I erroneously claimed in an earlier version of these notes for example it doesnrsquotpreserve the initial object) Indeed if I is any diagram shape then a quick calculation showsFun(I lowastAn) (const lowast)Fun(IAn) For the adjunction

colimI

Fun(IAn) An const

to descend to slice categories below const lowast and lowast we need that these objects are mapped toeach other and that unitcounit are equivalences on them If I is weakly contractible thencolimI(const lowast) |I| lowast by Proposition I36 and the required conditions are easily checkedTherefore the adjunction above descends to an adjunction

colimI

Fun(I lowastAn) (const lowast)Fun(IAn) lowast An const

in this case If you think about this for a moment this is exactly what we need to show thatlowastAn An commutes with colimits over I

Similarly once we have chosen a basepoint x isin X it doesnrsquot matter whether we take thedefining pullback of ΩxX in lowastAn or in An (and then equip it with its natural basepoint)This can be seen as above or simply by the fact that lowastAn An has a left adjoint so wecan apply Observation I41 However in contrast to Σ we canrsquot define Ω as a functor on Ansince there is no canonical choice of basepoint

68


Proof sketch of Proposition II10 The existence of a left adjoint would be immediate fromPropositions II8 and II9 but we can give a direct argument that also yields the explicitdescription We have seen in II6 that the diagram

Grp(An) An

(lowastAn)gt1

B

ev1

Ω

commutes Observe that (minus)+ = minus t lowast An lowastAn is a left adjoint of the forgetful functorlowastAn An (this is basically trivial) and that Σ lowastAn lowast An Ω are adjoints (this weleave as an exercisemdashjust play around with the defining pushoutspullbacks which can betaken in lowastAn by Remark II10a) Finally Ω (lowastAn)gt1 Grp(An) B is an adjunction(even an equivalence) by II6 The assertion now follows from the fact that left adjointscompose

II11 Example mdash (a) The evaluation functor ev1 Mon(An) An factors canonicallyover lowastAn An and as it turns out the resulting functors

ev1 Mon(An) minus lowastAn and ev1 Grp(An) minus lowastAn

both have left adjoints For a pointed anima (Xx) we denote the corresponding leftadjoint objects by FreeMon(Xx) and FreeGrp(Xx) and think of them as the ldquofreeE1-monoidgroup with unit xrdquo We have

FreeMon(Xx)infin-grp FreeGrp(Xx) ΩΣX

The first equivalence follows from the fact that left adjoints compose The secondequivalence can be seen along the lines of the proof of Proposition II10 without usneeding to know that FreeMon(Xx) exists

If (Xx) is connected one actually has FreeMon(Xx) FreeGrp(Xx) Indeed aswersquove seen in the proof of Proposition II5 we only need to check that the ordinarymonoid π0 FreeMon(Xx)1 is already an ordinary group Playing around with universalproperties we see that π0 FreeMon(Xx)1 is the free ordinary monoid on the set π0Xwith unit [x] hence just [x] itself since X is connected hence it already is a group

(b) By Proposition II9 FreeMon(S1) ∐ngt0 Tn is an infinite union of tori Tn =

prodni=1 S1

and thus a reasonable geometric object However Proposition II10 implies

FreeGrp(S1) ΩΣ(S1+) Ω(S2 or S1)

and the homotopy groups of the right-hand side are not fully known to this day Similarlyusing the pointed variant from (a) above we get

FreeMon(S1 lowast) FreeGrp(S1 lowast) ΩΣ(S1 lowast) ΩS2

(c)Lecture 101st Dec 2020

It is completely formal to see that ev1 Mon(An) An and ev1 Grp(An) An arerepresented by FreeMon(lowast) and FreeGrp(lowast) FreeMon(lowast)infin-grp respectively Whatrsquos notformal however is that FreeMon(lowast) N and FreeGrp(lowast) ΩS1 Z regarded as discretesimplicial anima This really needs Propositions II9 and II10

69


Moreover wersquoll see (pages 88 and 101) that the analogous functors are no longerrepresented by N and Z once we impose commutativity even though they are the freeE1-monoidgroup on a point and happen to be commutative In fact they are ldquotoocommutative to be free commutative monoidsrdquo (sphere spectrum intensifies)

(d) Letrsquos for some reason compute the pushout

N N

lowast N2

middot2

in Mon(An) The answer is N2 ΩRP2 To see this first observe that the functorπ0 Mon(An) Mon(Set) commutes with colimits so π0(N2) = Z2 since we knowhow to compute pushouts in Mon(Set) But Z2 is a group so N2 isin Grp(An) byProposition II5 In particular since the group completion functor (minus)infin-grp commuteswith colimits we get N2 Z2 ie the diagram

Z Z

lowast N2

middot2

is a pushout in Grp(An) Translating to lowastAn via Proposition II5 gives a diagram

S1 S1

D2 RP2

middot2

(we have replaced lowast by the disk D2 to make one leg of the pushout into a cofibration sothat it can be computed by an ordinary pushout of CW complexes by Theorem I34)and translating back using II6 shows N2 ΩRP2 as claimed

(e) Every monoidal 1-category (Cotimes) gives rise to a monoidal infin-category ie an object inMon(Cat(2)

1 ) sube Mon(Catinfin) We will produce this by straightening a suitable cocartesianfibration potimes Cotimes ∆∆op of 1-categories Define Cotimes isin Cat(2)

1 as follows Its objects are

obj(Cotimes) =∐ngt0

obj(C)n

ie given by tuples (n x1 xn) with [n] isin ∆∆ and xi isin C Wersquoll usually abbreviatex = (x1 xn) and simply write (n x) Morphisms are defined as

HomCotimes((n x) (m y)

)=

(α f)

∣∣∣∣∣∣α [m] [n] in ∆∆ and f = (f1 fm)where fj xα(jminus1)+1 otimes middot middot middot otimes xα(j) yjare morphisms in C for all j = 1 m

If α(j minus 1) + 1 gt α(j) for some j then xα(jminus1)+1 otimes middot middot middot otimes xα(j) is the empty tensorproduct which is the tensor unit 1C isin C by definition

70


The composition of (α f) (n x) (m y) and (β g) (m y) (k z) is given asfollows Its first component is α β [k] [n] For the second component observe thatwe can write

xαβ(jminus1)+1 otimes middot middot middot otimes xαβ(j) =β(j)minusβ(jminus1)otimes

i=1

(xα(β(jminus1)+iminus1)+1 otimes middot middot middot otimes xα(β(jminus1)+i)

)

The ith tensor factor on the right maps to yβ(jminus1)+i via fβ(jminus1)+i Tensoring themtogether and postcomposing with gj yβ(jminus1)+1otimesmiddot middot middototimeszj provides the desired compositionof f and g This finishes the construction of Cotimes

The functor potimes Cotimes ∆∆op just extracts [n] from (n x) and α from (α f) The fibresCotimesn of potimes are then obviously given by Cn To show that p is a cocartesian fibration of1-categories we must provide cocartesian lifts of morphisms in ∆∆op Given α [m] [n]in ∆∆ and (n x) isin Cotimes we claim that the identities can be regarded as a morphism

αlowast (n x) minus(mxα(0)+1 otimes middot middot middot otimes xα(1) xα(mminus1)+1 otimes middot middot middot otimes xα(m)

)

Itrsquos straightforward to check that αlowast is a cocartesian lift of α (note that the dauntinghomotopy pullbacks in Definition I24 become just good old pullbacks of setsdiscreteanima since we are dealing with 1-categories)

It remains to check that St(potimes) satisfies the conditions from Definition II1 As seenabove St(potimes)0 Cotimes0 C0 lowast is a point as desired To verify the Segal condition letrsquosorganize the world a little bit Recall that a map in ∆∆ is called inert if it is the inclusionof an interval and active if it preserves the largest and the smallest element The Segalmaps ei [1] [n] are precisely the inert maps with source [1] We claim that for ageneral inert map α [m] [n] the induced functor (given by taking cocartesian lifts)

αlowast Cn Cotimesn minus Cotimesm Cm

simply sends (x1 xn) isin Cn to (xα(1) xα(m)) isin Cm with no tensor productsoccuring Thus the Segal condition follows immediately

As a slogan ldquoinerts induce forgetful functors on fibresrdquo whereas ldquoactives donrsquotforget they just tensorrdquo

(f) The construction from (e) also works if C is Kan enriched In this case one can againconstruct a cocartesian fibration potimes Cotimes ∆∆op but of Kan-enriched categories thistime (where ∆∆op receives its discrete enrichment) Then Nc(Cotimes) Nc(∆∆op) ∆∆op

straightens to a monoidalinfin-category as well Fabian wouldnrsquot tell us what a cocartesianfibration of Kan-enriched categories is though since this would takemdashnot a lot but toomuchmdashtime

Example II11(e) allows us to construct many interesting examples All examples willturn out to be commutative (see Example II21a) but that doesnrsquot matter for now

II12 Algebraic K-Theory mdash Let R be a ring and consider Mod(R) as a monoidalcategory under oplus Moreover let Proj(R) denote the sub-groupoid of spanned by finiteprojective R-modules (ldquovector bundles on SpecRrdquo) and their isomorphisms This inherits a(symmetric) monoidal structure by Example II11(e) so we obtain

Proj(R) isin Mon(Grpd(2)

1)sube Mon(An)

71


II12a Definition (Quillen) mdash The projective class anima (or algebraic K-theory space)of a ring R is defined as

k(R) = Proj(R)infin-grp isin Grp(An)

and the higher projective class groups (or higher K-groups) of R are given by

Ki(R) = πi(k(R)1 lowast

)

The reason we write k(R) is that the notation K(R) is reserved for the K-theory spectrumwhich we will eventually define Note immediately that

K0(R) = π0(

Proj(R)infin-grp) =(π0 Proj(R)

)grp =

iso classes of finiteprojective R-modules

grp

where (minus)grp Mon(Set) Grp(Set) denotes the ordinary group completion This showsthat K0(R) is precisely what we wanted it to be way back in the introduction Definition 06We will soon(ish Corollary III7) see that also K1(R) can be described as in Definition 08

II13 Hermitian K-TheoryGrothendieckndashWitt Theory mdash Let R be a ringM anRotimesR-module and σ M M an involution that is flip-linear ie σ((aotimesb)m) = (botimesa)σ(m)Moreover assume that M is finite projective when considered as an R-module via themap R Z otimes R R otimes R (one can check that it doesnrsquot matter if we choose this map orR Rotimes Z RotimesR) and that the map

R simminus HomR(MM)r 7minus

(m 7 (r otimes 1)m

)is an isomorphism A pair (Mσ) subject to these conditions is called an invertible modulewith involution over R (beware that this is non-standard terminology)

For a finite projective R-module P isin Proj(R) consider the group HomRotimesR(P otimes PM) ofM -valued R-bilinear forms on P This is acted upon by Z2 via conjugation with the flip ofP otimes P and σ on M Define the groups of symmetric and quadratic forms on P as

SymR(PM) = HomRotimesR(P otimes PM)Z2

QuadR(PM) = HomRotimesR(P otimes PM)Z2

where (minus)Z2 denotes the invariants of the Z2-action and (minus)Z2 denotes the coinvariants(ie take the coequalizer rather than the equalizer) Finally let the group of even forms onP be

EvenR(PM) = im (Nm QuadR(PM) SymR(PM))

where Nm denotes the norm map which can be defined on general abelian groups with aZ2-action If X isin Z2-Ab is an abelian group with Z2 acting via an involution τ X X

Nm XZ2 minus XZ2

[x] 7minus x+ τ(x)

A symmetric form q isin SymR(PM) is called unimodular if the induced map

qlowast P simminus HomR(PM) = DM (P )

72


is an isomorphism The right-hand side should be thought of the ldquoM -valued dualrdquo ofP This is reasonable because the assumptions on M are chosen precisely to ensure thatDM Proj(R)op simminus Proj(R) is an equivalence with inverse Dop

M Similarly an even or quadratic form is called unimodular if their image in SymR(PM)

(via the inclusion or via Nm) is unimodular For r isin sym quad even we denote

Unimodr(PM) =

groupoid of unimodular r-forms ie pairs(P q) where q is a unimodular r-form on P

and isomorphisms between them

This is a symmetric monoidal 1-groupoid via oplus hence induces a symmetric monoidal animaby Example II11(e)II13a Definition (Karoubi) mdash For r isin sym quad even the GrothendieckndashWittanima of (RM) is given by

gwr(RM) = Unimodr(RM)infin-grp

Lecture 113rd Dec 2020

To make the rather technical definitions of symmetric quadratic and even forms abit clearer we discussed some examples Choose R a commutative ring let M = R andmake it an R otimes R-module via the multiplication map micro R otimes R R Choosing eitherσ = idR or σ = minus idR we we see that SymR(PM) returns the usual notion of symmetric orskew-symmetric R-bilinear forms on P

If R = C we can also equip M = C with a Cotimes C-module structure via

Cotimes C idotimes(minus)minusminusminusminusminus Cotimes C micro

minus C

where (minus) C C denotes complex conjugation as usual In this case SymC(PC) amountsto hermitian or skew-hermitian forms on P depending on whether σ = idC or σ = minus idCII13b Exercise mdash Show that QuadR(PM) is in bijection with the set of all pairs(b q) isin SymR(PM)timesHomSet(PMZ2) satisfying

b(p p) = Nm(q(p)

)and q(rp) = (r otimes r)q(p)

for all p isin P and r isin R Under this bijection the map Nm QuadR(PM) SymR(PM)simply forgets q

What Exercise II13b means depends a little on R We always assume that R is commu-tative(a) If σ = idM then MZ2 = M = MZ2 and Nm is simply multiplication by 2 So if M

has no 2-torsion then q is already defined by b and the equation b(p p) = Nm(q(p))Hence the norm map Nm QuadR(PM) SymR(PM) is an isomorphism onto itsimage which means that quadratic forms and even forms are in bijection in this case

(b) If σ = minus idM then MZ2 = M2 and MZ2 = M [2] = 2-torsion of M HenceNm MZ2 MZ2 is the zero map IfM [2] = 0 then b(p p) = 0 for all b isin SymR(PM)and all p isin P hence Nm QuadR(PM) SymR(PM) is surjective because underthe identification from Exercise II13b (b 0) is always a preimage of b Thus all sym-metric forms are even but in general an even form may have lots of quadratic formsthat map to it

(c) If 2 isin Rtimes is a unit and σ = idM then Nm QuadR(PM) SymR(PM) is anisomorphism and the notions of symmetric quadratic and even forms all agree

73


II14 Topological K-Theory mdash Letrsquos define four ordinary categories as follows

VectR =

finite-dim R-vector spacesR-linear homeomorphisms

EuclR =

finite-dim R-vector spacesarbitrary homeomorphisms

VectC =

finite-dim C-vector spacesC-linear homeomorphisms

SphR =

finite-dim R-vector spacesproper homotopy equivrsquos

We didnrsquot include VectC in the lecture but in hindsight we really should have These fourcategories can be Kan-enriched For each category C among them we let FC(U V )n be the setof continuous maps |∆n| times U V such that d times U V is a morphism in C for all pointsd isin |∆n| of the topological n-simplex Moreover all of them carry a Kan-enriched (symmetric)monoidal structure via the usual product times Hence we may apply the construction fromExample II11(f) to obtain monoidal anima

VectR VectC EuclR and SphR

Observe that while VectR VectC and EuclR are already groupoids SphR is not Butit becomes an anima after taking the coherent nerve since homotopy equivalences havehomotopy inverses Also observe that the one-point compactification (minus)lowast of topologicalspaces induces equivalences

HomSphR(U V ) Homcore(lowastAn)(Ulowast V lowast)

for all finite-dimensional R-vector spaces U V isin SphR Writing U = Rn V = Rm we getthat Ulowast = Sn and V lowast = Sm are actually spheres Hence the name SphR

II14a Definition mdash Define the following E1-groups

ko = Vectinfin-grpR ku = Vectinfin-grp

C ktop = Euclinfin-grpR and ksph = Sphinfin-grp

R

Before we indulge ourselves in their properties Irsquod like to do two reality checks The firstone is an alternative construction of VectR and VectC

II14b Lemma mdash Let O(n) sube GLn(R) and U(n) sube GLn(C) be the nth orthogonal andthe nth unital group Then

VectR ∐ngt0

BO(n) and VectC ∐ngt0

BU(n)

The groups O(n) GLn(R) U(n) and GLn(C) all admit CW structures hence they defineE1-groups via

Grp(CW) minus Grp(

Nc(CW)) Grp(An)

Thus it makes sense to talk of BO(n) BGLn(R) BU(n) and BGLn(C) We can also findsimilar descriptions of EuclR and SphR but let me not get into that

Proof of Lemma II14b Itrsquos well-known that O(n) and U(n) are deformation retracts ofGLn(R) and GLn(C) respectively (see [Hat02 Section 3D] for example) Hence it sufficesto prove stuff for the latter

Let VectnR sube VectR denote the connected component corresponding to n-dimensionalvector spaces By II6 and Theorem I14 constructing an equivalence BGLn(R) simminus VectnRis the same as constructing an equivalence

GLn(R) simminus ΩRnVectnR HomVectR(RnRn) FVectR(RnRn)

74


But we easily get FVectR(RnRn)i = HomTop(|∆i| |GLn(R)|) from the definition above(where we write |GLn(R)| to indicate that we really mean the topological space not theassociated anima) hence FVectR(RnRn) = Sing |GLn(R)| hence the unit map

GLn(Rn) simminus Sing |GLn(R)|

provides the desired equivalence This is also an equivalence of E1-groups since both Sing and| | preserve finite products hence the Segal condition The complex case works completelyanalogous

II14c Lemma mdash If G is a topological group with the homotopy type of a CW complexthen BG (as constructed in II6) and BG (the classifying space for principal G-bundles)coincide

Proof There are probably lots of ways to see this but what convinced me was thatΩBG G also holds on the topological side of things (see [Die08 Example (1447)]) henceboth BGs are mapped to G under the equivalence Ω (lowastAn)gt1

simminus Grp(An)

From Lemmas II14b II14c and the classification of principal G-bundles we find that

π0 HomAn(XVectR) = π0VectR(X) and π0 HomAn(XVectC) = π0VectC(X)

are the sets of isomorphism classes of real and complex vector bundles on X whenever X is aCW complex (it doesnrsquot really make sense to talk about vector bundles over anima) Takingthis a step further we can generalise our constructions from above and produce monoidalanima

VectR(X) VectC(X) EuclR(X) and SphR(X)

of real vector bundles complex vector bundles Rn-fibre bundles and spherical fibrationsover X respectively These satisfy (either by construction and the argument above or simplyby definition itrsquos up to you to decide)

VectR(X) HomAn(XVectR) EuclR(X) HomAn(X EuclR) VectC(X) HomAn(XVectC) SphR(X) HomAn(XSphR)

Now observe that HomAn(Xminus) An An preserves products hence Segal conditions Thusit preserves E1-monoids and by an easy check also E1-groups (since we only need thatπ0 HomAn(Xminus) sends E1-groups to ordinary groups) Hence Definition II14a and theuniversal property of group completion induces comparison maps

VectR(X)infin-grp minus HomAn(X ko) EuclR(X)infin-grp minus HomAn(X ktop) VectC(X)infin-grp minus HomAn(X ku) SphR(X)infin-grp minus HomAn(X ksph)

Be aware that in general these arenrsquot equivalences in general This is only true when X is afinite CW complex (or a retract of such) as we will see in Corollary III15 once we have thegroup completion theorem available

Now put kolowast(X) = πlowastHomAn(X ko) and define kulowast(X) ktoplowast(X) and ksphlowast(X)similarly These groups are pretty important invariants in geometric topology For examplekolowast(X) and kulowast(X) are whatrsquos usually called real and complex K-theory In the special case

75

Einfin-Monoids and Einfin-Groups

X = lowast they are completely known thanks to the famous Bott periodicity theorem (whichwersquoll see again in a refined version in Theorem III16)

πn(ku) Z if n equiv 0 mod 20 if n equiv 1 mod 2

and πn(ko)

Z if n equiv 0 mod 8Z2 if n equiv 1 mod 8Z2 if n equiv 2 mod 80 if n equiv 3 mod 8Z if n equiv 4 mod 80 if n equiv 5 mod 80 if n equiv 6 mod 80 if n equiv 7 mod 8

The homotopy groups of ksph are also ldquoknownrdquo in that πn(ksph) πsn is the nth stablehomotopy group of spheres Finally we wonrsquot give a description of the homotopy groups ofktop but we can at least their ldquodifferencesrdquo to those of ko and ksph For n 6= 4 we have that

πn(

fib(ko ktop)) Θn

is the group of exotic spheres in dimension n ie exotic smooth structures on Sn Moreover

πn(

fib(ktop ksph)) Lq

n(Z)

Z if n equiv 0 mod 40 if n equiv 1 mod 4Z2 if n equiv 2 mod 40 if n equiv 3 mod 4

yields Wallrsquos quadratic L-groups of the integers

Einfin-Monoids and Einfin-GroupsThe ldquoideardquo how to impose commutativity is to replace ∆∆op by some other category Lop whichhas ldquoflipsrdquo in it Lurie uses Finop

lowast to denote this category but Fabian decided to honourSegalrsquos original notation

II15 Definition mdash We denote by Lop the 1-category whose objects are finite sets andwhose morphisms are partially defined maps Write 〈n〉 = 1 n There is a functor

Cut ∆∆op minusLop

defined on objects via Cut([n]) = 〈n〉 On a morphism α [m] [n] in ∆∆ which thencorresponds to a morphism αop in ∆∆op pointing in the other direction Cut(αop) 〈n〉〈m〉is defined via

Cut(αop)(i) =

undefined if i 6 α(0)j if α(j minus 1) lt i 6 α(j)undefined if α(n) lt i

In more invariant words Cut sends a finite non-empty totally ordered set I isin ∆∆op to its setof Dedekind cuts (ie partitions into two non-empty intervals) and a map α I J in ∆∆

76


is sent to Cut(αop) = αlowast Cut(J) Cut(I) which is given by taking preimages (wheneverthese are non-empty otherwise itrsquos undefined)

If now C is an infin-category with finite products then a cartesian commutative monoid inC is a functor X Lop C such that X Cut ∆∆op C is a cartesian monoid in the sense ofDefinition II1 If it is even a cartesian group then X is called a cartesian commutative groupin C We let CMon(C)CGrp(C) sube Fun(Lop C) denote the full sub-infin-categories spanned bycartesian commutative monoidsgroupsII15a mdash Time to unwind The functor Cut ∆∆op

Lop takes the Segal maps ei [1] [n]in ∆∆ to the maps ρi 〈n〉〈1〉 defined by

ρi(j) =

1 if i = j

undefined else

Moreover the face maps d0 d1 d2 [1] [2] in ∆∆ are sent to lm r 〈2〉〈1〉 the uniquemaps defined on 1 1 2 and 2 respectively So composition in a cartesian commutativemonoid becomes

micro X1 timesX1 X2mminus X1sim

(lr)

But consider the flip map flip 〈2〉〈2〉 that swaps 1 and 2 Then m flip = m r flip = land l flip = r hold in Lop so the diagram

X1 timesX1

X1

X1 timesX1

flip factors

micro

micro

commutes So it really makes sense to think of X as commutativeII15b Exercise mdash If C is a 1-category show that Cutlowast Fun(Lop C) Fun(∆∆op C)restricts to a fully faithful functor CMon(C) Mon(C) Note that this is false for generalinfin-categories even for C = Cat(2)

1 II15c mdash Similar to Example II11(e) any symmetric monoidal 1-category (Cotimes) givesrise to an object in CMon(Cat(2)

1 ) sube CMon(Catinfin) Similar to the monoidal case this isconstructed from a cocartesian fibration potimes Cotimes Lop

Letrsquos briefly sketch the construction Objects of Cotimes are pairs (n x1 xn) where〈n〉 isin

Lop and x1 xn isin C Morphisms (n x) (m y) comprise the data of a mapα 〈n〉〈m〉 in Lop together with maps fj

otimesiisinαminus1(j) xi yj for all j = 1 m

Composition is defined in the ldquoobviousrdquo way using the braiding σ and to see that compositionis associative one has to use σ2 = id If you are confused by this braiding stuff In a nutshella braiding of a monoidal category is a choice of isomorphisms σxy xotimesy simminus yotimesx subject tosome naturality conditions In this terminology a symmetric monoidal category is a braidedmonoidal category for which σxy σyx = id

One can check that the cocartesian fibration potimesmon Cotimesmon ∆∆op from Example II11(e)fits into a pullback diagram

Cotimesmon Cotimes

∆∆op Lop

potimesmon

potimes

Cut

77


It turns out that the diagram

CMon(Cat(2)

1)

Mon(Cat(2)

1)

SymMonCat(2)1 MonCat(2)

1

Cutlowast

sim sim

forget

commutes (we never defined the bottom categories but if you sit down for a while you willsurely come up with suitable a definition of the 2-category of symmetricarbitrary monoidal1-categories) However a monoidal category can have many symmetries so the horizontalfunctors are not at all fully faithful For example let (Cotimes) be the category of Z-gradedR-modules equipped with the graded tensor product Any unit u isin Rtimes defines a braiding τuon C via τu X otimes Y simminus Y otimesX given by τu(xotimes y) = u|x||y|y otimes x on elementary tensors Ifu2 = 1 then this braiding defines a symmetric monoidal structure In particular there canbe many of these So commutativity is a structure not a property

Lecture 128th Dec 2020

II15d Semi-Additive and Additive infin-Categories mdash Aninfin-category is called semi-additive if it admits finite products and finite coproducts its initial and terminal object agreeand for any x y isin C the natural map(

idx 00 idy

) x t y simminus xtimes y

is an equivalence Here 0 x 0 y denotes the unique (up to contractible choice) map inHomC(x y) factoring through an initialterminal object which we also denote 0 isin C If C issemi-additive then wersquoll usually denote its unified product and coproduct by oplus

We call C additive if also the shear map(idx idx0 idx

) xoplus x simminus xoplus x

is an equivalence for all x isin C The typical source of examples is everything that has to dowith abelian groups so for example K(Z) and the derived category D(Z) are additive

II16 Proposition mdash If C is a semi-additive infin-category then the forgetful maps

CMon(C) simminus Mon(C) simminus C

are equivalences If C is additive then the same holds true for

CGrp(C) simminus Grp(C) simminus C

Before we get into the proof letrsquos show a consequence

II17 Corollary mdash If C is an additive infin-category then there is a canonical lift

HomC Cop times C minus CGrp(An)

If C is only semi-additive one still gets a lift to CMon(An)

78


Proof In general if F C D is a finite product-preserving functor then the inducedfunctor Flowast Fun(Lop C) Fun(LopD) preserves the Segal condition and also the con-dition from Definition II4 hence it restricts to functors Flowast CMon(C) CMon(D) andFlowast CGrp(C) CGrp(D)

We canrsquot apply this directly to our situation since HomC CoptimesC An doesnrsquot preservefinite productsmdashor rather it is too good at respecting products as

HomC

(noplusi=1

xi

noplusi=1

yi

)

nprodij=1

HomC(xi yj)

of which the diagonalprodni=1 HomC(xi yi) is only a factor

At least HomC(xminus) C An preserves products for all x isin C whence it induces afunctor HomC(xminus)lowast CGrp(C) CGrp(An) But if C is additive then CGrp(C) C byProposition II16 which provides the desired lift

HomC(xminus) C minus CGrp(An)

To paste these lifts together into a functor HomC Cop times C CGrp(An) consider thecomposite

Fun(CAn) minus Fun(

Fun(Lop C)Fun(LopAn))minus Fun

(CGrp(C)Fun(LopAn)

)

If we restrict to the full subcategory Funtimes(CAn) sube Fun(CAn) of finite product-preservingfunctors it lands in Fun(CGrp(C)CGrp(An)) as argued above Moreover if C is additivethen CGrp(C) C by Proposition II16 Now HomC Cop Fun(CAn) has image inFuntimes(CAn) hence we may compose it with the above to obtain the desired functor

HomC Cop minus Fun(CCGrp(An)

)

If C is only semi-additive then the argument still works with CMon(minus) instead of CGrp(minus)everywhere Moreover one can check (but we wonrsquot do that here) that if we we use the(semi-)additive structure on Cop instead to do the constructions above we still end up withthe same functor HomC Cop times C CGrp(An)

Proof sketch of Proposition II16 The key to the proof is the following claim() Let I isin ∆∆op

Lop Given a functor X I C satisfying X0 0 then X satisfies theSegal condition iff it is left-Kan extended from X|I61 along p I61 I

To conclude the statement of the proposition from () write Fun(I61 C)0 sube Fun(I61 C)for the full subcategory of functors F satisfying F (0) 0 Since 0 isin C is initial and terminalone easily checks that C simminus Fun(I61 C)0 sending x isin C to the unique (up to contractiblechoice) functor Fx satisfying Fx(0) 0 and Fx(1) x is an equivalence Moreover

Fun(I61 C)0 sube Fun(I61 C)Lanpminusminusminus Fun(I C)

is fully faithful by Corollary I54 and () precisely says that its essential image is Mon(C)for I = ∆∆op or CMon(C) for I = Lop This shows Mon(C) C CMon(C) as required If Cis additive then the image of x isin C under the above composite is automatically a cartesiangroup Indeed after an easy unravelling the condition from Definition II4 translates into

79


the condition that the shear map on x be an equivalence which is guaranteed by C beingadditive

To prove () first note that we get a natural counit transformation LanpX|I61 rArr X forfree so whether this is an equivalence can be checked on objects Recall from Theorem I52that

LanpX|I61(n) colimiisinI61n

Xi

(it will follow from the discussion below that these colimits exist in C so the theorem isindeed applicable) Therefore we have to understand the slice categories

Lop61〈n〉 and ∆∆op

61[n] ([n]∆∆61)op

The slice category [n]∆∆61 Letrsquos describe [n]∆∆61 first in words and then in a picture(whichever you start with will probably confuse you and then the other thing will hopefullyunconfuse you) Its vertices are given by maps [n] [1] and [n] [0] in ∆∆ Therersquos only oneof the latter kind and it is denoted 0 (in the picture itrsquos depicted as a black dot on the right)There are n+ 2 maps [n] [1] One that is constantly 0 one that is constantly 1 (depictedon the top and on the bottom) and n surjective maps (depicted on the left) Moreovertherersquos a bunch of non-degenerate edges The pink ones are those induced by 0 [0] [1] and0 [1] [1] the purple ones are those induced by 1 [0] [1] and 1 [1] [1] and finallythe black ones are induced by the unique morphism [1] [0]

0 010 11

01 1

1 1

0 0

0

However the punchline is that all of this doesnrsquot really matter since the dotted partconsisting of the vertices on the left and 0 on the right turns out to be final (in the lecture weclaimed that already the vertices on the left are final but I think thatrsquos not true) To provethis verify the criterion from Theorem I43(b) and use that the slice categories in question(in fact slice categories between slice categories) contain the respective arrow starting at 0as a terminal object After passing to ∆∆op

61[n] instead of [n]∆∆61 the dotted part becomescofinal hence the colimit in question can be taken over the dotted part only But the 0-vertexon the right is mapped to X0 0 and there are no edges between the vertices on the lefthence itrsquos easy to see that the colimit over the dotted part is just Xoplusn1 This is preciselywhat we wanted to show

The slice category Lop61〈n〉 The vertices of Lop

61〈n〉 consist of partially defined maps〈0〉〈n〉 and 〈1〉〈n〉 As 〈0〉 = empty there is only one such map of the first kind depictedby the vertex on the right in the picture below The vertex in the middle represents theunique nowhere-defined map 〈1〉〈n〉 and the vertices on the right correspond to the maps〈1〉〈n〉 which are actually defined on 1

80


12

n

empty empty

Again the dotted part (ie everything except the vertex in the middle) is cofinal and thecolimit over it evaluates to Xoplusn1 since the vertex on the right is sent to X0 0

Since Dgt0(Z) is additive a simple consequence of Proposition II16 is that the EilenbergndashMacLane functor from Very Long Example I56 lifts to a functor

K Dgt0(Z) minus CGrp(An)

Indeed K is a right-adjoint by construction hence it preserves products so we can use theargument from the proof of Corollary II17 to produce the desired lift We will soon (inExample II25(c)) extend this even further

So EilenbergndashMacLane anima give rise to Einfin-groups Another source of examples forEinfin-monoidsgroups is

CMon(Kan) minus CMon(An) and CGrp(Kan) minus CGrp(An)

(again note that Kan An preserves products) but these are essentially the same examplesIt is a theorem of DoldndashThom that a connected commutative monoid in Kan complexes hasunderlying anima a gem (ldquogeneralized EilenbergndashMacLane spacerdquo we defined these things inVery Long Example I56) The same then holds for not necessarily connected commutativegroups in Kan complexes since all connected components are equivalent

II18 Proposition mdash The realisation functor | | sAn An preserves finite products

Proof We compute

|M timesN | colim[n]isin∆∆op

(Mn timesNn) colim([n][m])isin∆∆optimes∆∆op

(Mn timesNm)

colim[n]isin∆∆op

(Mn times colim

[m]isin∆∆opNm

)(


Mn

)times(

colim[m]isin∆∆op

Nm

) |M | times |N |

For the second equivalence see Exercise II18a below The third equivalence follows fromProposition I42 together with the fact that Mn times minus commutes with arbitrary colimitsIndeed for any X isin An the functor X times minus An An is a left adjoint of HomAn(Xminus)Applying the same argument to minustimes colim∆∆op Nm gives the fourth equivalence

II18a Exercise mdash Show that the diagonal ∆ ∆∆op ∆∆op times∆∆op is cofinal as a map ofinfin-categories In other words ∆∆op is sifted

81


Proof By Theorem I43(b) we need to check that ∆∆op times∆∆optimes∆∆op ([m] [n])(∆∆op times∆∆op) isweakly contractible for all [m] [n] isin ∆∆op Note that we may form the slice category as a1-category since the nerve functor commutes with limits Also note that we may replace theslice category by its opposite Smn = ∆∆ times∆∆times∆∆ (∆∆ times∆∆)([m] [n]) for convenience

Unravelling definitions we find that objects in Smn can be thought of as maps of posetsα [i] [m]times [n] Any such map factors uniquely into a composition [i] [j] [m]times [n] ofa surjective and an injective map This defines a functor r Smn Cmn into the categoryCmn of non-empty linearly ordered subposets of [m] times [n] (Cmn is partially ordered byinclusion) But conversely any such subposet is given by an injective map [j] [m]times [n]for some j 6 m+ n+ 1 hence we get an inclusion s Cmn Smn in the reverse directionClearly r s = idCmn whereas the surjections [i] [j] define a natural transformationidSmn rArr s r

This shows that Smn and Cmn are weakly (and even strongly) homotopy equivalent aftertaking nerves Now if we recall [HCI Definition V47] we recognize Cmn as the barycentricsubdivision sd(∆m times∆n) which is weakly contractible because ∆m times∆n is

II18b mdash Recall from Corollary II8a that we have an adjunction


Now B = | | and Ω both preserve finite productsmdashthe former by Proposition II18 thelatter since it is a right-adjoint So both functors preserve the Segal condition Therefore ifwe apply Fun(Lopminus) to both sides of the adjunction above then the resulting adjunctionrestricts to another adjunction

B CMon(Mon(An)

)CMon(lowastAn) Ω (II18b1)

Now it is easy to see that the forgetful map CMon(lowastC) CMon(C) is an equivalence forany infin-category C with finite products (and thus a terminal object lowast isin C) We claim thatalso

CMon(Mon(C)

)CMon(C)

Mon(CMon(C)

)

sim

forgetlowastsim

simforget

(II18b2)

commutes (the forgetful functors are given by evaluation in degree 1 of course) Once wehave this we will obtain the following statement (which comes only later in Fabianrsquos noteshence the gap in numbering)

II21 Corollary mdash There is a commutative diagram of horizontal adjunctionts

CMon(An) CMon(An)

Mon(An) lowastAn

B

CutlowastΩ

ev1

B

Ω

Moreover we have the following analogues of II6 and Proposition II8

82


(a) Restricting B to the full subcategory CGrp(An) sube CMon(An) gives a fully faithfulfunctor B CGrp(An) CMon(An)

(b) Both functors BΩ CMon(An) CMon(An) actually take values in CGrp(An)(c) ΩB CMon(An) CGrp(An) is left-adjoint to CGrp(An) sube CMon(An)

Proof We construct the diagram and prove commutativity first The bottom row is alreadyknown After identifying CMon(lowastAn) CMon(An) and CMon(An) CMon(Mon(An))the upper row becomes (II18b1) To see that the diagram commutes we first check thatthe these identifications transform the forgetful functor CMon(Mon(An)) Mon(An) intoCutlowast CMon(An) Mon(An) Indeed this follows easily from (II18b2) Now both verticalarrows are of the form forget CMon(minus) (minus) from which commutativity of the diagramis clear

Next we check (b) Let M isin CMon(An) To check ΩM isin CGrp(An) it suffices to checkCutlowast ΩM isin Grp(An) (thatrsquos just how Einfin-groups are defined) But Cutlowast Ω Ω ev1 andthe right-hand side takes values in Grp(An) by II6 To show BM isin CGrp(An) it similarlysuffices to check that the ordinary monoid π0(BM)1 is an ordinary group as argued in theproof of Proposition II5 But in fact π0(BM)1 lowast since B Mon(An) lowastAn takes valuesin (lowastAn)gt1 and the diagram above commutes This shows (b)

In particular the top row adjunction restricts to an adjunction

B CGrp(An) CGrp(An) Ω

To show (a) it suffices therefore to check that the unit idrArr ΩB is an equivalence Equiva-lences of Einfin-groups can be detected on underlying E1-groups (in fact even on underlyinganima ie after applying ev1 since the Segal condition will do the rest) So it suffices thatCutlowast rArr CutlowastΩB is an equivalence But CutlowastΩB Ω ev1B ΩB Cutlowast and if G is anEinfin-group then

ΩB CutlowastG (CutlowastG)infin-grp CutlowastG

by Proposition II8 This finishes the proof of (a)Let C sube CGrp(An) denote the essential image of B|CGrp(An) Then B CGrp(An) simminus C

is an equivalence hence its right adjoint Ω C CGrp(An) is an equivalence too ForM isin CMon(An) and G isin CGrp(An) we may thus compute

HomCGrp(An)(ΩBMG) HomCGrp(An)(ΩBMΩBG) HomC(BMBG) HomCMon(An)(MΩBG) HomCMon(An)(MG)

where we also used G ΩBG twice This shows (c)

II21a Example mdash Some consequences(a) Corollary II21(c) shows that group completion in Mon(An) and CMon(An) is compati-

ble Hence all examples from II12 to II14 ie

k(R) gwr(R) ko ku ktop and ksph

are canonically Einfin-groups Indeed they arise as group completions of certain E1-monoids all of which have canonical Einfin-monoid refinements by II15c

83


(b) It follows from Corollary II21(a) that ΩBX X for all X isin CGrp(An) Since itrsquoswell-known that Ω shifts homotopy groups down we see that B shifts homotopy groupsup That is if e isin X1 denotes the identity element then

πi((BX)1

)

πiminus1(X1 e) if i gt 0lowast if i = 0

(in particular we donrsquot need to specify a basepoint for (BX)1 since it is connected)(c) We get a commutative solid diagram

Dgt0(Z) CGrp(An)

Dgt0(Z) CGrp(An)

[1]=Σ

K

BΩ

K

Ω

in Catinfin So B preserves EilenbergndashMacLane anima and only shifts the homotopygroups up by 1

To see commutativity first note that the diagram with the dotted vertical arrowscommutes (where Ω = (τgt1minus)[minus1] Dgt0(Z) Dgt0(Z) is defined as in PhilosophicalNonsense I18) Indeed Ω is given by a pullback which is preserved by the right-adjointfunctor K (being a right adjoint is still true for the lift K Dgt0(Z) CGrp(An)simply by construction and our Observation I40 that adjunctions extend to functorcategories) By abstract nonsense commutativity of the dotted diagram induces anatural transformation BK rArr K(minus[1]) Whether this is an equivalence can be checkedon objects C isin Dgt0(Z) But whether B(KC) K(C[1]) is an equivalence can inturn be checked on homotopy groups of underlying anima But from (b) we see thathomotopy groups of both B(KC) and K(C[1]) are those of KC shifted up by one

Now towards the proof of (II18b2) Letrsquos denote by Cattimesinfin sube Catinfin the (non-full) sub-infin-category spanned by infin-categories with finite products and functors preserving these Letfurthermore Catadd

infin sube Catsemi-addinfin sube Cattimesinfin denote the full sub-infin-categories spanned by the

additive and semi-additive infin-categories respectively

II19 Theorem mdash If an infin-category C has products then CMon(C) and CGrp(C) aresemi-additive and additive respectively Furthermore the functors

CMon Cattimesinfin minus Catsemi-addinfin

CGrp Cattimesinfin minus Cataddinfin

are right-adjoint to the inclusions Catsemi-addinfin sube Cattimesinfin and Catadd

infin sube Cattimesinfin respectively

Using Theorem II19 we get an easy proof of (II18b2) First note that the two maps

CMon(CMon(C)

)CMon(C)

simforgetlowast

simforget

agree and are equivalences Indeed CMon(minus) (minus) is the counit of the adjunction fromTheorem II19 hence an equivalence since the left adjoint Catsemi-add

infin sube Cattimesinfin is fully faithful

84


This shows that both functors are equivalences Moreover by writing down the triangleidentities we find that both maps are inverses (once from the right once from the left) to theunit CMon(C) CMon(CMon(C)) hence they must agree This almost shows (II18b2)except that we want to replace one CMon by a Mon To do this note that Proposition II16implies CMon(CMon(C)) Mon(CMon(C)) Moreover CMon(Mon(C)) Mon(CMon(C))follows by straightforward inspection as both sides can be interpreted as full subcategoriesof Fun(∆∆op times

Lop C) Hence we are doneNote that our argument can not be extended to show that the two forgetful functors

Mon(Mon(C)) Mon(C) agree This is still true but much harder to show Moreover if Cis a 1-category then Mon(Mon(C)) CMon(C) a fact that is usually called the EckmannndashHilton trick This is false for infin-categories What is true instead is

CMon(C) limnisinNop

Mon(n)(C)

where Mon(n)(minus) = Mon(Mon( (Mon(minus)) )) is the n-fold iteration of Mon(minus) Thetransition morphisms Mon(n)(C) Mon(nminus1)(C) are given by forgetting one Mon Thereare n choices to do so but it turns out that they all agree generalizing our claim aboutMon(Mon(C)) Mon(C)

For the proof of Theorem II19 we need a criterion for semi-additivity

II20 Lemma (compare [HA Proposition 24319] which is stronger) mdash Let C be acategory with finite products Then C is semi-additive if the following conditions hold(a) The terminal object lowast isin C also is initial(b) We have a natural transformation micro (∆ x 7 xtimesx)rArr idC such that both compositions

x xtimes lowast idxtimes0minusminusminusminus xtimes x microxminus x

x lowast times x 0timesidxminusminusminusminus xtimes x microxminus x

are homotopic to idx for all x isin C and the diagram

(xtimes x)times (y times y) (xtimes y)times (xtimes y)

xtimes ymicroxtimesmicroy

idxtimesfliptimesidy

microxtimesy

commutes up to homotopy for all x y isin C

II20a Remark mdash Lurie only requires that the first of the two compositions inLemma II20(b) is homotopic to idx and Fabian may or may not have done the samein the lecture However the second one doesnrsquot follow from the first and wersquoll really needboth to be homotopic to idx in the proof In fact if only the first composition were to behomotopic to idx we could take microx to be pr1 xtimes x x for all x isin C (itrsquos clear that thisalso makes the diagram above commute) and then we could show that every infin-categorieswith finite products and a zero object is semi-additive This canrsquot be true

Proof of Lemma II20 Let c d isin C We want to show that the maps

c ctimes lowast idctimes0minusminusminusminus ctimes d and d lowast times d 0timesiddminusminusminusminus ctimes d

85


exhibit c times d as a coproduct of c and d That is we need to show that the naturaltransformation

ϕ HomC(ctimes dminus) sim=rArr HomC(cminus)timesHomC(dminus)induced by these maps is an equivalence This can be checked on objects Given a isin C weconstruct an inverse ψa to ϕa as follows

ψa HomC(c a)timesHomC(d a) timesminus HomC(ctimes d atimes a) microaminus HomC(ctimes d a)

Note that we do not claim that the ψa combine into a natural transformation ψ which isinverse to ϕ only that ψa is inverse to ϕa since it suffices to have ϕ a pointwise equivalenceLetrsquos prove ϕa ψa id first This follows essentially from the first condition in (b) butsince this sparked some confusion in the lecture Irsquoll try to give a much more detailed proofthan in the lecture and be as precise as possible (hopefully this doesnrsquot make things worse)We begin by drawing a tiny diagram

HomC(c a)timesHomC(d a)

HomC(ctimes lowast atimes lowast)timesHomC(lowast times d lowast times a)

HomC(ctimes d atimes a) HomC(ctimes lowast atimes a)timesHomC(lowast times d atimes a)

HomC(ctimes d a) HomC(ctimes lowast a)timesHomC(lowast times d a) HomC(c a)timesHomC(d a)

simtimes

idsim

(I) (III)

microa (II) microatimesmicroa

sim

If we can show that this diagram commutes up to homotopy then walking around itsperimeter will show ϕa ψa id The cell labelled (III) commutes up to homotopybecause the composition of the three downward pointing arrows and the rightward pointingarrow is induced by microa (idatimes0) a a times lowast a times a a in the first factor and bymicroa (0times ida) a lowast times a atimes a a in the second factor both of which are homotopic toida by assumption

To see that the cell labelled (II) commutes up to homotopy it suffices to check thisafter composition with pr1 HomC(c times lowast a) times HomC(lowast times d a) HomC(c times lowast a) and aftercomposition with pr2 HomC(c times lowast a) times HomC(lowast times d a) HomC(lowast times d a) We only givethe argument for pr1 since the argument for pr2 will be analogous After composition withpr1 we end up with two maps HomC(c times d a times a) HomC(c times lowast a) Both are given byprecomposition with ctimes lowast ctimes d and postcomposition with microa atimes a a but in differentorders However it doesnrsquot matter (up to homotopy) in which order we postcompose andprecompose which proves commutativity of (II)

Finally to see that (I) commutes we look at the diagram (and its obvious analogue wherectimes lowast is replaced by lowast times d)

HomC(c a)timesHomC(d a) HomC(ctimes d atimes a)

HomC(c a)timesHomC(lowast a) HomC(ctimes lowast atimes a)

HomC(c a)timesHomC(lowast lowast) HomC(ctimes lowast atimes lowast)

times

times

sim

times

86


This commutes ldquoby naturality of the times-construction on morphism animardquo The bottom leftvertical arrow is an equivalence since lowast is initial so HomC(lowastminus) lowast In particular this showsHomC(c a)timesHomC(lowast lowast) HomC(c a) Under this identification the bottom row becomesthe morphism HomC(c a) HomC(c times lowast a times lowast) from (I) and the two left vertical arrowscompose to the projection pr1 HomC(c a)timesHomC(d a) HomC(c a) This shows that (I)commutes indeed and therefore that ϕa ψa id

Now we prove ψa ϕa id First we check that ψa ϕa is homotopic to the map inducedby precomposition with

ctimes d (ctimes lowast)times (lowast times d) minus (ctimes d)times (ctimes d) microctimesdminusminusminus ctimes d

To show this we draw another moderately large diagram

HomC(c a)timesHomC(d a) HomC(ctimes d a)

HomC(ctimes d atimes a) HomC((ctimes d)times (ctimes d) atimes a)

)HomC(ctimes d a) HomC

((ctimes lowast)times (lowast times d) a

)HomC

((ctimes d)times (ctimes d) a

)

(IV) ∆microctimesd

(V)

microa (VI)microa

The cell labelled (V) commutes up to homotopy because micro ∆rArr id is a natural transformation(and I believe Fabian is right that we really need this heremdashif we were to use Luriersquos weakercondition it would only commute on π0) If we choose the dotted arrow to be induced byctimesd (ctimeslowast)times (lowasttimesd) (ctimesd)times (ctimesd) then also (VI) commutes up to homotopy becauseit doesnrsquot matter in which order precomposition with said morphism and postcompositionwith microa are applied Finally (IV) commutes by inspection Walking around the perimeterof this diagram now proves that ψa ϕa is really induced by precomposition with the mapfrom above

Now we finally use the second condition from (b) to obtain a homotopy-commutativediagram

ctimes d (ctimes lowast)times (lowast times c) (ctimes d)times (ctimes d) ctimes d

(ctimes lowast)times (lowast times d) (ctimes c)times (dtimes d)

sim

sim

microctimesd

idctimesfliptimesidc sim microctimesmicrod

in C Using the first condition from (b) we see that the composition of the lower three arrowsis homotopic to idctimes idd Hence the morphism we precompose with is homotopic to theidentity on ctimes d which shows ψa times ϕa id as required

Proof of Theorem II19 We first prove that CMon(C) is semi-additive Of course we wouldlike to apply Lemma II20 so we must produce the corresponding data on CMon(C) First ofall note that const lowast Lop C is both terminal and initial in CMon(C) It is even terminalin Fun(Lop C) because lowast isin C is terminal and limits in functor categories are computedpointwise (Lemma I39) To see that it is initial let X isin CMon(C) and compute

Nat(const lowast X) HomC(lowast lim〈n〉isin

LopXn

) HomC(lowast X0) HomC(lowast lowast) lowast

87


where we used that 〈0〉 isin Lop is an initial object and thus spans a final subset in the sense ofDefinition I44 (and yes our terminology here isnrsquot optimal but then again there is no goodterminology at all)

Next we need to construct functorial maps microM MtimesM M for allM isin CMon(C) Con-sider the map times Loptimes

Lop Lop given by crossing the sets and crossing the morphisms (note

that this wouldnrsquot work with ∆∆op) It induces a functor Fun(Lop C) Fun(LopFun(Lop C))and one easily checks that the Segal condition is preserved ie we get a functor

Double CMon(C) minus CMon(CMon(C)

)

Unravelling definitions we find that Double(minus)1 CMon(C) CMon(C) is naturally equiva-lent to the identity and Double(minus)2 CMon(C) CMon(C) is naturally equivalent to thediagonal ∆ M 7M timesM Hence we can take microM M timesM M to be the multiplicationm Double(M)2 Double(M)1 ie the map induced by m 〈2〉〈1〉 in Lop (see Defini-tion II15) Since the microM are induced by a map in Lop itrsquos clear that they assemble into anatural transformation micro Also the required conditions for micro are straightforward to check

This shows that CMon(C) is semi-additive by Lemma II20 Hence the same holds forits full subcategory CGrp(C) sube CMon(C) since this is closed under finite products To showthat CGrp(C) is additive we need to check that the shear map is an equivalence for allobjects G isin CGrp(C) But in view of the fact that the codiagonal GtimesG G is induced bymicroG (this follows by unravelling the explicit inverse in the proof of Lemma II20) hence bymultiplication on Double(G) this gets translated into the fact that Double(G) is a cartesiancommutative group itself which is again easy to check

Finally we need to show that CMon Cattimesinfin Catsemi-addinfin and CGrp Cattimesinfin Catadd

infin areright adjoints If we denote R = CMon and η RrArr id the forgetful natural transformation(given by evaluation in degree 1) then Proposition II16 and the fact that CMon(C) issemi-additive show that ηR R R sim=rArr R and Rη R R sim=rArr R are natural equivalencesThus we may apply the dual of Proposition I61a to see that R is a right adjoint to theinclusion im(R) sube Cattimesinfin of its essential image But im(R) = Catsemi-add

infin Indeed ldquosuberdquo waschecked above whereas ldquosuperdquo follows from Proposition II16 again The argument for CGrp iscompletely analogous


Free Einfin-Monoids mdash Similar to Proposition II9 the forgetful functor CMon(An) Anas a left adjoint FreeCMon An CMon(An) On objects X isin An it is given by

FreeCMon(X)1 ∐ngt0

XnhSn

ie we take free words in X but we need to homotopy-quotient out the action of thesymmetric group Sn which the notation (minus)hSn is supposed to do In general given afunctor F BG C where C is an infin-category with colimits and G is a discrete group weput

FhG colimBG

F

On Xn the Sn-action is given by ldquopermuting factorsrdquo More precisely we may apply II6 tosee that giving a functor BSn core(An) sending the unique 0-simplex of BSn to Xn isthe same as giving a functor Sn ΩXn core(An) Homcore(An)(Xn Xn) But each factorpermutation is a map in Homcore(An)(Xn Xn) hence therersquos indeed a canonical map fromthe discrete anima Sn into it

88

Spectra and Stable infin-Categories

To prove the formula above we can proceed as in the proof of Proposition II9 Call amap in Lop inert if it is a bijection (where it is defined) and active if it is defined everywhereWith this terminology the proof of Proposition II9 can be copied except for the followingsmall change In the decomposition Lop

int〈1〉 ∐α Cα the categories Cα no longer have the

active map α 〈n〉〈1〉 as a terminal object instead α 〈n〉〈1〉 together with itsautomorphisms spans cofinal subcategory This subcategory is equivalent to the anima BSnas follows immediately from II6 Thus we get indeed FreeCMon(X)1

∐ngt0X

nhSn

As an example

FreeCMon(lowast)1 ∐ngt0

BSn (finite sets bijectionst

)= S

where the right-hand side is the symmetric monoidal 1-groupoid of finite sets and bjectionsbetween them whose tensor product is given by disjoint union In particular even thoughFreeMon(lowast) N is the free monoid on a point and commutative itrsquos not the free commutativemonoid on a point

Spectra and Stable infin-CategoriesLet CGrp(An)gti sube CGrp(An) be the full sub-infin-category spanned by those X with πjX1 = 0for j lt i We have seen in Corollary II21 that B CGrp(An) CGrp(An) is fully faithfulwith right adjoint Ω Moreover wersquove seen during the proof that B actually has essentialimage in CGrp(An)gt1 In fact its essential image is CGrp(An)gt1 To see this it suffices tocheck that the counit c BΩX simminus X is an equivalence for all X isin CGrp(An)gt1 This canbe checked on underlying anima ie after ev1 and then again on homotopy groups Since Ωshifts homotopy groups down and B shifts them up we get that clowast πi(BΩX)1

simminus πiX1 isan isomorphism for i gt 0 For i = 0 it is one as well as both sides are connected

Iterating this argument we find that the i-fold application B(i) = B middot middot middot B is anequivalence

B(i) CGrp(An) simminus CGrp(An)gti Moreover since ΩB id by Corollary II21(a) we get ΩB(i+1) B(i) This yields a fullyfaithful functor

Binfin CGrp(An) minus limNop

(

Ωminus CGrp(An) Ω

minus CGrp(An) Ωminus CGrp(An)

)(the limit being taken in Catinfin of course) with essential image

limNop

(

Ωminus CGrp(An)gt2

Ωminus CGrp(An)gt1

Ωminus CGrp(An)

)

II22 Definition mdash We say that aninfin-category C has finite limits if it has finite productsand pullbacks (one can check that this is equivalent to having limits over all finite simplicialsets) If this is the case then a suspension functor Ω lowastC lowastC can be defined and we put

Sp(C) = limNop

(

Ωminus lowastC Ω

minus lowastC)

called the infin-category of spectrum objects in C If C = An we obtain Sp = Sp(An) theinfin-category of spectra

89


II23 Proposition mdash Suppose C has finite limits Then the forgetful functor

Sp(CGrp(C)

) simminus Sp(C)

is an equivalence

Proof sketch Topologists hate this proof First observe that Sp(CGrp(C)) CGrp(Sp(C))This is because

limNop

(

Ωminus Fun(Lop lowastC) Ω

minus Fun(Lop lowastC)) Fun

(Lop limNop

(

Ωminus lowastC Ω

minus lowastC))

(here we use that Fun(Dminus) Catinfin Catinfin commutes with limits since it is right-adjointto D timesminus and also that Ω in functor categories can be computed pointwise by Lemma I39)and one checks that the Segal condition and the group condition are preserved

Therefore we will be done by Proposition II16 once we show that Sp(C) is an additivecategory Of course we will employ Lemma II20 again Clearly (lowast lowast ) isin Sp(C) is bothinitial and terminal Also Sp(C) again has finite limits inherited from lowastC because limitsin limits can be computed degreewise (combine the fact that Hom anima in limits arelimits of Hom anima with Corollary I50 and Proposition I42) Moreover we know thatΩ Sp(C) simminus Sp(C) is an equivalence by construction and factors as

Ω Sp(C) minus Grp(Sp(C)

) ev1minus Sp(C)

(I didnrsquot find this quite obvious so I decided to sketch a proof in Remark II23a below)Finally to apply Lemma II20 we must produce functorial morphisms microX X timesX Xwith the properties stated there For this we use X Ω(Ωminus1X)1 and take microX to be inducedby the multiplication on Ω(Ωminus1X) isin Grp(Sp(C))

Up to checking the various conditions this shows that Sp(C) is semi-additive Foradditivity we must check that the shear map is an equivalence on every X isin Sp(C) Similarto the proof of Theorem II19 this follows from the fact that Ω(Ωminus1X) is a cartesian grouprather than just a monoid

II23a Remark mdash For any infin-category D with finite limits and a zero object lowast isin D(ie an object that is both initial and terminal) the functor Ω D D factors as

Ω D minus Grp(D) ev1minus D

This can be bootstrapped from the case D = lowastAn which we already know from II6 Thefirst step is to enhance the Yoneda embedding to a fully faithful functor D Fun(Dop lowastAn)To this end we consider Fun(DopAn) Fun(Ar(Dop)Ar(An)) Fun(lowastDopAr(An)) andcheck that its image lands in the ldquocorrect fibrerdquo ie in Fun(lowastDop lowastAn) if we restrict thesource to finite limits-preserving functors (where the Yoneda embedding lands)

On Fun(Dop lowastAn) we have loop functor Ω Fun(Dop lowastAn) Fun(Dop lowastAn) as welland it is simply given by postcomposition with Ω lowastAn lowastAn Hence it factors as

Fun(DopGrp(lowastAn)

)Grp

(Fun(Dop lowastAn)

)Fun(Dop lowastAn) Fun(Dop lowastAn)

ev1

sim

Ω

Ω

Since the Yoneda embedding preserves limits we get a lift Ω D Grp(Fun(Dop lowastAn))and one checks that it lands in the full sub-infin-category Grp(D) as required

90


II23b Homotopy Groups of Spectra mdash Because Nop sube Zop is final in the sense ofDefinition I44 we can also write Sp(C) limZop(lowastC Ω ) This gives evaluation functors

Ωinfinminusi Sp(C) minus C

for all i isin Z Note ΩΩinfinminusiX Ωinfinminus(iminus1)X ΩinfinminusiΩX so this somewhat weird notationactually makes sense In the case C = An we can now define homotopy groups for X isin Sp as

πi(X) = π0(Ωinfin+iX)

If we think of a spectrum X as a sequence X = (X0 X1 X2 ) where Xi = ΩinfinminusiXtogether with equivalences Xi ΩXi+1 then we see that πi(X) π0(Xminusi) πj(Xjminusi) forall j gt 0 (we donrsquot need to specify any base points since the Xi are pointed anima) becauseΩ lowastAn lowastAn shifts homotopy groups down

Itrsquos probably clear to all of you but let me also mention that equivalences of spectra canbe detected on homotopy groups

II23c Lemma mdash A map f X Y of spectra is an equivalence in Sp iff it inducesisomorphisms flowast πlowast(X) simminus πlowast(Y ) on homotpy groups

Proof Write Xi = Ωinfinminusi and Yi = Ωinfinminusi as above Then f is an equivalence in Sp iff eachΩinfinminusif Xi Yi is an equivalence in An Indeed the inverses of Ωinfinminusif will again assembleinto a compatible system of morphisms and hence into an inverse of f Whether Ωinfinminusif isan equivalence in An can be detected on homotopy groups In fact both Xi is an elementof CGrp(An) so all its connected components are equivalent and the same is true for YiHence instead of considering homotopy groups with arbitrary base points it suffices to checkthat π0(X) π0(Y ) is a bijection and πj(Xi) πj(Yi) (with the canonical choice of basepoints) are group isomorphisms for all j gt 0

Now πj(Xi) πjminusi(X) for all j gt 0 and the same holds for Y hence it suffices to haveisomorphisms flowast πlowast(X) simminus πlowast(Y ) as claimed

II23d Summary of What We Know So Far mdash Letrsquos explicitly record the followingfacts we already learned in the proof of Proposition II23 and afterwards(a) Sp(C) has finite limits (because lowastC has) and limits in Sp(C) are computed degreewise

ie Ωinfinminusi Sp(C) C preserves limits(blowast) The only reason this doesnrsquot work for colimits as well is that the degreewise colimits

might not be compatible with the Ω-maps any more so they donrsquot necessarily definean element of Sp(C) If that happens to be the case then the degreewise colimit is infact the colimit in Sp by the same argument as for limits (combine the fact that Homanima in limits are limits of Hom anima with Corollary I50 and Proposition I42)

(c) Ω Sp(C) simminus Sp(C) is an equivalence(d) Sp(C) is additive

II24 Corollary (Recognition principle for infinite loop spaces BoardmanndashVogt MaySegal) mdash The functor

Binfin CGrp(An) minus Spis fully faithful with essential image those X isin Sp with πiX = 0 for i lt 0 (as a sloganEinfin-groups are spectra with no negative homotopy groups) Also πi(BinfinX) = πi(X1 lowast) forall Einfin-groups X isin CGrp(An) and all i gt 0

91


Proof The target of Binfin is Sp(CGrp(An)) Sp by Proposition II23 We have seenthat Binfin is fully faithful and identified its essential image before Definition II22 Clearlyany element X of the limit characterising the essential image satisfies πiX = 0 for i lt 0Conversely if X satisfies this condition then πjΩinfinminusiX πjminusiX which vanishes for j lt iRemembering Sp Sp(CGrp(An)) this shows ΩinfinminusiX isin CGrp(An)gti so X lies in theessential image of Binfin The final assertion is clear since πi(BinfinX) πi(ΩinfinBinfinX) πi(X1 lowast)holds by definition

II25 Example mdash Let C always have finite limits Here are ldquomy first spectrardquo(a) If Ω lowastC simminus lowastC is an equivalence then Sp(C) C (because in that case wersquore

taking a constant limit over Nop which is weakly contractible so that limit is C byProposition I36) For example Sp(D(Z)) D(Z) or even more silly Sp(Sp) Sp

(b) Letrsquos compute Sp(Dgt0(R)) for a ring R Recall that the loop functor on Dgt0(R) isdefined as ΩC τgt0(C[minus1]) isin Dgt0(R) Then the functor

D(R) simminus Sp(Dgt0(R)

)C 7minus

(τgt0C (τgtminus1C)[1] (τgtminus2C)[2]

)is an equivalence The inverse functor sends a sequence (C0 C1 ) with equivalencesσi Ci (τgt1Ci+1)[minus1] to colimN Ci[minusi] The structure maps Ci[minusi] Ci+1[minus(i+ 1)]in this colimit are constructed from the σi in the obvious way

(c) Consider the EilenbergndashMacLane functor K Dgt0(Z) An Being a right adjoint itpreserves finite limits and in particular loop spaces Hence it upgrades canonically to afunctor

Sp(Dgt0(Z)

)Sp(An)

D(Z) Sp

Sp(K)

H

the EilenbergndashMacLane spectrum functor By construction the diagram

D(Z) Sp

CGrp(An)K(τgtminusiminus)[i]

H

Ωinfinminusi

commutes If C isin Dgt0(Z) then HC = (K(C)K(C[1])K(C[2]) ) and we haveπi(HC) = Hi(C) If C is an abelian group A concentrated in degree 0 we obtain theEilenbergndashMacLane spectrum HA = (K(A 0)K(A 1)K(A 2) )

(d) We have Sp(Anop) lowast because it is a sequential limit over (emptyAnop) (Anempty)op lowast(since there is no map X empty in An except idempty empty empty) Also

Sp((lowastAn)op) lim

Nop

(

Ωminus (lowastAn)op Ω

minus (lowastAn)op)

limNop

(

Σminus lowastAn Σ

minus lowastAn)op lowast

since only contractible spaces can be an infinite suspension (as all homotopy groupsmust be vanish)

92


II26 Homology and Cohomology of Spectra mdash Via the EilenbergndashMacLane spec-trum functor H the adjunction Cbull lowastAn Dgt0(Z) K from Very Long Example I56 canbe upgraded to another adjunction

Cbull Sp D(Z) H

The new Cbull is defined by the formula Cbull(X) colimiisinN Cbull(ΩinfinminusiX)[minusi] on objects To seewhere the transition maps in this colimit come from recall that Ω = [minus1] is the loop functor inD(Z) Together with C(lowast) 0 this provides a natural transformation Cbull(Ωminus)rArr Cbull(minus)[minus1]which yields the required transition maps

To check that Cbull is a left adjoint it suffices to do so on objects by Corollary I32 (inparticular this will show that Cbull is a functor at all) where we can compute

HomD(Z)(Cbull(X) D

) limiisinNop

HomD(Z)(Cbull(ΩinfinminusiX) D[i]

) limiisinNop

HomlowastAn(ΩinfinminusiXK(D[i])

) limiisinNop

HomlowastAn(ΩinfinminusiXΩinfinminusiHD)

and the last term is HomSp(XHD) since Hom anima in Sp (in fact in arbitrary limits ofinfin-categories) can be computed as a limit over the Hom anima in each degree Also allequivalences are clearly functorial in D isin D(Z)

Now define the homology and the cohomology of X isin Sp with coefficients in A isin Ab to be

Hlowast(XA) = Hlowast(Cbull(X)otimesLZ A

)= colim

iisinNHlowast(ΩinfinminusiXA)

Hlowast(XA) = Hlowast(RHomZ(Cbull(X) A)

)

In the special case X S[Y ] for some anima Y it turns out that Hlowast(S[Y ] A) = Hlowast(YA)and Hlowast(S[Y ] A) = Hminuslowast(YA) coincide with singular homology and (up to a sign swap)singular cohomology See Lemma II31d below In Corollary II55b wersquoll also find out thatHlowast(XA) = TorSlowast(XHA) and Hlowast(XA) ExtlowastS(XHA) can be interpreted as ldquoTor and Extover the sphere spectrumrdquo

Warning mdash Donrsquot confuse the homology or cohomology of a spectrum with thehomology or cohomology theory associated to a spectrum which wersquoll introduce in Defini-tion II35 below Also neither Hlowast(HCA) for a complex C isin D(Z) nor Hlowast(BinfinMA) forsome M isin CGrp(An) are easy to compute in general

II27 PropositionDefinition mdash Suppose C is an infin-category with a zero object (iea simultaneously initial and terminal object 0 isin C) Then the following conditions areequivalent(a) C has finite limits and Ω C simminus C is an equivalence(b) C has finite colimits and Σ C simminus C is an equivalence(c) C has finite limits and finite colimits and a commutative square in C is a pushout square

iff it is a pullback squareSuch infin-categories are called stable infin-categories

Proof The implications (c) rArr (a) (b) are clear Applying the pushout-pullback conditionto the pushout square defining Σ and the pullback square defining Ω shows that the unit

93


id sim=rArr ΩΣ and counit ΣΩ sim=rArr id are natural equivalences so Σ and Ω must be equivalencesof infin-categories

For (a) rArr (c) the same argument as in the proof of Proposition II23 shows that C isadditive (write X as ΩY in Grp(C) and apply Lemma II20) So we only need to check thatpushouts exist and coincide with pullbacks Let P sube Fun(∆1times∆1 C) be the full subcategoryspanned by pullback squares We claim() The restriction P simminus Fun(Λ2

0 C) is an equivalenceTo prove () we construct an inverse functor Given a diagram F Λ0 C which we canview as a span c a b in C we construct the following moderately large diagram

Ωa Ωc 0

Ωb x g 0

0 f a b

0 c

All squares are pullbacks as indicated The fact that Ωa Ωb and Ωc appear in the top leftcorner follows by combining suitable pullback squares into larger pullback rectangles

The inverse functor now sends

F =

a b

c

7minus Ωminus1

Ωa Ωb

Ωc x

(technically we have only defined the inverse on objects but its clear how to make it functorialsince limits are functorial)

To construct pushouts using () let F Λ20 C be a span c a b as above We know

that it can be uniquely (up to contractible choice) extended to a pullback square where thebottom right corner is some object d isin C The same goes for the trivial span consisting ofidentities x = x = x for some x isin C and in this case the object we have to add is x again ofcourse Hence

Nat

a b

c

x x

x

Nat

a b

c d

x x

x x

HomC(d x)

because ∆1 times∆1 has terminal vertex (1 1) This computation shows that d is a pushout ofthe given span Moreover we get the property that pushouts agree with pullbacks for freeThis finishes the proof of (a) rArr (c) For (b) rArr (c) just dualize everything

An immediate consequence of PropositionDefinition II27 is that Sp(C) is stable for allinfin-categories C with finite limits In particular it holds for Sp and D(R) for any ring R byExample II25(b) (but itrsquos also true for K(R)) Moreover Fun(I C) is stable whenever C is

94


II28 Corollary mdashLecture 1415th Dec 2020

For a functor F C D between stable infin-categories the followingare equivalent(a) F preserves finite limits(b) F preserves finite colimits

Proof Since C and D are additive F preserves finite coproducts iff it preserves finiteproducts Since also pushout squares and pullback squares coincide F preserves the formeriff it preserves the latter This suffices to get all finite colimits and limits

II29 Definition mdash (a) Let F C D be a functor between infin-categories with finitelimits We call F left exact if it preserves finite limits

(b) We denote by Catlexinfin sube Catinfin the (non-full) sub-infin-category spanned by these Dually

Catrexinfin sube Catinfin is the full sub-infin-category spanned by right-exact functors ie finite

colimit-preserving functors between infin-categories with finite colimits(c) Finally we denote by Catst

infin the full subcategory of both Catlexinfin and Catrex

infin spanned bythe stable infin-categories

For example the functor Ωinfin Sp(C) C is left exact for every C with finite limits Alsonote that Ωinfin factors canonically over CGrp(C) This needs doesnrsquot need an argument as inRemark II23a simply recall that Sp(C) Sp(CGrp(C)) by Proposition II23

II29a Remark mdash Before we move on I would like to point out a subtlety thatrsquosperhaps easy to overlook In Theorem II30 below wersquoll investigate the functor

Sp(minus) Catlexinfin minus Catst

infin

But why is it a functor though My own knee-jerk reaction was ldquowell limits are functorialrdquobut thatrsquos not the problem The problem is that Ω lowastC lowastC is also supposed to befunctorial in C Essentially this leads to the following question() Let CatI-ex

infin sube Catinfin be the (non-full) subcategory spanned by infin-categories with I-shaped limits and functors preserving these Why do the maps limI Fun(I C) C forC isin CatI-ex

infin assemble into a natural transformation

limI

Fun(Iminus) =rArr (minus)

of functors CatI-exinfin Catinfin

There are a few more steps involved to show that Ω is natural too Namely one has toproduce natural transformations (minus)rArr lowast(minus)rArr Fun(Λ2

2 lowast(minus)) but Irsquoll leave that to youTo show () letrsquos first produce the ldquoadjointrdquo transformation const (minus) rArr Fun(Iminus)

Since the adjunctionminustimes I Catinfin Catinfin Fun(Iminus)

extends to Fun(CatI-exinfin Catinfin) by Observation I40 we may equivalently give a transformation

minustimes I rArr (minus) Here the projection to the first factor is an obvious candidate and it is indeedthe correct one

We may view the natural transformation const as a functor CatI-exinfin Ar(Catinfin) Its

image lies in the full subcategory spanned by those [1] Catinfin that represent left-adjointfunctors (because limI is a pointwise right adjoint) But in fact more is true (and thatrsquos

95


the crucial point) Consider objects (L C D) and (Lprime Cprime Dprime) in Ar(Catinfin) as well as amorphism between them which we view as a solid diagram

C Cprime

D DprimeL LprimeR Rprime

If L and Lprime are left adjoints then the dotted arrows exist but the resulting new square onlycommutes up to a natural transformation not up to natural equivalence in general We letArL(Catinfin) be the non-full sub-infin-category spanned by those squares for which the adjointsquare does in fact commute Then CatI-ex

infin Ar(Catinfin) already factors over ArL(Catinfin)since morphisms in CatI-ex

infin preserve I-shaped limitsNow note that under the equivalence Uncocart Ar(Catinfin) simminus Cocart([1]) the non-

full sub-infin-category ArL(Catinfin) corresponds to Bicart([1]) the infin-category of bicartesianfibrations over ∆1 with morphisms that preserve both cocartesian and cartesian edges (weknow from [HCII Proposition XI10] that bicartesian fibrations correspond to adjunctionsand the additional condition unravels to the condition that cartesian edges are preserved aswell) Now we can apply the cartesian straightening Stcart Bicart([1]) Fun([1]opCatinfin)to get a functor CatI-ex

infin Fun([1]opCatinfin) After currying around this gives a map[1]op Fun(CatI-ex

infin Catinfin) which finally provides the desired transformation

II30 Theorem mdash The functor Sp(minus) Catlexinfin Catst

infin is right-adjoint to the inclusionCatst

infin sube Catlexinfin the counit being given by Ωinfin Sp(C) C In particular if C is a stable

infin-category we get a canonical lift

Sp

Cop times C AnΩinfin

HomC

homC

of the HomC functor

Fabian remarks that Catstinfin sube Catlex

infin also has a left adjoint given by colim(lowastC Ω ) butwe probably wonrsquot ever use this More interestingly we can apply Theorem II30 to thestable infin-categories D(R) and K(R) for any ring R

II30a Corollary mdash The Hom functors on D(R) and K(R) have canonical refinementsover Sp given by

homD(R)(CD) H(RHomR(CD)

)and homK(R)(CD) H

(HomR(CD)

)

Proof sketch To prove the left equivalence it suffices to check that we get HomD(R)(CD)after applying Ωinfin (and that everything is functorial in C and D but this will be clear) Sowe compute

ΩinfinH(RHomR(CD)

) K

(RHomR(CD)

) HomAn

(lowastK(RHomR(CD))

) HomD(Z)

(Z[0] RHomR(CD)

) HomD(R)

(Z[0]otimesLZ CD

) HomD(R)(CD)

96


using Cbull(lowast) Z and the derived tensor-Hom adjunction The right equivalence can be provedanalogously but one should first check that Cbull An Dgt0(Z) already factors over Kgt0(Z)and gives rise to the same EilenbergndashMacLane functor K Kgt0(Z) An We havenrsquot donethis but one can simply modify the construction in Very Long Example I56 (details left tothe reader)

Proof of Theorem II30 By the dual of Proposition I61a all we need to show is that thetwo functors Ωinfin Sp(Sp(C)) simminus Sp(C) and Ωinfinlowast Sp(Sp(C)) Sp(C) are equivalences Butwe know that Ωinfin is one by Example II25(a) Also Ωinfinlowast is an equivalence too because it is aldquocoordinate fliprdquo of Ωinfin By that we mean the following Plugging in the definitions we canwrite Sp(Sp(C)) as a limit indexed by Nop times Nop Using the dual of Proposition I42 we canflip that diagram around swapping Ωinfin and Ωinfinlowast which proves the latter is an equivalenceas well

For the ldquoin particularrdquo we need another small argument similar to the proof of Corol-lary II17 since HomC CoptimesC An isnrsquot left exact Every HomC(xminus) C An is howeverhence the adjunction provides us with canonical lifts homC(xminus) Sp for all x isin C whenC is a stable infin-category To show that these combine into a functor homC Cop times C Anconsider the full subcategory Funlex(CAn) sube Fun(CAn) spanned by left exact functors Weclaim that

Ωinfinlowast Funex(CSp) simminus Funlex(CAn)is an equivalence Indeed it becomes one after applying core(minus) by the adjunction wersquove justproved To show that it is an equivalence after core Ar(minus) we can use the same argumentonce we convince ourselves that Ar Funex(CSp) Funex(CAr(Sp)) (use that limits in arrowcategories are pointwise by Lemma I39) and that Ar(Sp) Sp(Ar(An)) (use that Ar(minus)commutes with limits)

Now the Yoneda embedding Cop Fun(CAn) has image in Funlex(CAn) Funex(CSp)and we get our desired homC after currying

II31 Spectra and Prespectra mdash We already know that Sp is complete and Ωinfinminusi SpAn preserves limits for all i see the summary before Corollary II24 We also know fromPropositionDefinition II27 that Sp has finite colimits because it is stable But in fact Spis even cocomplete To see this we will write Sp as a Bousfield localisation of a suitablecocomplete infin-category of prespectra and apply Corollary I61

Prespectra can be introduced in general So let C be an infin-category with finite limitsand (thus) a terminal object lowast isin C We let

PSp(C) sube Fun(N2 lowastC)

be the full subcategory of functors that ldquovanish away from the diagonalrdquo So the objects ofPSp(C) can be pictured as lattice diagrams

lowast lowast X2 middot middot middot

lowast X1 lowast middot middot middot

X0 lowast lowast middot middot middot

97


The diagonal squares induce morphisms ΣXi Xi+1 (provided C has pushouts) or equiva-lently Xi ΩXi+1 If the latter were to be isomorphisms we would recover Sp(C) as wersquollsee in a moment

To construct a fully faithful embedding Sp(C) PSp(C) write N2 = colimnisinN N26(nn)

as an ascending union of finite pieces N26(nn) and consider the diagram

PSp(C) limnisinNop

Fun(N26(nn) lowastC

)

Sp(C) limnisinNop

lowastC

sube

sim

η=lim ηn

We need to explain where the upward-pointing arrow η = lim ηn comes from The morphisms

ηn lowastC minus Fun(N26(nn) lowastC

)send an object X isin lowastC to the right Kan extension of the following functor

lowast middot middot middot lowast

lowast lowast

lowast lowast

lowast middot middot middot lowast X

︸︷︷︸n+1

︸︷︷

︸n+1N2

6(nn)

Note the blank spots along the diagonal except for the (n n)th entry So we right-Kanextend along the inclusion N2

6(nn)r(0 0) (nminus1 nminus1) sube N26(nn) Since this inclusion

is fully faithful Corollary I54 (combined with PropositionDefinition I58(a)) shows that ηnis fully faithful and thus also the limit η over all ηn gives a fully faithful map between thelimits One easily checks that the image of η lands in the full subcategory PSp(C) and thuswe get the desired fully faithful functor Sp(C) PSp(C)

As another consequence of the maps ηn being defined by right Kan extension we obtaincanonical equivalences

HomPSp(C)(XY ) limnisinNop

HomlowastC(Xn Yn)

for all X isin PSp(C) and Y isin Sp(C) (just compute the left-hand side as a Hom anima inlimnisinNop Fun(N2

6(nn) lowastC) and use the universal property of right Kan extension)Note that PSp(C) inherits all limits and colimits from lowastC Also Sp(C) sube PSp(C) is

closed under limits but not colimits To show that Sp(C) is cocomplete (under appropriateconditions) we need to show that it is in fact a Bousfield localisation of PSp(C) This isdone by the following proposition

98


II31a Proposition mdash If lowastC has sequential colimits and Ω lowastC lowastC commutes withthem then the inclusion Sp(C) sube PSp(C) has a left adjoint (minus)sp PSp(C) Sp(C) Onobjects it is given by

Ωinfinminusi(Xsp) colimnisinN

ΩnXn+i

where the transition maps are induced by the morphisms Xi ΩXi+1 that where constructedon the previous page

Before we prove Proposition II31a wersquoll discuss some consequences First to be able toapply it in the case C = An we show

II31b Lemma mdash The loop space functor Ω lowastAn lowastAn for pointed anima commuteswith sequential colimits In particular Sp is a Bousfield localisation of PSp and thereforecocomplete by Corollary I61

Since we are already at it here are two more useful assertions that werenrsquot mentioned inthe lecture(alowast) The functors Ωinfinminusi Sp lowastAn commute with sequential colimits for all i isin Z(blowast) The functors πi Sp Ab for i isin Z and πi lowastAn Set for i gt 0 commute with

sequential colimits In the latter case also recall that sequential colimits in Grp (fori = 1) or Ab (for i gt 2) are computed on underlying sets

Proof First of N is weakly contractible hence the colimits can be formed in An instead(Remark II10a) We can replace any sequential diagram (X1 X2 ) in An bya diagram (X prime1 X prime2 ) Kan such that all transition maps are cofibrations thenTheorem I34 allows us to compute the colimit in Kan instead Now write Ω Flowast((S1 lowast)minus)as the pointed mapping space out of S1 (or any simplicial model for it it doesnrsquot even needto be Kan) and observe that the right-hand side commutes with filtered colimits in Kanbecause S1 is compact This shows the first assertion

Now that we know Ω lowastAn lowastAn commutes with filtered colimits we may applythe summary before Corollary II24 to see that sequential colimits in Sp can be computeddegreewise proving (alowast) Hence in (blowast) it suffices to show the assertion for lowastAn We canwrite πi π0 Flowast((Si lowast)minus) As in the first part Flowast((Si lowast)minus) commutes with filteredcolimits in Kan and π0 Kan Set preserves arbitrary colimits since it is a left adjoint

II31c Remark mdash Both assertions of Lemma II31b remain true if we replace ldquosequentialcolimitsrdquo by ldquofiltered colimitsrdquo where an indexing infin-category I is filtered if every mapK I from a finite simplicial set K extends to a map K I from its cone

In fact the proof can more or less be carried over Filtered infin-categories are weaklycontractible (one can use the filteredness condition to show πi(Exinfin I lowast) = 0 for all i andall basepoints) we can again replace the given diagram by some cofibrant diagram in thecorrect model structure on Fun(C[I] sSet) and then use that Flowast((Si lowast)minus) preserves filteredcolimits by compactness

Similarly to our construction of the embedding Sp(C) sube PSp(C) one can show that wehave an adjunction

Σinfin lowastC PSp(C) ev(00)

if C has finite colimits On objects c isin lowastC (which is all we need by Corollary I32) we defineΣinfinc as follows Recall

PSp(C) sube limnisinNop

Fun(N26(nn) lowastC

)

99


so we may imagine Σinfinc as a sequence ((Σinfinc)0 (Σinfinc)1 (Σinfinc)2 ) The nth piece (Σinfinc)nis then defined to be the left Kan extension of the of the following picture

c lowast middot middot middot lowast

lowast lowast

lowast lowast


︸︷︷︸n+1

︸︷︷

︸

n+1N26(nn)

Again note the blank spots along the diagonal so the Kan extension is along the inclusionN26(nn) r (1 1) (n n) sube N2

6(nn) this timeCombining this with Proposition II31a we see that Ωinfin Sp lowastAn has a left ad-

joint Σinfin = (Σinfin)sp lowastAn Sp Plugging in the explicit formulas for Kan extension(Theorem I52) to compute Σinfin we get

ΩinfinΣinfin(Xx) colimnisinN

ΩnΣn(Xx)

for all (Xx) isin lowastAn The left-hand side is sometimes denoted Q(Xx) but that has nothingto do with the Quillen Q-construction from page 56)

We will denote the composite

S[minus] An (minus)+minusminusminus lowastAn Σinfin

minus Sp

In particular we obtain S = S[lowast] Σinfin(S0 lowast) the legendary sphere spectrum Plugging inthe adjunctions we know gives HomSp(Sminus) HomAn(lowastΩinfinminus) Ωinfin so S represents thefunctor Ωinfin Sp An Using Lemma II31b(b) the homotopy groups of S are given by

πi(S) = colimnisinN

πiΩnΣn(S0 lowast) = colimnisinN

πi+n(Sn lowast)

ie the stable homotopy groups of spheres In contrast to the vastly complicated problem ofcomputing these the homology of S (defined as in II26) is rather boring

II31d Lemma mdash For any abelian group A the homology of S with coefficients in A isgiven by

Hi(S A) =A if i = 00 else

More generally we have Cbull(S[X]) Cbull(X) for all X isin An (where the Cbull(minus) on the left-handside is the one from II26 and the Cbull(minus) on the right-hand side was defined in Very LongExample I56) so in particular

Hlowast(S[X] A

)= Hlowast(XA) and Hlowast

(S[X] A

)= Hminuslowast(XA)

100


Proof Fabian claimed the proof needs a bit of homotopy theory to compute Hlowast(ΩnSn A)in low degrees but I believe there is also a straightforward way that doesnrsquot need anythingBy construction (see II26) we have

CbullS colimiisinN

Cbull(ΩinfinminusiS)[minusi] colimiisinN

colimnisinNgti

Cbull(ΩnminusiSn)[minusi]

where we used that Cbull lowast An D(Z) commutes with colimits Moreover the homologyfunctors Hlowast D(Z) Ab commute with filtered colimits hence

Hlowast(S A) Hlowast(Cbull(S)otimesLZ A

) colim

iisinNcolimnisinNgti

Hlowast+i(ΩnminusiSn A)

colimnisinN

colimiisinN6n


colimnisinN

Hlowast+n(Sn A)

where we used that i = n is terminal in N6n to get the third isomorphism The bottom termclearly vanishes for lowast 6= 0 and equals A for lowast = 0 as we wished to show The additionalassertion follows from the fact that Cbull(minus) An D(Z) and Cbull(S[minus]) An D(Z) bothpreserve colimits and agree on lowast isin An (as we just checked) hence they must be equal byTheorem I51

II32 Corollary (BarattndashPriddyndashQuillen) mdash The functor ev1 CGrp(An) lowastAn hasa left adjoint FreeCGrp lowastAn CGrp(An) given by

FreeCMon(Xx)infin-grp FreeCGrp(Xx) ΩinfinΣinfin(Xx)

Here Ωinfin Spgt0simminus CGrp(An) denotes the inverse of Binfin CGrp(An) simminus Spgt0 which we

recall is an equivalence by Corollary II24

Proof We didnrsquot spell this out in the lecture since there isnrsquot much to do Itrsquos evidentthat Σinfin lowastAn Sp lands in the full subcategory Spgt0 hence itrsquos also a left-adjoint toΩinfin Spgt0 lowastAn After applying the other Ωinfin (ie the inverse of Binfin) we obtain thesecond equivalence For the first equivalence recall that Corollary II21 actually shows thatthe E1-group completion of an Einfin-monoid has a canonical Einfin-group structure

Applying Corollary II32 to the unnumbered example on page 88 we see that

Sinfin-grp =(finite sets bijectionst

)infin-grp FreeCMon(lowast)infin-grp ΩinfinS

an example that Fabian already terrified us with in the introduction In particular we seethat even though FreeGrp(lowast) Z is a free group on a point and commutative it fails ratherdrastically to be the free commutative group on a point

Proof of Proposition II31a By Corollary I32 it suffices to check that Xsp as defined thereis really a spectrum and a left-adjoint object to X for all X isin PSp(C) To check that Xsp isa spectrum we compute

ΩXspi+1 Ω colim

nisinNΩnXn+i+1 colim

nisinNΩn+1Xn+i+1 colim

nisinNΩnXn+i Xsp

i

101


(and all of this is functorial) since Ω commutes with sequential colimits To check that Xsp

is really a left-adjoint object of X we compute

HomSp(C)(Xsp Y ) limiisinNop

HomlowastC (Xspi Yi)

limiisinNop

(colimnisinN

ΩnXn+i Yi

) limiisinNop

HomlowastC(

colimnisinNgti

ΩnminusiXnΩnminusiYn)

limiisinNop

limnisinNop

gti

HomlowastC(ΩnminusiXnΩnminusiYn

) limnisinNop

limjisinN6n

HomlowastC(ΩjXnΩjYn

) limnisinNop

HomlowastC(Xn Yn)

HomPSp(C)(XY )

The first two equivalences are just plugging in definitions For the third equivalence we doan index shift by minusi and use ΩnminusiYn Yi since Y is a spectrum The fourth equivalence isCorollary I50 For the fifth equivalence we substitute j = nminus i and use that limits commute(by the dual of Proposition I42) Since j has ldquonegative slope in irdquo we really take the limitover j isin N6n the missing (minus)op is not a typo (and crucial for the next argument) The sixthequivalence now follows from the fact that j = 0 is initial in 0 isin N6n Finally the seventhequivalence is the formula for HomPSp(C)(XY ) that we deduced just before the formulationof Proposition II31a

Our next goal is to define a more ldquoSegal stylerdquo model for Sp(C) To do this we constructanother infin-category op Fabian remarks that this symbol is not the mathbb-version of theRoman letter F but of the archaic Greek letter Digamma which was the letter after Gammaand Delta in the ancient Greek alphabet so Lop ∆∆op and op fit together nicely Wikipediaclaims however that Digamma was only the sixth letter and that Epsilon came between itand Delta Nevertheless Irsquoll obediently stick to Fabianrsquos notation

II33 Definition mdash Let op sube lowastAn be the smallest sub-infin-category containing S0 andclosed under finite colimits Therersquos a functor Lop op sending 〈n〉 to 〈n〉+ (ie we adda basepoint to the discrete anima 〈n〉) and any partially defined map α 〈m〉〈n〉 to themap α+ 〈m〉+ 〈n〉+ which is α wherever it is defined and sends everything else to thebasepoint

Finally if C is an infin-category with finite limits and (thus) a terminal object lowast isin C wedefine Sp(C) sube Fun( op lowastC) to be the full subcategory of excisive and reduced functors iethose taking pushouts to pullbacks and lowast = 〈0〉+ to lowast isin lowastC We might as well define Sp(C)as the full sub-infin-category of Fun( op C) (rather than Fun( op lowastC)) spanned by reducedand excisive functors since every reduced functor F op C lifts canonically to lowastC

Note that Sp(C) is stable It has finite limits inherited from lowastC and Ω is an equivalencewith inverse induced by precomposition with Σ op op Hence the criterion fromPropositionDefinition II27(a) is fulfilled

102


II34 Proposition mdash With assumptions as in Definition II33 there is a natural equiva-lence of infin-categories

Sp(C) simminus Sp(C)F 7minus

(F (S0) F (S1)

)

Proof sketch There are multiple ways to prove this One can directly show that Sp(C)satisfies the universal property from Theorem II30 Or one can simply check that both

evS0 Sp(Sp(C)

) simminus Sp(C) and evS0lowast Sp(Sp(C)

) simminus Sp(C)

are equivalences and then apply the same reasoning (in particular use Proposition I61a) asin the proof of that theorem

We will sketch a third way by constructing an inverse functor For this we use the followinggeneral construction Let Anfin sube An denote the smallest sub-infin-category containing lowast andclosed under finite colimits Let D be any infin-category with finite limits and finite colimitsFor d isin D we put

X otimes d = colimX

const d and dX = limX

const d

These exist by assumption and are functorial in X and d If (Xx) isin op and D is pointedie has a zero object then we put

(Xx)otimes d = cofib(

colimx

const d colimX

d

)

The inverse functor Sp(C) Sp(C) now sends a spectrum E isin Sp(C) to the functorΩinfin(minusotimesE) op lowastC Note that Sp(C) has finite colimits since it is stable so the functorminusotimes E op Sp(C) is well-defined

Herersquos a proposition that was added in the 16th lecture

II34a Proposition mdash In addition to the assumptions from Definition II33 supposethat lowastC has sequential colimits and that Ω lowastC lowastC commutes with them Then theinclusion Sp(C) sube Funlowast( op lowastC) into the reduced functors has a left adjoint sending areduced functor F op lowastC to

F sp colimnisinN

ΩnF (Σnminus)

The transition maps in this colimit are induced by the natural transformations F rArr ΩF (Σminus)which exist since F is reduced

Proof sketch This essential argument is copied from Fabianrsquos notes [AampHK Chapter IIp 71] As usual by Corollary I32 all we need to do is to verify that the formula aboveindeed defines a left-adjoint object of F If E op lowastC is already reduced and excisivethen E ΩE(Σminus) Using this and a calculation as in the proof of Proposition II31a oneeasily shows Nat(FE) Nat(F sp E)

103


It remains to see however that F sp is indeed excisive (and thatrsquos the hard part of theproof) Since Ω commutes with filtered colimits we get F sp ΩF sp(Σminus) Now consider anarbitrary pushout square in op and extend it to a diagram

A C lowast

B D Q lowast

lowast P ΣA ΣB

lowast ΣC ΣD

The upper left 2times 2-square induces a map F sp(B)timesF sp(D) Fsp(C) ΩF sp(ΣA) F sp(A)

and one can check that this is an inverse to the canonical map in the other direction Thisshows that F sp turns pushouts into pullbacks as required

The functors minusotimes E and E(minus) for spectra that were constructed in the proof of Proposi-tion II34 are interesting in their own right Observe for example that

X otimesHA H(Cbull(X)otimesLZ A

)since one can (and we will) show both sides are colimit-preserving functors An Sp withvalue HA on lowast isin An hence they are equal by Theorem I51 and the fact that Sp is cocompleteThis motivates the following definition

II35 Definition mdash Let E isin Sp and X isin An The homology and cohomology theoryassociated to E are defined by

Elowast(X) = πlowast(X otimes E) = πlowast

(colimX

E)

and Elowast(X) = πlowast(EX) = πlowast

(limXE)

Wersquoll check in Corollary II55b that Elowast(minus) and Elowast(minus) satisfy the EilenbergndashSteenrodaxioms Itrsquos also a common convention to define the cohomology theory associated to E asElowast(X) = πminuslowast(EX) instead However we will see that Fabianrsquos sign convention is superior

Some Foreshadowing mdash We can also extend the tensor product (or smash product)minusotimesminus to all of Sp by defining

E otimes Eprime = colimiisinN

((ΩinfinminusiEprime)otimes E

)[minusi]

(where [minusi] = Ωi is the intrinsic loop functor on Sp) It turns out that this defines a symmetricmonoidal structure on Sp behaving like minusotimesLZ minus on D(Z) although not even symmetry is inany way obvious from the definition Proving this will be the content of the next few lecturesAlong the way (see II53) we will also see that X otimesE and EX can be equivalently describedas S[X]otimes E and homSp(S[X] E) respectively

If M is an Einfin-monoid itrsquos usually very hard to understand its group completionMinfin-grp But we can understand S[Minfin-grp] isin Sp This is still good enough to computeHlowast(Minfin-grp A) or even Elowast(Minfin-grp) which will be the content of the ldquogroup completiontheoremrdquo Theorem III6 Once we have this wersquore in business to compute some K-groups

104

An Interlude on infin-Operads

An Interlude on infin-Operadsinfin-Operads and Symmetric Monoidal infin-Categories


Recall that a morphism in Lop is inert if it is a bijection (where it is defined) and activeif it is defined everywhere This yields two subcategories Lop

act and Lopint Recall also that

symmetric monoidal infin-categories are precisely the objects of CMon(Catinfin) which is a fullsubcategory of Cocart(Lop) via cocartesian unstraightening

II36 Definition mdash A (multicoloured and symmetric) infin-operad (in anima) is a functorp O Lop of infin-categories such that the following conditions hold(a) Every inert in Lop has cocartesian lifts with arbitrary sources (in the invariant sense of

Definition I24(b))(b) The functor O Lop

int Catinfin arising by cocartesian straightening satisfies the Segalcondition That is O0 lowast and On

prodni=1O1 via the Segal maps ρ1 ρn 〈n〉〈1〉

(c) Let x y isin O with p(x) = 〈m〉 and p(y) = 〈n〉 By (b) we can write y (y1 yn) forsome yi isin O1 Then we require that the diagram

HomO(x y)nprodi=1

HomO(x yi)

HomLop(〈m〉〈n〉

) nprodi=1

HomLop(〈m〉〈1〉

)

(ρ1ρn)

p p

(ρ1ρn)

is a pullback Note that ince the bottom arrow is an injective map of discrete sets thetop arrow must be an inclusion of path components

We call a morphism f x y in O inert if it is a p-cocartesian lift of an inert morphism inL We call f active if it is a (not necessarily p-cocartesian) lift of an active morphism Wealso call O1 the underlying infin-category of O

Finally theinfin-category Opinfin ofinfin-operads is the (non-full) sub-infin-category of CatinfinLop

spanned by the infin-operads and by functors which preserve inert morphisms

II37 First Steps with infin-Operads mdash Some trivial examples of infin-operads are givenby

Lopint minus

Lop lowast 〈0〉minusminusLop and id Lop minus

Lop

In general a rich source of further examples is given by symmetric monoidal infin-categories

II37a Symmetric Monoidal infin-Categories as infin-Operads mdash The cocartesianunstraightening induces a functor

(minus)otimes = Uncocart CMon(Catinfin) minus Opinfin

To see this note that conditions Definition II36(a) and (b) are trivially satisfied so we onlyhave to check (c) Wersquove already remarked that the bottom row of the diagram in questionis an injection of discrete anima hence we must show that the top row is an inclusion ofthe corresponding path components So let α 〈m〉〈n〉 be some morphism and denote

105


by HomαO(x y) the path component over α Let x αlowast(x) be a p-cocartesian lift of α and

consider the diagram

HomOn(αlowastx y)nprodi=1

HomO1

((αlowastx)i yi

)

HomαO(x y)

nprodi=1

HomρiαO (x y)

sim

sim sim

That the vertical arrows are equivalences follows easily from the fact that x αlowastx isp-cocartesian and the pullback diagram in Definition I24(a) The top arrow is an equivalenceby the Segal condition Hence the bottom arrow is an equivalence as well which is what wewanted to show Thus (minus)otimes has its image in contained in Opinfin sube Catinfin

Lop as claimedHowever (minus)otimes is not fully faithful and itrsquos a good thing it isnrsquot since it allows us to

define lax symmetric monoidal functors Morphisms in CMon(Catinfin) are strongly symmetricmonoidal functor ie they have to preserve the tensor product In contrast to that wedefine a lax symmetric monoidal functor C D to be a map of infin-operads Cotimes Dotimes Wecan now define infin-categories SymMonCatinfin CMon(Catinfin) and SymMonCatlax

infin sube Opinfin tobe the essential image of (minus)otimes (with the convention that essential images are always fullsub-infin-categories)

A quick reality check As the terminology suggests a lax monoidal functor F Cotimes Dotimescomes with a natural transformation

F (minus)otimesD F (minus) =rArr F (minusotimesC minus)

in Fun(C times CD) This is a consequence of the following random lemma (which wasnrsquot inthe lecture)

II37b Lemma mdash Let p C ∆1 and q D ∆1 be cocartesian fibrations Theirstraightenings correspond to functors Stcocart(p) α C0 C1 and Stcocart(q) β D0 D1in Catinfin Let furthermore functors F0 C0 D0 and F1 C1 D1 be given Then there is apullback diagram

Nat(β F0 F1 α) HomCatinfin∆1(CD)

lowast HomCatinfin(C0D0)timesHomCatinfin(C1D1)

(F0F1)

Proof sketch This should be a consequence of [Lur09] but I think I can also give a direct(but terribly uninvariant) proof A map C D in Catinfin∆1 is the same as a map C[ Din sSet+(∆1)] or more precisely

HomCatinfin∆1(CD) core Fm(∆1)](C[D)

is the core of the simplicial set of marked maps The idea is now to construct a nicemarked simplicial set which is marked equivalent to C[ in the cocartesian model structure onsSet+(∆1)] Let

Cyl[(α) = C[0 times (∆1)[ tC[0times1 C[1 times 1

106


be the mapping cylinder of α and let Cyl(α) be defined analogously but with (∆1)] insteadof (∆1)[ Then Cyl(α) is marked equivalent to C (but not fibrant in the cocartesian modelstructure) Indeed the straightening functor

St+ sSet+(∆1)] minus FunsSet(C[∆1] sSet+)

is a left adjoint hence commutes with colimits so St+(Cyl(α)) is the pushout (and homotopypushout) of (empty C0)rArr (id C0 C0) along (empty C0)rArr (empty C1) and this pushout clearlyagrees with St+(C) α C0 C1 Since St+ is a Quillen equivalence this shows thatCyl(α) and C are indeed marked equivalent

This implies that Cyl[(α) and C[ are marked equivalent too To make this precise eitheruse that the underlying simplicial set of St+(X) only depends on that of X and then analysethe markings on St+(C[) and St+(Cyl[(α)) Or use that Cyl(α) C is the inclusionof a fibrewise left deformation retraction (up to replacing it by a cofibration see [HCIILemma X37]) and observe that in this particular case this stays true if we remove themarkings of the non-equivalence edges

So we may replace C[ by Cyl[(α) Now consider the following inclusions of elements insSet+(∆1)] where marked edges are highlighted in yellow

C0 C1Cyl[(α)

0 1(∆1)

subeC0 C1

C0

Cyl[(α)

C0 times (∆1)]

0 1(∆1)

subeC0 C1

C0

Cyl[(α)

Cyl[(α)C0 times (∆1)]

0 1(∆1)

supeC0 C1

C0


0 1(∆1)

All of these are marked equivalences Hence we may further replace Cyl[(α) by the markedsimplicial set on the right which we denote X The given map F0 C0 D0 extends uniquely(up to contractible choice) to a marked map F C0 times (∆1)] D which automaticallysatisfies F|C0times1 β F0 Hence maps C[ D with given values on C0 and C1 correspondto maps X D with given values on C0 times (∆1)] and C1 which in turn correspond to mapsCyl[(α) D1 with given values on C0 times0 and C1 times1 If you think about this and makeit a little more precise you get the desired pullback diagram

II37c A Recognition Criterion mdash How can we decide whether a given infin-operad is asymmetric monoidal infin-category ie in the essential image of (minus)otimes CMon(Catinfin) OpinfinWe must check that p O Lop has cocartesian lifts of active maps By the Segal conditionand the fact that has p-cocartesian lifts for all inert morphisms it suffices to check that ithas p-cocartesian lifts for the unique active maps fn 〈n〉〈1〉 (we leave it as an exercise towork out a proper argument) This can in turn be reformulated as follows For elements(a1 an) isin On and b isin O1 define the multi-morphism space

HomactO((a1 an) b

)HomO

((a1 an) b

)lowast HomLop

(〈n〉〈1〉

)

fn

107


If a morphism g (a1 an) a over fn is p-cocartesian then the precompositionglowast HomO1(aminus) sim=rArr Homact

O ((a1 an)minus) is an equivalence of functors O1 An Theconverse however is not quite true in the sense that if glowast is an equivalence then g is onlylocally p-cocartesian by the criterion from [HCII Chapter IX p 23] The way to think aboutthis is that ldquoa a1 otimes middot middot middot otimes anrdquo So if p is a locally cocartesian fibration then these tensorproducts exist But for them to be associative we need (after some unravelling) that locallyp-cocartesian lifts compose By [HCII Proposition IX13] this is precisely the condition weneed for the locally cocartesian fibration p to be a cocartesian fibration

Summarising we obtain the following criterion() An infin-operad p O Lop is a symmetric monoidal infin-category iff the following two

conditions hold(a) For all 〈n〉 isin Lop and all (a1 an) isin On there exists an ldquoa a1 otimes middot middot middot otimes anrdquo in

O1 representing the multi-morphism space ie therersquos a map g (a1 an) aover fn 〈n〉〈1〉 such that

g HomO1(aminus) sim=rArr HomactO((a1 an)minus

)is an equivalence

(b) The ldquotensor productrdquo from (a) is associative ie

a1 otimes a2 otimes a3 (a1 otimes a2)otimes a3 a1 otimes (a2 otimes a3)

The moral of this story is that intuitively an infin-operad O is the data you need to makean ldquoinfin-category in which you specify what a map out of a tensor product should berdquo Whethersuch a category exists should equivalent to whether O Cotimes for some C isin CMon(Catinfin)

II37d Dual infin-Operads mdash There is an entirely dual theory of dual infin-operads (beaware thatrsquos not the same thing as infin-cooperads in fact all four combinations of (emptydual)infin-(emptyco)operads exist and are distinct things) A dualinfin-operad is a functor to L = (Lop)op

satisfying Definition II36(a) (b) and (c) for cartesian lifts The resulting infin-category dOpinfinis equivalent to Opinfin via (minus)op Opinfin

simminus dOpinfin Note however that this doesnrsquot preserveunderlying infin-categories Instead it fits into a diagram

Opinfin dOpinfin

Catinfin Catinfin

sim

(minus)1 (minus)1

(minus)op

The image of the cartesian unstraightening Uncart CMon(Catinfin) dOpinfin is denotedSymMonCatoplax

infin thought of as symmetric monoidal infin-categories with oplax monoidalfunctors Just as in II37a an oplax symmetric monoidal functor F Cotimes Dotimes comes with anatural transformation F (minusotimesC minus)rArr F (minus)otimesD F (minus)

II37e Sub-infin-Operads mdash Let O Lop be an infin-operad and D sube O1 a full sub-infin-category Define Dotimes sube O to be the full sub-infin-category spanned by the d (d1 dn)with di isin D Then Dotimes

Lop is an infin-operad with underlying infin-category DSince I didnrsquot find this completely obvious that Dotimes is aninfin-operad herersquos a full argument

The conditions from Definition II36(b) and (c) are immediately inherited from O To checkthe remaining condition (a) we must check that whenever d dprime is a cocartesian lift of an

108


inert morphism in Lop with d isin Dotimes then also dprime isin Dotimes Say d isin Dotimesn and let d (d1 dn)Then the components di isin Dotimes1 are precisely the endpoints of cocartesian lifts d di ofρi 〈n〉〈1〉 Since cocartesian lifts compose (see [HCII Proposition IX5]) we see that thecomponents of dprime are a subset of the components of d hence dprime isin Dotimes as well This showsthat Dotimes is indeed an infin-operad

In the case where O Cotimes is a symetric monoidal infin-category itrsquos naturally to askwhether Dotimes is one as well Fabian gave two criteria for this in the lecture(a) Suppose that dotimesdprime isin D for all d dprime isin D and 1C isin D Then Dotimes is a symmetric monoidal

infin-category and the inclusion D sube C is strongly monoidal If it has moreover a rightadjoint R C D then R can be extended to a map of infin-operads Rotimes Cotimes Dotimes suchthat Rotimes is right-adjoint to the inclusion Dotimes sube Cotimes In particular R has a canonical laxsymmetric monoidal refinement

(b) Suppose that D sube C has a left adjoint L C D with the property that wheneverL(f) L(x) simminus L(y) is an equivalence then L(f otimes idz) L(x otimes z) simminus L(y otimes z) is anequivalence too for all z isin C Then Dotimes is symmetric monoidal and L extends to a mapLotimes Cotimes Dotimes of infin-operads such that Lotimes is left-adjoint to Dotimes sube Cotimes Moreover L isstrongly symmetric monoidal and D sube C is lax symmetric monoidal

Irsquod like to add the following closely related assertion which wasnrsquot mentioned in the lecturealthough wersquoll need it later It roughly says that whether a left adjoint is strongly monoidalcan be checked on objects(clowast) Let Rotimes Dotimes Cotimes be a map of infin-operads If the underlying functor R D C has a

left adjoint L C D satisfying

L(1C) 1D and L(c1 otimesC c2) L(c1)otimesD L(c2)

for all c1 c2 isin C then L extends to a map Lotimes Cotimes Dotimes of infin-operads which isleft-adjoint to Rotimes and strongly symmetric monoidal

The proofs of (a) (b) and (clowast) are pretty straightforward but a bit tedious

Proof of (a) The fact that Dotimes is a symmetric monoidal infin-category and that D sube C is astrongly monoidal functor (ie Dotimes sube Cotimes preserves all cocartesian edges) follows from therecognition criterion sketched in II37c To construct Rotimes we employ Corollary I32 as usualto see that Rotimes can be constructed pointwise So let c isin Cotimes and write c = (c1 cn) Weput Rotimes(c) = (R(c1) R(cn)) and claim that the natural map ηc c Rotimes(c) (induced bythe unit maps ηci ci R(ci) for R) witnesses it as a right-adjoint object of c That is wemust show that

ηclowast HomCotimes(d c) simminus HomDotimes(dRotimes(c)

)is an equivalence for all d isin Dotimes By Definition II36(c) it suffices to show the same forηcilowast HomCotimes(d ci) simminus HomDotimes(dR(ci)) But whether this is an equivalence of animacan be checked on (derived) fibres over HomLop(〈n〉〈1〉) Given α 〈n〉〈1〉 in Lopchoose a cocartesian lift d αlowastd to Dotimes which is also cocartesian in Cotimes by constructionThen the fibres over α are given by Homα

Cotimes(d ci) HomC(αlowastd ci) and HomαDotimes(dR(ci))

HomD(αlowastd ci) hence ηcilowast is indeed an equivalence since R C D is a right adjoint of theinclusion D sube C This shows that Rotimes exist as a map in Catinfin Itrsquos clear from the constructionthat itrsquos also a map in Catinfin

Lop Finally since cocartesian lifts of inert morphisms justldquoforgetrdquo some of the factors they are preserved by Rotimes whence it is a map of infin-operadsThis shows (a)

109


Proof of (b) First observe that the condition on L can be strengthened as follows If weare given equivalences L(fi) L(xi) simminus L(yi) in D for all i = 1 n then also

L(f1 otimes middot middot middot otimes fn) L(x1 otimes middot middot middot otimes xn) simminus L(y1 otimes middot middot middot otimes yn)

is an equivalence To show that Dotimes is a symmetric monoidal infin-category we use therecognition criterion from II37c Given (d1 dn) isin Dotimesn and any d isin D we have

HomactDotimes((d1 dn) d

) Homact

Cotimes((d1 dn) d

) HomC(d1 otimes middot middot middot otimes dn d) HomD

(L(d1 otimes middot middot middot otimes dn) d

)

hence composing (d1 dn) d1otimesmiddot middot middototimesdn with the unit map d1otimesmiddot middot middototimesdn L(d1otimesmiddot middot middototimesdn)in Cotimes gives a locally cocartesian lift of fn 〈n〉〈1〉 to Dotimes which verifies the first conditionof the recognition criterion For the second condition we must check L(L(x otimes y) otimes z

)

L(xotimes y otimes z) But xotimes y L(xotimes y) becomes an equivalence after applying L as L2 Lhence this follows from the assumption on L This proves that Dotimes is symmetric monoidal

To construct Lotimes we proceed as in (a) and define it pointwise via Lotimes(c1 cn) =(L(c1) L(cn)) By the same tricks as above showing that this is indeed a right-adjointobject of (c1 cn) reduces to showing

HomαCotimes((c1 cn) d

) Homα

Dotimes((L(c1) L(cn)) d

)for all d isin D and all α 〈n〉〈1〉 Let (c1 cn) c be a cocartesian lift of α to Cotimes ThenHomα

Cotimes((c1 cn) d) HomC(c d) and also c ci1 otimes middot middot middot otimes cim where i1 im isin 〈n〉 arethe elements where α is defined The assumption on L then implies

L(c) L(L(ci1)otimes middot middot middot otimes L(cim)

)

because each cij L(cij ) becomes an equivalence after applying L Hence a cocartesianlift of α to Dotimes is given by (L(c1) L(cn)) L(c) and Homα

Dotimes((L(c1) L(cn)) d) HomD(L(c) d) Now HomC(c d) HomD(L(c) d) because L C D is left-adjoint to theinclusion D sube C which finally shows the desired equivalence Itrsquos clear from the constructionthat Lotimes preserves cocartesian edges whence L is indeed strongly symmetric monoidal

Proof of (clowast) The condition on L implies L(c1 otimesC middot middot middot otimesC cn) L(c1) otimesD middot middot middot otimesD L(cn) forall c1 cn isin C We define Lotimes pointwise as in (b) By the same arguments as given thereall we need to check is that

HomαCotimes((c1 cn) R(d)

) Homα


)for all c1 cn isin C d isin D and all α 〈n〉〈1〉 If α is defined at i1 im isin 〈n〉 then(c1 cn) ci1 otimesC middot middot middot otimesC cim is a cocartesian lift of α to Cotimes hence


) HomC

(ci1 otimesC middot middot middot otimesC cim R(d)

)

The same argument for Dotimes together with the condition on L implies

HomαDotimes((L(c1) L(cn)) d

) HomD

(L(c1 otimesC middot middot middot otimesC cn) d

)

whence we are done by the fact that L and R are adjoints This shows that Lotimes is an adjointof Rotimes as desired Itrsquos straightforward to check that Lotimes is a map over Lop and preservescocartesian edges hence L is indeed strongly symmetric monoidal

110


Fabian also mentioned that in general there is an equivalence of infin-categories

oplax monoidal left adjoints C D lax monoidal right adjoints D C

This is not trivial since lax and oplax structures are defined in different fibration pictures Thefirst written proofs appeared in November 2020 by Rune Haugseng [Hau20] and independentlyby Fabian Sil and Joost Nuiten [HLN20]

II37f Derived Tensor Products mdash Letrsquos apply II37e(a) to get a symmetric monoidalstructure on the derived category D(R) of a commutative ring R One can check that theKan-enriched category Ch(R) from Example I15(e) is symmetric monoidal under minusotimesR minusas a Kan-enriched category Hence its coherent nerve K(R) is a symmetric monoidal infin-category Now recall from Example I62(c) that D(R) is a Bousfield localisation of K(R)More precisely D(R) can be embedded fully faithfully into K(R) as the full sub-infin-categoryD(R)K-proj sube K(R) of K-projective complexes ie those C such that HomK(R)(Cminus) invertsquasi-isomorphisms Clearly the tensor unit R[0] is K-projective (as is any bounded belowdegree-wise projective complex) and if C and D are K-projective then so is C otimesR D asHomK(R)(C otimesR Dminus) HomK(R)(CHomR(Dminus)) inverts quasi-isomorphisms too

We may thus apply II37e(a) to obtain a symmetric monoidal structure minus otimesLR minus onD(R) The localisation functor K(R) D(R) (ldquotaking K-projective resolutionsrdquo) is laxsymmetric monoidal in the sense explained in II37a In particular there are canonical mapsC otimesLR D C otimesR D for all CD isin K(R)

The Cocartesian and Cartesian Symmetric Monoidal StructuresIf an infin-category C has finite products (and hence in particular a terminal object) thentaking products should define a symmetric monoidal structure on C called the cartesianmonoidal structure Similarly on an infin-category with finite coproducts there should be acocartesian monoidal structure Constructing these guys will be our next immediate goal Ofcourse all of this is part of our greater plan to construct the symmetric monoidal structureon Sp

While Lurie writes down explicit simplicial sets to construct the desired infin-operads wewill cheat a bit and use complete Segal spaces instead That this works is based on thefollowing lemma

II38 Lemma mdash Let E be an infin-category For a presheaf X isin P(E) consider the slicecategory EX with respect to the Yoneda embedding Y E E P (E) and X P(E) (in thenotation of Lemma I53a this would have been Y EX instead) The canonical functor

P(EX) simminus P(E)X

constructed as the unique colimit-preserving extension of Y E EX P(E)X via Theo-rem I51 is an equivalence of infin-categories

Proof Before we start observe that P(E)X is indeed cocomplete since one immediatelychecks that it inherits colimits from P(E) We will use the following criterion() Let F C D be a functor from a small infin-category to a cocomplete infin-category Let

| |F P(C) D SingF denote the adjunction resulting from Theorem I51 Then | |Fis an equivalence if the following three conditions hold

111


(a) F is fully faithful(b) HomD(F (c)minus) D An commutes with colimits for all c isin C(c) SingF D P(C) is conservative

We prove () first Condition (b) implies that SingF commutes with colimits Indeed thiscan be checked pointwise by Theorem I19(b) since there always is a natural transformationcolimI SingF rArr SingF colimI Moreover colimits in P(C) are computed pointwise Pluggingin the explicit description of SingF that Corollary I55 offers us we must thus check thatcolimI HomD(F (c) di) HomD(F (c) colimI di) for all c isin C and di isin D This is (b)

Now since F is fully faithful the unit idP(C) rArr SingF | |F is an equivalence on repre-sentables Indeed the right-hand side sends c isin C to HomD(F minus F (c)) HomC(minus c) byCorollary I55 again Now both sides preserve colimits and every presheaf on C can be writtenas a colimit of representatives by Lemma I53a hence the unit is an equivalence everywhereBy the triangle identities and the fact that SingF is conservative by (c) this implies that thecounit |SingF minus|F rArr idD is an equivalence as well This finishes the proof of ()

Now to check that () is applicable for C = EX and D = P(E)X We already arguedthat P(E)X is cocomplete Itrsquos easily checked that EX P(E)X is fully faithful since itrsquosinduced by the fully faithful Yoneda embedding so (a) holds To check (c) let α F F prime

be a morphism in P(E)X such that αlowast HomP(EX)(minus F ) HomP(EX)(minus F prime) is anequivalence on representable presheaves We must check that α is an equivalence itself But ifαlowast is an equivalence on representables then the Yoneda lemma implies that α F (e) F prime(e)is an equivalence for all e isin E (except those e isin E for which X(e) = empty since these arenrsquotcontained in the image of EX E but in that case we have F (e) = empty = F prime(e) too since Fand F prime come with a map to X) hence α is an equivalence of presheaves

Finally we have to check (b) Given a representable object HomE(eminus) X and anarbitrary object F X of P(E)X the Yoneda lemma and [HCII Corollary VIII6] providea pullback square

HomP(E)X(

HomE(eminus) F)

HomP(E)(

HomE(eminus) F)

F (e)

lowast HomP(E)(

HomE(eminus) X)

X(e)

sim

sim

Clearly F (e) commutes with colimits in F because colimits in functor categories are computedpointwise (Lemma I39) Hence it suffices to check that colimits in An commute with pullbacksIn other words if f K L is any morphism of anima then flowast AnL AnK preservescolimits But Right(K) AnK since every morphism to K can be factored into a rightanodyne and a right fibration and likewise for L so we are done since we verified inLemma I42a that flowast Right(L) Right(K) has a right adjoint flowast

II38a mdash Applying Lemma II38 to E = ∆∆ and X = Nr(C) isin sAn for some infin-categoryC we obtain P(∆∆Nr(C)) sAnNr(C) We will usually abbreviate the left-hand side asP(∆∆C) If you think about it this makes a lot of sense Objects of ∆∆Nr(C) are maps ∆n Nr(C) of simplicial anima But Nr Catinfin sAn is fully faithful by TheoremDefinition I64so we may equivalently think of maps [n] C

We can now specify objects of (or functors to) CatinfinC as objects of (or functors to)P(∆∆C) and then compose with

P(∆∆C) sAnNr(C) asscatminusminusminusminus CatinfinC

112


This will be done quite a number of times on the next few pages

Lecture 1622nd Dec 2020

II38b mdash To check that an object on the left is complete Segal we will check the relativeSegal condition A map Y X in sAn is called relative Segal if the diagram

HomsAn(∆n Y ) HomsAn(In Y )

HomsAn(∆n X) HomsAn(In X)

is a pullback in An If X is a Segal anima then the lower row is an equivalence Thus Y is aSegal anima too if Y X is relative Segal

Likewise a morphism Y X is called relatively complete Segal if it is relative Segal andthe diagram

HomsAn(∆0 Y ) HomsAn(∆3(∆02 cup∆13) Y

)HomsAn(∆0 X) HomsAn

(∆3(∆02 cup∆13) X

)

Again if this holds and X is a complete Segal anima then so is Y (here we use a slightreformulation of TheoremDefinition I64(b))

With that letrsquos dive into cocartesian monoidal structure

II39 Construction mdash Let Lopt sube 〈1〉

Lop be the full subcategory spanned by theldquodefined mapsrdquo (for every n therersquos a unique nowhere-defined map 〈1〉〈n〉 this is theonly one we exclude) For any infin-category C define pt Ct Lop via II38a applied to thepresheaf FCt ∆∆Lop An which is given by

FCt([n] Lop) HomCatinfin

([n]timesLop

Lopt C

)on objects (and turned into a functor of infin-categories in the obvious way)

II40 Proposition mdash With notation as in Construction II39(a) The presheaf FCt defines a complete Segal anima over Nr(Lop) via Lemma II38(b) pt Ct Lop is always an infin-operad with underlying category C(c) An active morphism (a1 an) b in Ct is locally pt-cocartesian iff the maps ai b

exhibit b as the coproduct of the ai(d) In particular pt Ct Lop is a symmetric monoidal category iff C has finite coproducts

Proof sketch The proof of (a) was left as an exercise so herersquos what I figured out To showthat FCt is Segal consider the diagram

HomsAn(∆n FCt) HomsAn(In FCt)

HomsAn(∆nNr(Lop)

)HomsAn

(InNr(Lop)

)sim

113


The bottom row is an equivalence since Nr(Lop) is Segal To show that the top row is anequivalence as well it suffices to check that it induces equivalences on all (derived) fibresUnravelling what that means (using the computation of Hom anima in slice categories from[HCII Corollary VIII6]) we need to check that

HomsAnNr(Lop)(∆n FCt) simminus HomsAnNr(Lop)(In FCt) nprodi=1

HomsAnNr(Lop)(∆1 FCt)

is an equivalence for all choices of ∆n Nr(Lop) Recall that sAnNr(Lop) P(∆∆Lop)by Lemma II38 Using this together with the Yoneda lemma and the definition of FCt ourassertion translates into

HomCatinfin([n]timesLop

Lopt C

) simminusnprodi=1

HomCatinfin([1]timesLop

Lopt C

)

In other words we must prove that [n] timesLopLopt is an iterated pushout of [1] timesLop

Lopt in

Catinfin One way to do this is to simply check by hand that

In timesN(Lop) N(Lopt ) minus ∆n timesN(Lop) N(Lop

t )

is an inner anodyne map of simplicial sets This shows that FCt is Segal Completeness canbe shown analogously whence we have proved (a)

To prove (b) we must investigate the fibres of pt Ct Lop So fix some 〈n〉 isin Nr(Lop)Since FCt is complete Segal by (a) we get FCt Nr(asscatFCt) Nr(Ct) Because Nr

preserves limits we may as well compute the corresponding fibre of FCt Nr(Lop) andthen take asscat(minus) again To do this we claim that the following diagram commutes

P(∆∆Lop) sAnNr(Lop)

sAn

sim

evaluation atconstant maps

minustimesNr(Lop)〈n〉

The right diagonal arrow sends a simplicial anima X over Nr(Lop) to X timesNr(Lop) 〈n〉 (ieto the fibre wersquore interested in) The left diagonal arrow sends a presheaf F ∆∆Lop Anto a simplicial anima Y defined by Yk F (const 〈n〉 [k] Lop) To show commutativitywe may use Theorem I51 whence it suffices to check that both ways around the diagramagree on representable presheaves and preserve colimits The former is straightforward tocheck (although it took me some time to wrap my head around this) For the latter itrsquosclear that the left diagonal arrow preserves colimits and for the right diagonal arrow we canuse that colimits in anima commute with pullbacks (as seen in the proof of Lemma II38)and that colimits and pullbacks in sAn are computed pointwise

Letrsquos apply our newfound knowledge to pt Ct Lop Itrsquos straightforward to check thatthe diagram

[k]times 〈n〉 Lopt

[k] Lop

pr1

const 〈n〉

114


is a pullback diagram In particular the fibre of pt Ct Lop over 〈n〉 is the associatedcategory of a simplicial anima Y given by

Yk FCt(

const 〈n〉 [k] Lop) HomCatinfin([k]timesLop

Lopt C

) HomCatinfin

([k]times 〈n〉 C

) HomCatinfin

([k] Cn

)Up to checking some functoriality stuff this shows Y Nr(Cn) so the fibre of pt Ct Lop

over 〈n〉 is Cn This shows that pt satisfies Definition II36(b)Two more conditions are to check Let x y isin Ct and let α pt(x) pt(y) be some

morphism in Lop Then the fibre HomαCt(x y) = HomCt(x y) timesHomLop (pt(x)pt(y)) α of

HomαCt(x y) over α has a particularly simple description If we think of α as a functor

α [1] Lop then HomαCt(x y) fits into a pullback

HomαCt(x y) HomCatinfin

([1]timesLop

Lopt C

)lowast HomCatinfin

([0]timesLop

Lopt C

)timesHomCatinfin

([0]timesLop

Lopt C

) (d1d0)

(xy)

To prove this use Yonedarsquos lemma the computation of Hom anima in slice categories from[HCII Corollary VIII6] and the fact that Nr Catinfin sAn is fully faithful to compute


Lopt C

) FCt(α [1] Lop) HomP(∆∆Lop)

(α [1] minus Lop FCt

) HomsAnNr(Lop)

(Nr(α) ∆1 Nr(Lop) FCt

) HomsAn(∆1 FCt)timesHomsAn(∆1Nr(Lop)) Nr(α) HomCatinfin

([1] Ct

)timesHomCatinfin ([1]

Lop) α

With this and an analogous computation for HomCatinfin([0]timesLopLopt C) the above pullback

square is straightforward to verify Letrsquos use this to check that pt Ct Lop has cocartesianlifts for inert morphisms So let α 〈m〉〈n〉 be inert and x isin Ct a lift of 〈m〉 Then xcorresponds to a map [0]timesLop

Lopt C and a lift of α corresponds to a map [1]timesLop

Lopt C

where the pullback is formed using α [1] Lop So we need to solve a lifting problem

[0]timesLopLopt C

[1]timesLopLopt

d1

in such a way that the solution is a pt-cocartesian lift of α Since α is inert ie an isomorphismwhere it is defined [1] timesLop

Lopt consists of a disjoint union of copies of [1] (their number

corresponds to the number of elements of 〈m〉 where α is defined) and [0] (correspondingto the elements of 〈m〉 where α isnrsquot) We define the desired lift [1] timesLop

Lopt C to be

degenerate on every copy of [1] noting that it is already defined on their starting points andon the additional copies of [0]

To show that this gives a cocartesian lift we must check the condition from Defini-tion I24(a) ie that

HomCt(y z) simminus HomCt(x z)timesHomLop (〈n〉〈k〉) HomLop(〈m〉〈k〉

)

115


is an isomorphism for all z isin Ct and pt(z) = 〈k〉 This can be checked on (derived) fibresover any β isin HomLop(〈m〉〈k〉) The fibre of the left-hand side is Homβ

Ct(y z) The fibreof the right-hand side is the same as that of HomCt(x z) HomLop(〈n〉〈k〉) over β αwhich is Homβα

Ct (x z) That these two are equivalent is straightforward to check from theconstructions using the pullback diagram above This proves that pt Ct Lop satisfiesDefinition II36(a) The condition from Definition II36(c) can be checked in the same wayThis proves (b)

For (c) it suffices to check the recognition criterion from II37c By definition we have

HomactCt((a1 an)minus

) Homfn

Ct((a1 an)minus

)

The pullback [1]timesLopLopt formed using fn [1] Lop is precisely the cone over a discrete set

with n elements Using this and the pullback diagram above one easily verifies

HomfnCt((a1 an)minus

)

nprodi=1

HomC(aiminus)

Hence any representing object of HomactCt ((a1 an)minus) is a coproduct of the ai whence

(c) follows Part (d) is an immediate consequence of (c) and the discussion in II37c ascoproducts are clearly associative

Proposition II40 allows us to define the cocartesian monoidal structure on aninfin-categorywith finite coproducts To construct the cartesian monoidal structure once can consider((Cop)t)op isin dOpinfin If C has finite products so Cop has finite coproducts then the dual infin-operad ((Cop)t)op defines a symmetric monoidalinfin-category and we can define ptimes Ctimes Lop

viaCtimes = Uncocart (Stcart((Cop)t)op)

II41 Construction mdash There is also a direct construction of ptimes Ctimes Lop but this isa bit tricky which perhaps isnrsquot very surprising since infin-operads would like to specify mapsout of a tensor product whereas products would like to have maps into them specified Wedefine a 1-category Lop

times as follows Its objects are pairs (n S) where S sube 〈n〉 is a subsetand morphisms (n S) (mT ) are given by morphisms α 〈n〉〈m〉 in Lop such thatαminus1(T ) = S Let ptimes Ctimes Lop be given by II38a applied to the presheaf FCtimes∆∆Lop Anwhich is in turn given by

FCtimes([n] Lop) HomCatinfin

([n]timesLop

Loptimes C

)

The same analysis as in the proof of Proposition II40(a) shows that the fibre of ptimes Ctimes Lop

over 〈n〉 is given byCtimesn Fun

((power set of 〈n〉)op C

)

Now let Ctimes be the full subcategory of Ctimes which is fibrewise spanned by those functorsF (power set of 〈n〉)op C satisfying F (S)

prodiisinS F (i)

II42 Proposition mdash With notation as in Construction II39(a) The presheaf FCtimes defines a complete Segal anima via Lemma II38(b) ptimes Ctimes Lop is always a cocartesian fibration with Ctimes1 C (but not necessarily an

infin-operad)

116


(c) If C has finite products then ptimes Ctimes Lop satisfies the Segal condition hence it is asymmetric monoidal infin-category with underlying infin-category C Moreover in this casewe have

HomactCtimes((a1 an)minus

) HomCtimes(a1 times middot middot middot times anminus)

Proof Similar to Proposition II40 and omitted Fabianrsquos script [AampHK Proposition II42]has some details

Monoids over an infin-Operad mdash If p O Lop and pprime Oprime Lop are infin-operads letFunOpinfin(OOprime) be the full sub-infin-category of FunLop(OOprime) spanned by the infin-operad mapsHere FunLop(OOprime) is given by the pullback

FunLop(OOprime) Fun(OOprime)

lowast Fun(OLop)

pprimelowast

p

as usual If C has products then FunOpinfin(Lop Ctimes) CMon(C) In fact this can be statedin much greater generality which wersquoll do in Theorem II43 below Define the infin-categoryof O-monoids to be the full subcateroy OMon(C) sube Fun(O C) spanned by those functorsF O C satisfying the Segal condition That is the maps (a1 an) ai in O aresupposed to induce equivalences F (a1 an) simminus

prodni=1 F (ai)

Also note that there is a section Lop Loptimes sending 〈n〉 7 (n 〈n〉) It induces canonical

maps HomCatinfin([n] timesLopLoptimes C) HomCatinfin([n] timesLop

Lop C) HomCatinfin([n] C) hence amap FCtimes Nr(C) of simplicial anima and therefore a map Ctimes C

II43 Theorem mdash If C has finite products then the map Ctimes C constructed aboveinduces an equivalence

FunOpinfin(O Ctimes) simminus OMon(C)

is an equivalence If C has finite coproducts and HomactO (empty x) lowast for all x isin O (so that O is

unital see LemmaDefinition II45a below) then the restriction

FunOpinfin(O Ct) simminus Fun(O1 C)

is an equivalence as well

Proof See [HA] The first part follows from Proposition 2425 the second part fromPropositions 24316 and 23111

Algebras over infin-OperadsTheorem II43 is supposed to motivate the following general definition

II44 Definition mdash (a) If O is an infin-operad and Cotimes is a symmetric monoidal structureon an infin-category C then we put

AlgO(Cotimes) = FunOpinfin(O Cotimes)

and call this the infin-category of O-algebras in C If the symmetric monoidal structureon C is clear from the context wersquoll often write AlgO(C) instead

117


(b) More generally let Oprime be another another infin-operad with a map α Oprime O and letC be an O-monoidal infin-category ie an object of OMon(Catinfin) Again we denote by(minus)otimes OMon(Catinfin) OpinfinO the functor induced by cocartesian unstraighteningThen AlgOprimeO(Cotimes) is defined as the pullback

AlgOprimeO(Cotimes) FunOpinfin(Oprime Cotimes)

lowast FunOpinfin(OprimeO)

α

This recovers part (a) as the special caseAlgO(Cotimes) AlgOComm(Cotimes) where we denoteComm = id Lop

Lop (more on that in Example II45(c))

II45 Example mdash ldquoMy first algebras over infin-operadsrdquo

(a) If O is the infin-operad lowast 〈0〉minusminus Lop then AlgO(Cotimes) is easily identified with the fibre of Cotimesover 〈0〉 hence AlgO(Cotimes) lowast

(b) If O is Lopint

Lop then AlgO(Cotimes) C Hence thisinfin-operad will be denoted O = TrivWersquoll give a sketch of why this is true One easily finds

FunLop(Lop

int Cotimes) FunLop

int

(Lopint Cotimes timesLop

Lopint)


FunOpinfin(Triv Cotimes

) Γcocart

(Cotimes timesLop

Lopint)

where Γcocart is defined as in Proposition I36 In particular the right hand side isequivalent to lim(Stcocart(Cotimes timesLop

Lopint) Lop

int Catinfin) so we are to show that thislimit is C

To see this note that by definition of Lopint its opoosite category L

int can be viewedas the category of finite sets 〈n〉 with injective maps between them Hence there is aL

int Catinfin sending 〈n〉 to itself considered as a discrete infin-category Let U Lopint

be its cartesian unstraightening so that the fibre Un = U timesLopint〈n〉 is isomorphic to

〈n〉 Then Cotimesn Cn limUn C Hence we may apply the dual of Proposition I42 tosee that the limit wersquore looking for is given as limU C But now itrsquos straightforward tocheck that the unique point of the fibre U1 is an initial object of U hence limU C Cas required

(c) If O is id Lop Lop then we define AlgO(Cotimes) = CMon(Cotimes) If C has finite products

and Cotimes Ctimes is theinfin-operad giving the cartesian monoidal structure then this notationis consistent with our previous definition of CMon(C) by Theorem II43 Consequentlythe infin-operad id Lop

Lop will be denoted O = Comm (or sometimes Einfin)(d) In view of (c) a natural question is whether there exists an infin-operad Assoc satisfying

AlgAssoc(Ctimes) Mon(C) This is indeed true and we will construct Assoc soon (to beprecise in a future version of Example II45(d))

(e) Similarly there is an infin-operad LMod encoding pairs (AM) of associative monoids Aand an object M on which A acts from the left LMod will also be constructed in afuture version of II45(e)

118


(f) If p Cotimes Lop is any symmetric monoidal infin-category then we would expect thatthe tensor unit 1C isin C gives rise to an object in CMon(Cotimes) This is indeed true andworks in greater generality To give this result the space it deserves we outsource it toLemmaDefinition II45a below

II45a LemmaDefinition mdash For any infin-operad O the unique object lowastO isin O0 isterminal in O Moreover the following conditions are equivalent(a) Homact

O (empty x) lowast for all x isin O1(b) The unique object lowastO isin O0 is initial in OSuch infin-Operads are called unital If O is unital and p Cotimes O an O-monoidal infin-category(see Definition II44(b)) then the tensor unit of Cotimes ie the unique element of Stcocart(p)(lowastO)canonically gives rise to an initial object isin AlgOO(Cotimes) In the case O Comm theevaluation of at 〈1〉 isin Comm is indeed the usual tensor unit 1C isin C

Proof The construction of isin AlgOO(Cotimes) was done in the lecture (in the special caseO Comm) but the rest wasnrsquot

The fact that lowastO is terminal follows immediately from Definition II36(c) For theequivalence observe that there is only one map 〈0〉〈1〉 in Lop hence Homact

O (empty x) HomO(lowastO x) for all x isin O1 This immediately implies (b) rArr (a) To see (a) rArr (b) notethat HomO(lowastO x) lowast for all x isin O1 already suffices to show the same for all x isin O byDefinition II36(c)

Now let O unital infin-operad and p Cotimes O an O-monoidal infin-category To constructisin AlgOO(Cotimes) sube FunO(O Cotimes) our goal is to produce a canonical infin-operad map

O Cotimes

Oid p

We will choose it to be a map between cocartesian fibrations even After straighteninga map between cocartesian fibrations over O corresponds to a natural transformation inFun(OCatinfin) between const lowast and the functor Stcocart(p) O Catinfin Since lowastO isin O isinitial we get a natural transformation const lowastO rArr idO in Fun(OO) hence a naturaltransformation

const lowast const Stcocart(p)(lowastO)rArr Stcocart(p)

in Fun(OCatinfin) as required In the special case O Comm (to which our considerationsapply since 〈0〉 isin Comm0 is initial in Comm) the evaluation of at 〈1〉 is the endpoint of acocartesian lift of 〈0〉〈1〉 hence indeed 1C isin C

For simplicity let me only prove that is initial the special case O Comm sincethe general case works just the same but would require some new notation So consideranother commutative algebra A isin AlgComm(Cotimes) corresponding to a functor A Lop Cotimesthat preserves inerts and satisfies p A idLop Letrsquos also write An = A(〈n〉) Since Apreserves inerts we have An (A1 A1) via the Segal maps Also n (1C 1C) byconstruction We compute

HomAlgComm(Cotimes)( A) HomFunLop (CommCotimes)( A) Nat( A)timesNat(idLop idLop ) id

119


using that AlgComm(Cotimes) sube FunLop(Comm Cotimes) is a full sub-infin-category and that taking Homanima commutes with limits of the infin-categories they are taken in Now plug Corollary I49into the right-hand side and use the fact that limits commute (ie the dual of Proposition I42)to obtain

HomAlgComm(Cotimes)( A) lim(α 〈m〉〈n〉)isinTwAr(

Lop)HomCotimes( m An)timesHomLop (〈m〉〈n〉) α

lim(α 〈m〉〈n〉)isinTwAr(Lop)

HomαCotimes( m An)


HomαCotimes( m An)


HomαCn( n An)


HomC(1C A1)n

In the second-last step we used that m αlowast m n is a p-cocartesian lift of α Now thetarget projection t TwAr(Lop) Lop is final Indeed this can be shown as in the proof ofCorollary I50 (where we considered the source projection s TwAr(I) Iop but wersquoll seethat things dualize in the right way) t is cocartesian since (s t) TwAr(Lop) L

timesLop is

a left fibration and pr2 Ltimes Lop Lop is cocartesian Hence we may apply Exercise I50a

to get |t〈n〉| |tminus1〈n〉| But the fibres of t are given by tminus1〈n〉 L〈n〉 hence weakly

contractible as required by Definition I44 So t is indeed finalThe upshot is that we may now write

HomAlgComm(Cotimes)( A) lim〈n〉isin

LopHomC(1C A1)n HomC(1C A1)0 lowast

since 〈0〉 isin Lop is initial Wersquore done

Day ConvolutionWersquoll use the following construction due to Saul Glasman [Gla16]

II46 Construction mdash Let Cotimes be a symmetric monoidal infin-category and O an infin-operad Define F(CotimesO) Lop via II38a applied to the presheaf F ∆∆Lop An which isgiven by

F([n] Lop) HomCatinfin

Lop([n]timesLop CotimesO

)Then F defines a complete Segal anima over Nr(Lop) Indeed this can be shown as inthe proof of Proposition II42 the only difference being that one can no longer show byhand that In timesNr(Lop) Cotimes ∆n timesNr(Lop) Cotimes is a Joyal equivalence instead one can forexample use that pullback along cocartesian fibrations preserves Joyal equivalences by [HTTProposition 3313] or a stronger version of [HCII Theorem IX17])

Once we know F is complete Segal we may apply the same method as in the proof ofProposition II40 to compute

F(CotimesO)n Fun(Cotimesn On) Fun(CnOn1 )

Let the day convolution Day(CotimesO) sube F(CotimesO) be the (non-full) sub-infin-category spannedby those functors F Cotimesn On which split up to equivalence as products F F1 times middot middot middot times Fnand those morphisms that ldquosplit similarlyrdquo To make it more precise which morphisms we

120


allow let α 〈m〉〈n〉 be a morphism in Lop Regarding α as a map α [1] Lop we formthe pullbacks Cotimesα = [1]timesLop Cotimes and Oα = [1]timesLop O Let finally αi denote the active mapαminus1(i) 〈1〉 for all i isin 〈n〉 Then there are decompositions

Cotimesα CS times[1] Cotimesα1times[1] middot middot middot times[1] Cotimesαn and Oα OS1 times[1] Oα1 times[1] middot middot middot times[1] Oαn

where S sube 〈m〉 is the subset where α is undefined By definition a morphism in F(CotimesO)covering α is a map Cotimesα O in Catinfin

Lop or equivalently a map Cotimesα Oα in Catinfin[1]Such a morphism is permitted into Day(CotimesO) iff it respects the above decompositions Notethat this condition is always satisfied if α is active

II46a An Error in the Construction mdash I believe I mightrsquove found an error inConstruction II46 But before I explain what I think goes wrong let me already tell youthat even if there is indeed an error in [Gla16] it wonrsquot have any consequences neither forour lecture nor for mathematics as a whole as wersquoll see in II46b below

The issue is that I donrsquot see why Day(CotimesO) would satisfy the Segal condition and Ieven think that I have an argument why Day(CotimesO)n Fun(CO1)n is wrong in generalBy definition Day(CotimesO)n sube Fun(CnOn1 ) is spanned by those functors F which are up toequivalence of the form F1 times middot middot middot times Fn along with those transformations η F rArr G whichare up to equivalence of the form η1 times middot middot middot times ηn for some ηi Fi rArr Gi We can write

Fun(CnOn1 ) nprodi=1

Fun(CnO1)

which induces a similar decomposition

Day(CotimesO)n nprodi=1

Day(CotimesO)ni

where Day(CotimesO)ni sube Fun(CnO1) is spanned by those functors and those transforma-tions which factor up to equivalence over the projection pri Cn C For the Segalcondition to hold we would like that prlowasti Fun(CO1) Fun(CnO1) is an equivalenceonto Day(CotimesO)ni Now the inclusion Day(CotimesO)ni sube Fun(CnO1) of simplicial sets isan isofibration Indeed lifting against Λ2

1 ∆2 holds since Day(CotimesO)ni is closed undercompositions in Fun(CnO1) lifting against higher horn inclusions is trivial and lifting ofequivalences is due to the fact that Day(CotimesO)ni is closed under equivalences in Fun(CnO1)Moreover since Day(CotimesO)ni sube Fun(CnO1) is injective as a map of simplicial sets we geta pullback diagram

Day(CotimesO)ni Day(CotimesO)ni

Day(CotimesO)ni Fun(CnO1)

of simplicial sets But its legs are isofibrations hence this is also a pullback diagram inCatinfin Thus for prlowasti Fun(CO1) Fun(CnO1) to be an equivalence onto Day(CotimesO)niwe would need that

Fun(CO1) Fun(CO1)

Fun(CO1) Fun(CnO1)

prlowastiprlowasti

121


is a pullback diagram in Catinfin as well But this canrsquot be true in general For example if theabove diagram is a pullback for all infin-operads O then Corollary I50 together with Yonedarsquoslemma imply that

Cn C

C C

pri

pri

is a pushout (I believe not every infin-category D is of the form D O1 for some infin-operad Obut for the Yoneda argument to work it suffices to have a fully faithful functor D O1 forevery infin-category D which we always have just equip P(D) with the cocartesian monoidalstructure from Proposition II40) This pushout condition canrsquot be true in general Taken = 2 and let C be your favourite non-discrete symmetric monoidal anima Then the pushoutcondition would imply Σ(C) lowast

In a nutshell Day(CotimesO)ni sube Fun(CnO1) is spanned by those functors and transforma-tions which factor up to equivalence over pri Cn C whereas Fun(CO1) sube Fun(CnO1)is spanned by those which do so on the nose It seems obvious that these two (non-full)sub-infin-categories are equivalent but I think itrsquos not true

II46b Why we donrsquot need to worry mdash First of all Lurie [HA Subsection 226]has his own construction of Day convolution So infin-operad theory is not in jeopardy SecondBastiaan has pointed out that the issues from II46a donrsquot occur in the case where C isweakly contractible Indeed we claim() If C is weakly contractible then const D Fun(CD) is fully faithful for all infin-

categories DApplying () inductively to D Fun(CiO1) for i = 1 nminus 1 shows that Fun(CO1) subeFun(CnO1) is fully faithful and then everything works To show () we may assume thatD has small colimits since we can always replace D by P(D) Then const has a left adjointcolimC Fun(CD) D and the counit colimC const d simminus d is an equivalence for all d isin Dbecause the indexing infin-category C is weakly contractible Thus const is fully faithful byPropositionDefinition I58(a)

Now for all symmetric monoidal infin-categories Cotimes to which we are ever going to applythe Day convolution construction in this lecture C has an initial or terminal object so wewonrsquot get into trouble

II47 Proposition mdash Let Cotimes be a symmetric monoidal infin-category such that C is weaklycontractible (of course this condition is only added because of II46b) let O be an infin-operadand let Day(CotimesO) be constructed as in Construction II46 Then(a) Day(CotimesO) Lop is an infin-operad with underlying infin-category Fun(CO1)(b) If q Cotimes Cprimeotimes is an infin-operad map or p O Oprime is one so are

qlowast Day(CprimeotimesO) minus Day(CotimesO) and plowast Day(CotimesO) minus Day(CotimesOprime)

(c) If O Dotimes where D is a cocomplete symmetric monoidal category such that minusotimesD minuscommutes with colimits in each variable then Day(CotimesDotimes) represents a symmetricmonoidal structure on Fun(CD) with tensor product

(F1 otimesDay F2)(c) colimdotimesCdprimec

(F1(d)otimesD F2(dprime)

)

122


To be really precise the colimit on the right-hand side is taken over the slice category(minusotimesC minus)c where minusotimesC minus C times C C denotes the tensor product on C Moreover theunderlying infin-category of Day(CotimesDotimes) is again cocomplete and minus otimesDay minus commuteswith colimits in either variable

(d) In the situation of (c) let q C Cprime be a strongly symmetric monoidal functor Thenqlowast has a left adjoint (left Kan extension) which is strongly symmetric monoidal

Proof sketch Let α 〈m〉〈n〉 and define Cotimesα Oα as above Then HomCatinfinLop(Cotimesα O)

HomCatinfin[1](Cotimesα O) and likewise HomCatinfinLop(CotimesmO) HomCatinfin(CotimesmOm) Using this

and adapting the arguments from the proof sketch of Proposition II40 one shows that wehave pullback squares

HomαF(CotimesO)(FG) HomCatinfin[1](Cotimesα Oα)

lowast HomCatinfin(CotimesmOm)timesHomCatinfin(Cotimesn On)

(d1d0)

(FG)

for all FG isin F(CotimesO) with images 〈m〉 and 〈n〉 respectively (but we do not claim thatF(CotimesO) is an infin-operad) If we restrict each factor to the path components spanned bythose functors that meet the requirements from Construction II46 we obtain a similarpullback diagram for Homα

Day(CotimesO)(FG) Also note that we can plug in and pull out thedecompositions of Oα Om and On to make this pullback more confusing

To prove (a) we must provide cocartesian lifts of inert morphisms If α is inert thenlifting α to F(CotimesO) means (by unravelling the pullback diagram above) we need to solve alifting problem

Cotimesm Om

Cotimesα Oα

d1

Additionally we want that the solution lies in Day(CotimesO) and is cocartesian Since α is inertwe have Cotimesα CS times Cn times [1] and Oα OS1 timesOn1 times [1] where again S sube 〈m〉 is the subsetwhere α isnrsquot defined Moreover the top arrow in the diagram above represents an element ofDay(CotimesO)m hence a functor F Cm Om1 which decomposes as F F1times middot middot middot timesFm Thuswe can choose the dashed arrow in the diagram above to be Fi on those factors with i isin Sand to be Fi times id[1] on those factors with i isin S This clearly defines a map in Day(CotimesO)and to show that itrsquos cocartesian one can adapt the arguments from the proof sketch ofProposition II40 and use the description of Homα

Day(CotimesO)(minusminus) given above This showsthat Day(CotimesO) Lop satisfies Definition II36(a) For whether or not the Segal conditionfrom Definition II36(b) holds see our discussion in II46a and II46b Finally the conditionfrom Definition II36(c) can again be checked using our description of Homα

Day(CotimesO)(minusminus)This finishes part (a)

For (b) we get functors qlowast F(CprimeotimesO) F(CotimesO) and plowast F(CotimesO) F(CotimesOprime) sincethe formation of the presheaf F in Construction II46 is clearly functorial in Cotimes and OSo all we need to check is that qlowast and plowast preserve the decomposition conditions fromConstruction II46 as well as cocartesian lifts of inert morphisms But both things are clearfrom our constructions

123


Now letrsquos prove (c) Let FG isin F(CotimesO) lie over 〈m〉 and 〈n〉 Then F and G definefunctors F Cm Dm and G Cn Dn The following claim will be used several timesduring the proof() Let α 〈m〉〈n〉 be an active morphism By cocartesian unstraightening α induces

morphisms αC Cm Cn and αD Dm Dn Then

HomαF(CotimesO)(FG) Nat(αD FG αC)

Claim () follows easily from Lemma II37b and the pullback square at the beginning ofthe proof Also note that if F and G belong to Day(CotimesDotimes) then

HomαF(CotimesDotimes)(FG) Homα

Day(CotimesDotimes)(FG)

since Day(CotimesO) sube F(CotimesO) contains all lifts of the active morphism α as noted at the endof Construction II46 So () gives a way of computing Homα

Day(CotimesDotimes)(FG)Wersquoll use the recognition criterion from II37c to show that Day(CotimesDotimes) is symmetric

monoidal As usual let fn 〈n〉〈1〉 denote the unique active morphism Let F Cn Dnsuch that F F1timesmiddot middot middottimesFn and let G C D The maps (fn)C Cn C and (fn)D Dn Dare given by the n-fold tensor products otimesnC Cn C and otimesnD Dn D Thus () shows

HomactDay(CotimesO)

((F1 Fn) G

) Homfn

Day(CotimesO)(FG) Nat(otimesnD FG otimesnC)

Hence a lift F F prime of fn is locally cocartesian iff F prime C D is the left Kan extension ofotimesnD F Cn D along otimesnC Cn C Since D is cocomplete these Kan extensions exist byTheorem I52 In particular for F1 F2 C D we have that F1 otimesDay F2 is the left Kanextension of F1 otimesD F2 C2 D along otimes2

C C2 C which shows that the desired formulaholds true (but we donrsquot know yet that minusotimesDay minus defines a symmetric monoidal structure)

So far we havenrsquot used thatminusotimesDminus commutes with colimits in either variable but wersquoll needit to show that the locally cocartesian edges compose so that Day(CotimesO) Lop is not onlylocally cocartesian but honestly cocartesian by [HCII Proposition IX13] To show this it willsuffice to check that minusotimesDay minus is associative and unital For associativity it suffices to checkthat F1otimesD (F2otimesDayF3) C2 D is the left Kan extension of F1otimesD F2otimesD F3 C3 D alongidC timesotimes2

C C3 C2 (and likewise for (F1otimesDayF2)otimesDF3) since then the fact that Kan extensioncomposes will do the rest Given (c cprime) isin C2 we have (idC timesotimes2

C)(c cprime) idC c times otimes2Ccprime

Now c is cofinal in idC c hence c times otimes2Ccprime is cofinal in idC c times otimes2

Ccprime The value at

(c cprime) of the left Kan extension in question can therefore be computed as

colim(cdotimesCdprime)(ccprime)

(F1(c)otimesD F2(d)otimesD F3(dprime)

) F1(c)otimesD colim

dotimesCdprimecprime


) F1(c)otimesD (F2 otimesDay F3)(cprime)

as desired Here where we finally used that minusotimesD minus commutes with colimits In the exactsame way one shows F otimesDay 1Day F where 1Day C D is given by the left Kan extensionof 1D lowast D along 1C lowast C This concludes the proof that Day(CotimesDotimes) is symmetricmonoidal The additional assertion is clear since left Kan extension preserves colimits andcolimits in functor categories are pointwise This proves (c)

For (d) note that qlowast Fun(CprimeD) Fun(CD) has a left adjoint q given by left Kanextension Using the definition of minusotimesDayminus via left Kan extension and the fact that left Kanextension composes itrsquos straightforward to check q(F1 otimesDay F2) q(F1)otimesDay q(F2) andq(1Day) 1Day Hence II37e(clowast) can be applied to finish the proof

124


Stabilisation of infin-Operads and the Tensor Product on SpectraII48 Definition mdash Let O be an infin-operad and F I O1 be a diagram Then acolimitlimit of F is called operadic if for every x1 xn y isin O1 we have

HomactO

((colimI

F x2 xn

) y) limIop

HomactO((F (minus) x2 xn) y

)Homact

O

((x1 xn) lim

IF) limI

HomactO((x1 xn) F (minus)

)An infin-operad is pointedsemi-additiveadditivestable if O1 is and the required limits andcolimits are operadic Spelling this out explicitly we obtain(a) O is pointed if O1 has a zero object which is also both operadic initial and operadic

terminal We define Oplowastinfin sube Opinfin as the (non-full) sub-infin-category spanned by infin-operads with an operadic terminal object and maps preserving these We denote byOpptinfin sube Oplowastinfin the full sub-infin-category spanned by pointed infin-operads

(b) O is semi-additiveadditive if O1 is and all finite products and coproducts are operadicWe define Optimesinfin sube Opinfin as the (non-full) sub-infin-category spanned by infin-operads withfinite operadic products and maps preserving these We denote by Opsemi-add

infin Opaddinfin sube

Oplowastinfin the full sub-infin-categories spanned by semi-additive and additive infin-operadsrespectively

(c) O is stable if O1 is and all finite limits and colimits are operadic We define Oplexinfin sube Opinfin

as the (non-full) sub-infin-category spanned by infin-operads with finite operadic limits andmaps preserving these We denote by Opst

infin sube Oplowastinfin the full sub-infin-category spanned bystable infin-operads

II48a Remark mdash If O Cotimes is a symmetric monoidal infin-category then all limitsare operadic because Homact

Cotimes((x1 xn)minus) HomC(x1 otimes middot middot middot otimes xnminus) and a colimit isoperadic iff colimI(F (minus)otimes x) (colimI F )otimes x for all x isin C

In general a limit in O1 is operadic iff it is also a limit in O Indeed the limit conditionin O means that

limIop

HomαO

((x1 xn) lim

IF) limI

HomαO((x1 xn) F (minus)

)for every α 〈n〉〈1〉 in Lop The special case α = fn shows that if the limit overF I O1 sube O happens to lie inside O1 then it is operadic Conversely any α can befactored into α = fm ι where ι 〈n〉〈m〉 is inert If x ιlowastx is a cocartesian lift of ιstarting at x = (x1 xn) then Homα

O(xminus) HomactO (ιlowastxminus) hence operadic limits are

also limits in OGiven the limit case one might expect that a colimit is operadic iff (colimI F x2 xn)

is a colimit in O over (F (minus) x2 xn) I O for all x2 xn isin O1 But this is falseunless I is weakly contractible The reason why this fails is rather stupid Similar to abovewe must check the colimit condition on Homα

O for any α 〈n〉〈1〉 If we factor α = fm ιthen it may happen that ι isnrsquot defined at 1 isin 〈n〉 and in this case the colimit conditionreads Homact

O (ιlowastx y) limIop HomactO (ιlowastx y) which only holds in general if I is weakly

contractibleII49 Construction mdash Given aninfin-operad O with sufficient operadic limits and colimitsour goal is to construct new infin-operads

OOpinfinlowast CMonOpinfin(O) CGrpOpinfin(O) and SpOpinfin(O)

125


with underlying infin-categories lowastO1 CMon(O1) CGrp(O1) and Sp(O1) respectively Wewill tackle all four constructions at once Consider the following infin-operads

[1]min minusLop (Lop)times minus Lop and ( op)and minus Lop

The first one is given by the symmetric monoidal structure on [1] obtained by taking theminimum (in particular 1 isin [1] is the tensor unit) The second one is the cartesian symmetricmonoidal structure (as introduced in Proposition II42) on Lop The third one is given by thesmash product We didnrsquot construct this yet but it will arise as follows Starting from thecartesian symmetric monoidal structure Antimes on An the smash product will be defined as itsldquopointificationrdquo (lowastAn)and = (Antimes)Opinfin

lowast Then ( op)and sube (lowastAn)and can be defined via II37e(a)So to be super precise we would need to prove the assertions about (minus)Opinfin

lowast in Theorem II50below before we could even formulate anything about SpOpinfin(minus) but obviously thatrsquos notwhat wersquore going to do We will also check below Remark II51a that the smash productcoincides with the one you know from topology

If O isin Oplowastinfin we letOOpinfinlowast sube Day

([1]minO

)denote the infin-operad obtained via II37e from the full sub-infin-category (lowastO1) sube Fun([1]O1)spanned by those functors F [1] O1 with F (0) = lowast Similarly if O isin Optimesinfin we let

CGrpOpinfin(O) sube CMonOpinfin(O) sube Day((Lop)timesO

)be induced by CGrp(O1) sube CMon(O1) sube Fun(LopO1) ie the full sub-infin-categories offunctors satisfying the Segal condition andmdashin the case of CGrp(O1)mdashadditionally thecondition from Definition II4 Finally for O isin Oplex

infin we let

SpOpinfin(O) sube Day(( op)andO

)be given by the full sub-infin-category Sp(O1) sube Fun( opO1) spanned by reduced and excisivefunctors (see Definition II33)

These infin-operads come equipped with canonical maps

O minus OOpinfinlowast minus CMonOpinfin(O) minus CGrpOpinfin(O) Ωinfin

minus SpOpinfin(O)

Except for the fourth one which comes from the inclusion CMon(O1) supe CGrp(O1) thesemaps are induced via functoriality of Day convolution (see Proposition II47(b)) by theinfin-operad maps

Comm minus [1]min minus (Lop)times minus ( op)and All of these are induced by strongly monoidal functors between the underlying categories Thefirst one comes from 1 lowast [1] the second one from [1] Lop sending i 7 〈i〉 for i = 0 1and the third one comes from the functor (minus)+ Lop op introduced in Definition II33

With these constructions we can now formulate a striking theorem of Thomas Nikolaus

II50 Theorem ([Nik16 Theorem 411 and Theorem 57]) mdash The infin-operads OOpinfinlowast

CMonOpinfin(O) CGrpOpinfin(O) and SpOpinfin(O) from Construction II49 are pointed semi-additive additive and stable respectively Moreover the respective constructions assembleinto functors

(minus)Opinfinlowast Oplowastinfin minus Oppt

infin CMonOpinfin(minus) Optimesinfin minus Opsemi-addinfin

CGrpOpinfin(minus) Optimesinfin minus Opaddinfin SpOpinfin(minus) Oplex

infin minus Opstinfin

126


each of which is right-adjoint to the respective fully faithful inclusion in the other direction(ie they are all right Bousfield localisations)

Proof sketch Once we know that our four functors take values in the correct infin-categorieswe can proceed as in Theorem II19 and Theorem II30 to show that they are right Bousfieldlocalisations So we only need to prove the first part of the theorem

The most subtle part is to show that the required colimits in the infin-operads OOpinfinlowast

CMonOpinfin(O) CGrpOpinfin(O) and SpOpinfin(O) are actually operadic Thomas Nikolaus doesthis with a beautiful trick Letrsquos first assume O Cotimes where C is a symmetric monoidalinfin-category for which Proposition II51 below is applicable Then this proposition says thatin fact all small colimits are operadic and wersquore happy In general one can find a fully faithfulmap ofinfin-operads i O Cotimes such that Proposition II51 is applicable to Cotimes and i1 O1 Cpreserves the required finite limits see [Nik16 Proposition 27] and Remark II51a Therequired finite colimits in lowastO1 CMon(O1) CGrp(O1) and Sp(O1) are also finite colimitsin lowastC CMon(C) CGrp(C) and Sp(C) since each of them (zero objects finite coproducts insemi-additive infin-categories finite colimits in stable infin-categories) can be built from finitelimits which i1 O1 C preserves Hence they are operadic and we win

In general neither of the constructions from Construction II49 gives a symmetricmonoidal category even when O itself is symmetric monoidal The following propositiongives a criterion for the symmetric monoidal structure to be inherited

II51 Proposition mdash Suppose O Cotimes is a symmetric monoidal infin-categories withall small operadic colimits (this is a short way of saying that C is cocomplete and minusotimesC minuscommutes with colimits in either variable see Remark II48a) Suppose furthermore thatthe inclusions

lowastC sube Ar(C) CMon(C) sube Fun(Lop C) CGrp(C) sube Fun(Lop C) Sp(C) sube Fun( op C)

have left adjoints Then (Cotimes)Opinfinlowast CMonOpinfin(Cotimes) CGrpOpinfin(Cotimes) and SpOpinfin(Cotimes) are

symmetric monoidal infin-categories again and have all small operadic colimits Moreoverthere are maps infin-operads

(minus)+ Cotimes minus (Cotimes)Opinfinlowast FreeCMon Cotimes minus CMonOpinfin(Cotimes)

FreeCGrp(minus) Cotimes minus CGrpOpinfin(Cotimes) Σinfin Cotimes minus SpOpinfin(Cotimes)

which are strongly monoidal and left-adjoint to the respective maps in the other directionconstructed at the end of Construction II49 Similar assertions hold for other left adjointsamong these For instance in the case Cotimes = Antimes we get a fully faithful strongly monoidalleft adjoint Binfin CGrpOpinfin(Antimes) SpOpinfin(Antimes) of Ωinfin

Proof We only prove the assertions about Sp since the arguments can be copied verbatimfor the other cases By Proposition II47(c) Day(( op)and Cotimes) is symmetric monoidal andhas all small operadic colimits Since Sp(C) sube Fun( op C) has a left adjoint we mayapply II37e(b) to see that SpOpinfin(Cotimes) is again symmetric monoidal and its inclusion intoDay(( op)and Cotimes) has a left adjoint Day(( op)and Cotimes) SpOpinfin(Cotimes) which is strongly monoidalTogether with our computation of colimits in Bousfield localisations (Corollary I61) thisshows that SpOpinfin(Cotimes) has all small operadic colimits

127


Moreover to produce the strongly monoidal left adjoint Σinfin Cotimes SpOpinfin(Cotimes) it sufficesto construct a stronglyy monoidal left adjoint of the canonical map

Day(( op)and C

)minus Day(Comm Cotimes) Cotimes

induced by the strongly monoidal map Comm ( op)and But this is precisely what Proposi-tion II47(d) does The additional assertion about Binfin can be shown in the same way

II51a Remark mdash Proposition II51 is always applicable when Cotimes Day(DotimesAntimes)for some small symmetric monoidal infin-category D That is whenever C P(D) equippedwith the Day convolution structure Since Cotimes in the proof of Theorem II50 can be alwayschosen in this way (see the proof of [Nik16 Proposition 27]) this is all we need

So letrsquos prove that the assumptions of Proposition II51 are really satisfied for C P(D)First of all P(D) has all small operadic colimits by Proposition II47(c) so only the requiredleft adjoints have to be constructed Letrsquos first do the case of Sp By Proposition II34athe inclusion Sp(C) sube Funlowast( op lowastC) into the reduced functors has a left adjoint wheneverlowastC has sequential colimits and Ω lowastC lowastC commutes with them This is satisfied forlowastAn as we checked after Proposition II31a hence also for const lowastP(D) Fun(Dop lowastAn)since everything is pointwise Now Funlowast( op lowastC) Funlowast( op C) since every reducedfunctor F op C canonically lifts to lowastC so it suffices to construct a left adjoint ofFunlowast( op C) sube Fun( op C) If C has pushouts we can send an arbitrary F op C tocofib(F (lowast) F (minus)) op C to obtain the desired left adjoint Again P(D) clearly haspushouts whence we are done

Next we treat CMon Denote i Lop61

Lop If C has small limits then restrictionand right Kan extension along i gives a functor ilowastilowast Funlowast(

Lop C) CMon(C) This isa left Bousfield localisation which can be easily checked using Proposition I61a and anassertion similar to claim () from the proof of Proposition II16 Composing this with a leftBousfield localisation Fun(Lop C) Funlowast(

Lop C) as above gives the required localisationFun(Lop C) CMon(C) in the CMon case This immediately implies the CGrp case sinceCGrp(An) sube CMon(An) has a left adjoint by Corollary II21 hence the same is true forCGrp(P(D)) Fun(DCGrp(An)) sube Fun(DCMon(An)) CMon(P(D))

Irsquoll leave the final case ie showing that lowastC sube Ar(C) has a left adjoint in the specialcase C P(D) to you

II51b Example mdash Letrsquos discuss two special cases of Proposition II51 The first onewasnrsquot in the lecture but it will show up later(a) We claim that CGrpOpinfin(Settimes) Abotimes is the symmetric monoidal structure on the

1-category of abelian groups given by the usual tensor product Indeed let AB isin Abthought of as functors AB Lop Set Let A otimes B denote their Day convolutionDay((Lop)timesSettimes) By Proposition II47(c) we can compute it as the left Kan extensionof

Lop timesLop Settimes Set Set

Lop

timesLop

AtimesB timesSet

AotimesB

By the universal property of left Kan extension this means that

HomAb(AotimesBC) Nat(AotimesBC) Nat (timesSet (AtimesB) C timesLop)

128

Some Brave New Algebra

for all abelian groups C Unravelling we find that the right-hand side is given byZ-bilinear maps AtimesB C hence AotimesB is indeed the usual tensor product

The inclusion Set sube An induces inclusions Ab CGrp(Set) sube CGrp(An) andAbotimes CGrpOpinfin(Settimes) sube CGrpOpinfin(An) which are fully faithful again To see thisnote that the left adjoint π0 An Set induces left adjoints for each of the inclusionsand the condition that the counit transformation is an equivalence is inherited henceso is fully faithfulness by PropositionDefinition I58(a)

(b) Proposition II51 shows that lowastAn carries a pointed symmetric monoidal structurecalled the smash product via (lowastAn)and = (Antimes)Opinfin

lowast We wish to compute minus and minusexplicitly to show that it coincides with the smash product we know from topology

So let XY isin lowastAn sube Ar(An) Letrsquos first compute the Day convolution X timesDay Y Using the explicit formula from Proposition II47(c) we obtain

(X timesDay Y )(0) colim

lowast times lowast lowast times Y

X times lowast

X or Y



X times lowast X times Y

X times Y Since the left adjoint of lowastAn sube Ar(An) sends any T [1] An to cofib(T (0) T (1))we obtain X and Y cofib(X and Y X times Y ) after unravelling what the proofs ofProposition II51 and II37e(b) actually do Hence X and Y is what we would expect

II52 Definition mdash The tensor product (or traditionally smash product) of spectra isdefined by the infin-operad Spotimes = SpOpinfin(Antimes) which is a symmetric monoidal infin-categoryby Proposition II51

Some Brave New AlgebraLecture 177th Jan 2021

The term ldquobrave new algebrardquo roughly refers to the idea that Sp should behave like thecategory of abelian groups and more generally to notion of Einfin-ring spectra and modules overthem which should behave very much like ordinary commutative rings and modules Makingthis precise and proving it rigorously takes Lurie about 400 pages in [HA] Of course therersquosno way to even remotely cover this in this course So be warned that proofs will becomemore and more sparse towards the end of this final subsection of Chapter II NeverthelessFabian made it his goal to introduce all relevant definitions and theorems in the lecture andeven more material in the lecture notes [AampHK Chapter II pp 104ndash132] to give us at leasta guide for how to read Lurie Starting from Chapter III we will then just assume thathomological algebra works with Einfin-ring spectra just as it does for ordinary rings

II53 First Steps with the Tensor Product on Sp mdash Recall the symmetric monoidalstructure on Sp from Definition II52

The tensor unit 1Sp is given by the sphere spectrum S The quickest way to see this isthat S[minus] Antimes Spotimes is strongly symmetric monoidal because so are both its constituents

129


(minus)+ Antimes (lowastAn)and and Σinfin (lowastAn)and Spotimes by Proposition II51 and the tensor unitlowast isin An is sent to S

As an easy consequence we obtain

X otimes E S[X]otimes E and (Xx)otimes E Σinfin(Xx)otimes E

for all (Xx) isin lowastAn and E isin Sp where X otimes E and (Xx) otimes E are defined as in theproof of Proposition II34 Indeed for the left-hand equivalence itrsquos enough to show thatboth minus otimes E An Sp and S[minus] otimes E An Sp preserve colimits and that they agree onlowast isin An for then they must agree by Theorem I51 That both sides agree on lowast preciselysays that S is the tensor unit which we just checked That minus otimes E preserves colimits isclear by inspection whereas for S[minus]otimesE it follows from the fact that the tensor product onspectra commutes with colimits in either variable by Proposition II51 Hence the left-handequivalence holds The right-hand one follows as (Xx)otimesE cofib(xotimesE X otimesE) andΣinfin(Xx) cofib(S[x] S[X])

This can be used to derive a general formula for the tensor product of spectra Anyspectrum E can be written as

E colimnisinN

(Σinfin(ΩinfinminusnE)

)[minusn]

Indeed using the (ΣinfinΩinfin)-adjunction we can calculate

HomSp

(colimnisinN


)[minusn] Eprime

) limnisinNop

HomSp(Σinfin(ΩinfinminusnE) Eprime[n]

) limnisinNop

HomlowastAn(ΩinfinminusnEΩinfinminusnEprime)

and the right-hand side is HomSp(EEprime) since Hom anima in limits are limits of Hom animaHence the claimed equivalence follows from Yonedarsquos lemma

Together with the fact that the tensor product of spectra commutes with arbitrarycolimits this shows the general formula

E otimes Eprime colimnisinN


)[minusn]otimes Eprime colim

nisinN

(ΩinfinminusnE otimes Eprime

)[minusn]

as was already announced after Definition II35 Note that the tensor products on theright-hand side are no longer those in Sp but defined as in the proof of Proposition II34 sothis formula givesmdashat least in theorymdasha way to compute tensor products of spectra

In real life though the colimits in question are seldom computable To give an exampleof how complicated even simple tensor products can become write

TorSi (EEprime) = πlowast(E otimes Eprime) and ExtiS(EEprime) πi homSp(EEprime)

and consider TorSlowast(HFp HFp) = Aorp and ExtlowastS(HFp HFp) = Ap called the (dual and non-dual) Steenrod algebra These are infinite-dimensional graded Fp-algebras but completelycomputed (albeit not by the formula above) For example Aor2 = F2 [xi | i isin N] withdeg(xi) = 2i minus 1 is a polynomial ring in infinitely many variables

To show that our formula isnrsquot completely useless Irsquod like to use it to prove the tensorndashHom adjunction This wasnrsquot discussed in the lecture but Fabian has also written out aproof in the official lecture notes [AampHK Remarks II53(iv)]

130


II53a Lemma mdash For all EEprime Eprimeprime isin Sp we have an equivalence

homSp(E otimes Eprime Eprimeprime) homSp(EhomSp(Eprime Eprimeprime)

)It is functorial in all three variables ie an equivalence in Fun(Spop times Spop times SpSp)

Proof Letrsquos first check it in the special case Eprime S[X] Note that homSp(Sminus) Sp Spis the identity functor Indeed wersquove already verified HomSp(Sminus) Ωinfin (see beforeLemma II31d) and Ωinfin Sp An clearly lifts to idSp via Theorem II30 Hence the functors

homSp(E otimes S[minus] Eprimeprime)op homSp(EhomSp(S[minus] Eprimeprime)

)op An minus Spop

agree on lowast isin An and both clearly preserve colimits hence they must agree by Theorem I51This settles the special case Eprime S[X] More generally this shows

homSp(minusotimes S[minus]minus)op homSp(minushomSp(S[minus]minus))op

Indeed we can interpret both sides as functors SpoptimesSp Fun(AnSpop) Upon inspectionthey land in the full sub-infin-category Funcolim(AnSpop) of colimit-preserving functors ButFuncolim(AnSpop) Fun(lowastSpop) and if we interpret the two functors above as functorsSpop times Sp Fun(lowastSpop) then they clearly agree since both minus otimes S and homSp(Sminus) areequivalent to the identity The case Eprime Σinfin(Xx) and more generally the equivalence offunctors

homSp(minusotimes Σinfinminusminus) homSp(minushomSp(Σinfinminusminus)

)in Fun(Spop times (lowastAn)op times SpSp) immediately follows

Finally we can write Eprime colimnisinN(Σinfin(ΩinfinminusnEprime))[minusn] and then the formula from II53reduces the general case to the case Eprime Σinfin(Xx) This also proves the full functorialitystatement since we can similarly write idSp colimnisinN(Σinfin(Ωinfinminusnminus))[minusn]

Einfin-Ring SpectraII54 Definition and Examples mdash For any symmetric monoidal infin-operad we defineCAlg(Cotimes) = AlgComm(Cotimes) We already introduced the notation CMon(Cotimes) for this inExample II45(c) but nevermind In the special case Cotimes Spotimes we put

CAlg = CAlg(Spotimes) = AlgComm(Spotimes)

and call this the infin-category of Einfin-ring spectra Any Einfin-ring spectrum has an underlyingspectrum given by its image under ev〈1〉 CAlg FunOpinfin(CommSpotimes) Sp Occasionally(and abusingly) we wonrsquot distinguish between an Einfin-ring spectrum and its underlyingspectrum

In our picture where Sp corresponds to the 1-category Ab of abelian groups CAlg takesthe place of the 1-category CRing of commutative rings (which makes a lot of sense asCRing CAlg(Abotimes) by unwinding of definitions) Now come ldquomy first Einfin-ring spectrardquo(a) Recall from II53 that S[minus] Antimes Spotimes is strongly monoidal and that CAlg(Antimes)

CMon(An) by Theorem II43 Hence S[minus] upgrades to a functor

S[minus] CMon(An) minus CAlg

For M isin CMon(An) we think of S[M ] isin CAlg as the ldquospherical monoid ring on Mrdquo

131


(b) Every ordinary commutative ring defines an Einfin-ring spectrum and more generally sodoes every commutative differential graded algebra (CGDA in the following) We willdiscuss this in detail in Lemma II55 Corollary II55a and the space in between

Let me also mention another example that came only up at a later point in the course butfits here very well(clowast) Let R be a commutative ring and k(R) its algebraic K-theory space from Defini-

tion II12a We know from Example II21a(a) that it is an Einfin-group But Binfink(R)even has a canonical refinement as an Einfin-ring spectrum To see this put

CMon(Grpd)otimes = CMonOpinfin(Grpdtimes) and CMon(An)otimes = CMonOpinfin(Antimes)

The tensor product minusotimesR minus turns Proj(R) which is already an object of CMon(Grpd)via minus oplus minus into an object of CAlg(CMon(Grpd)otimes) sube CAlg(CMon(An)otimes) The func-tors (minus)infin-grp CMon(An)otimes CGrp(An)otimes and Binfin CGrp(An)otimes Spotimes are stronglymonoidal by Proposition II51 (the former is hidden under ldquosimilar assertions holdfor other left adjoints among theserdquo) hence Binfink(R) Binfin Proj(R)infin-grp is indeed anelement of CAlg

Letrsquos discuss now how to turn ordinary commutative rings and CDGAs into Einfin-ring spectraas promised in (b)

II55 Lemma mdash The EilenbergndashMacLane functor H D(Z) Sp refines to a map

H D(Z)otimesLZ minus Spotimes

of infin-operad ie it is a lax symmetric monoidal functor More generally for arbitrarycommutative rings R the functor Cbull(minus R) = Cbull(minus) otimesLZ R Sp D(R) has a left adjointH D(R) Sp which refines to a lax symmetric functor

H D(R)otimesLR minus Spotimes

Proof This isnrsquot anything special about D(R) but an instance of the following more generalprinciple(lowast) If Cotimes is a symmetric monoidal infin-category then HomC(1C minus) C An has a canonical

lax symmetric monoidal refinement If Cotimes is stable as an infin-operad then the same istrue for the enriched Hom functor homC(1C minus) C Sp

The derived base change minusotimesLZ R D(Z) D(R) has the forgetful functor D(R) D(Z) asa left adjoint hence Cbull(minus R) has indeed a right adjoint which deserves to be named H aswell since it really does nothing but forgetting to D(Z) and then applying H D(Z) SpCorollary II30a implies H homD(R)(R[0]minus) Since R[0] isin D(R) is the tensor unit wesee that (lowast) indeed implies the assertion of the lemma

To prove (lowast) letrsquos first consider an arbitrary symmetric monoidal infin-category D with allsmall operadic colimits Then the Day convolution Day(CotimesDotimes) is symmetric monoidal andits tensor unit is given by

HomC(1C minus)otimes 1D C minus D

Note that the tensor product here is neither otimesC nor otimesD but the one defined in theproof of Proposition II34 That is the functor under consideration is actually given bycolimHomC(1Cminus) const 1D and one immediately checks that this coincides with the left Kan

132


extension of 1D lowast D along 1C lowast C which is how we constructed the tensor unitin the proof of Proposition II47(c) Now per construction (just take a sharp look atConstruction II46 again) there is a map

AlgComm(

Day(CotimesDotimes))minus FunOpinfin(CotimesDotimes)

(it is even an equivalence but we wonrsquot need that) By LemmaDefinition II45a the tensorunit of Day(CotimesDotimes) defines canonically an element of the left-hand side hence also anelement of the right-hand side This shows that HomC(1C minus) otimes 1D has a canonical laxsymmetric monoidal refinement

Now plug in Dotimes Antimes Then 1D lowast and X otimes lowast colimX const lowast X for allX isin An hence we obtain that HomC(1C minus) C An is the Day convolution tensor unitand thus has a lax symmetric monoidal refinement HomC(1C minus) Cotimes Antimes as requiredMoreover it preserves limits (which are automatically operadic) hence the refinement liftsto homC(1C minus) Cotimes Spotimes by Theorem II50 if Cotimes is a stable infin-operad

Now we can resume our discussion on how to include commutative rings and CDGAsinto CAlg Observe that we have two more lax symmetric monoidal functors

Ch(Z)otimesZ minus K(Z)otimesZ minus D(Z)otimesLZ

where Ch(Z) is the 1-category of chain complexes over Z The left functor comes from thefact that K(Z) Nc(Ch(Z)) is the coherent nerve of the Kan-enrichment of Ch(Z) Theright functor was constructed in II37f Unwinding definitions we see that CAlg(Ch(Z)otimesZ) CDGAlg is the 1-category of commutative differential graded algebras Moreover we havean inclusion Abotimes sube Ch(Z)otimesZ of symmetric monoidal 1-categories given by interpretingabelian groups as chain complexes concentrated in degree 0 which gives a map CRing CAlg(Abotimes) CAlg(Ch(Z)otimesZ) CDGAlg Summarizing if we apply CAlg(minus) everywherewe obtain functors

CRing minus CDGAlg minus CAlg(D(Z)otimes

LZ)minus CAlg

In fact it turns out that we can completely import ordinary commutative ring theory

II55a Corollary mdash The above induces a fully faithful functor CRing CAlg withessential image those R isin CAlg for which πi(R) = 0 for all i 6= 0

Here we define the homotopy groups of an Einfin-ring spectrum as those of its underlying spec-trum Also note that in the composition above the middle functor CDGAlg CAlg(D(Z)otimesLZ )from the 1-category of CDGAs to the infin-category of Einfin-chain complexes is not fully faithfulso this is really a thing about the composition

Proof sketch of Corollary II55a The proof wasnrsquot discussed in the lecture but is takenfrom Fabianrsquos lecture notes [AampHK Remarks II53(ii)] Recall from Example II51b that

Abotimes CGrpOpinfin(Settimes) minus CGrpOpinfin(Antimes) Spotimesgt0

is fully faithful with left adjoint π0 The adjunction together with the condition that thecounit is an equivalence persists to taking CAlg(minus) everywhere This almost shows whatwe want except that we have to explain why the functor iotimes CRing CAlg(Spotimesgt0) sube CAlgthat we get out is the one induced by H ie the one from the formulation of the corollary

133


Itrsquos more or less clear from the constructions that the underlying functor of iotimes is indeedH homAb(Zminus) Ab Sp but we need to check that the lax symmetric monoidalrefinement is the right one

Fabian outlines two ways to do so One can use the additional result from II37ewhich says that lax monoidal refinements of right adjoints are equivalent to oplax monoidalrefinements of left adjoints so it suffices that both candidates induce the same refinement ofπ0

Alternatively one can argue as follows By construction and LemmaDefinition II45a thelax symmetric monoidal refinement of HomAb(Zminus) Ab An from the proof of Lemma II55is initial in

CAlg(

Day(AbotimesAntimes)) FunOpinfin(AbotimesAntimes)

(we didnrsquot need this to be an equivalence in the proof of Lemma II55) Hence there is aunique map to Ωinfiniotimes which must be an equivalence since it is an equivalence on underlyingfunctor Ab An (and then Segal conditions do the rest) Now use Theorem II50 to seethat our lax symmetric monoidal refinement of homAb(Zminus) coincides with iotimes

Before we go on and construct module categories Irsquod like to go on another detour andprove that the homology and cohomology theories associated to a spectrum really do whattheyrsquore supposed to do

II55b Corollary mdash The homology and cohomology associated to a spectrum E byDefinition II35 can be described as

Elowast(minus) = πlowast(S[minus]otimes E

)= TorSlowast

(S[minus] E

) and Elowast(minus) = πlowast homSp(S[minus] E

)= ExtlowastS

(S[minus] E

)

They satisfy the EilenbergndashSteenrod axioms Moreover the homology and cohomology theoryassociated to an EilenbergndashMacLane spectrum HA for some abelian group A coincide withsingular homology and (up to a sign swap) singular cohomology That is

HAlowast(minus) Hlowast(minus A) An minus Ab and HAlowast(minus) Hminuslowast(minus A) Anop minus Ab

Finally the homology and cohomology of a spectrum E itself as defined in II26 is given by

Hlowast(EA) πlowast(E otimesHA) and Hlowast(EA) πlowast homSp(EHA)

The sign swap in cohomology is deliberate Fabian believes it was a mistake to have everdefined singular cohomology in nonnegative degrees rather than in nonpositive ones (or infact to have ever defined cochain complexes) I was late to be convinced but in the end Iwas not at least because in III17 wersquoll encounter a situation where our sign convention isdefinitely preferable

Proof of Corollary II55b The description of Elowast(minus) follows from our computations inII53 Along the same lines one can also prove EX homSp(S[X] E) hence the descriptionof Elowast(minus)

Next we check the EilenbergndashSteenrod axioms To construct the long exact sequencesand excision first note that fibre and cofibre sequences in the stable infin-category Sp coincideand that to each such sequence there is an associated long exact sequence on homotopygroup (because the functors Ωinfinminusi Sp lowastAn preserve fibre sequences so the long exact

134


sequences from lowastAn are inherited) Now both S[minus] otimes E and homSp(S[minus] E) transformcofibre sequences in An into fibre sequences in Sp providing the long exact sequences forElowast(minus) and Elowast(minus)

It remains to see that they transform disjoint unions into direct sumsproducts ForElowast(minus) this is easy homSp(S[minus] E) sends disjoint unions to products and taking homotopygroups commutes with products For Elowast(minus) we can argue similarly that S[minus] otimes E sendsdisjoint unions to coproducts but now we need to show that πlowast sends coproducts of spectrato products Arbitrary coproducts can be written as filtered colimits of finite coproductsand πlowast commutes with filtered colimits (Remark II31c) hence it suffices to check the caseof finite coproducts But Sp is stable hence finite coproducts agree with finite products andπlowast preserves the latter

Finally letrsquos do our reality check for EilenbergndashMacLane spectra The cohomology caseis easy From the (Cbull H)-adjunction combined with Corollary II30a we get

homSp(EHA) homD(Z)(Cbull(E) A

) H

(RHomZ(Cbull(E) A)

)By Example II25(c) the homotopy groups of the right-hand side are the homology groups ofRHomZ(Cbull(E) A) which are the singular cohomology groups Hlowast(EA) (in negative degrees)by definition Since Hlowast(S[X] A) = Hminuslowast(XA) for all X isin An by Lemma II31d plugging inthe special case E S[X] also shows HAlowast(X) = Hminuslowast(XA)

For the homology case we actually need to use fact that H is lax symmetric monoidal byLemma II55 Combining this with the unit E HCbull(E) of the (Cbull H)-adjunction givesfunctorial maps

E otimesHA minus HCbull(E)otimesHA minus H(Cbull(E)otimesLZ A

)

If we can show that the composition is an equivalence taking homotopy groups on bothsides will show πlowast(E otimesHA) = Hlowast(EA) as desired and then the special case E S[X]together with Lemma II31d will also show HAlowast(X) = Hlowast(XA)

To show that the composition is indeed an equivalence first note that it is an equivalencefor E S by Lemma II31d and that both sides commute with shifts and colimits in E Forthis we need that H commutes with colimits which is perhaps a bit surprising at first giventhat H is only a right adjoint but we give an argument in the proof sketch of Theorem II57below (and once you know this theorem H preserving colimits is not that surprising anymore) Now all of Sp can be generated from S using shifts and colimits hence

E otimesHA simminus H(Cbull(E)otimesLZ A

)is an equivalence everywhere

infin-Categories of ModulesEinfin-ring spectra have categories of modules that behave like the derived categories of modulesover ordinary rings In particular as wersquoll see in Theorem II57 below the category of modulesover HR for some R isin CRing is nothing else but D(R)

As a first step wersquoll construct certain operads Assoc and LMod These have alreadybeen teasered in Example II45(d) and (e) so wersquoll hijack that numbering and define themproperly

135


II45 Example mdash What it should have been(d) We would like to have an infin-operad that describes associative monoids So we define

E1 = Assoc Lop as follows The objects of Assoc are finite sets and a morphism I Jis a map in Lop together with a total ordering on each preimage αminus1(j) Compositionis given by composition in Lop and lexicogaphic ordering If we equip the 1-categoryAssoc with the obvious functor Assoc Lop it becomes an infin-operad with underlyingcategory Assoc1 = lowast and

HomactAssoc

((lowast lowast)︸︷︷︸n entries

lowast)

= total orders on 〈n〉

(just as there are n ways to multiply n factors in an associative but not necessarilycommutative monoid)

(e) We would also like to have aninfin-operad that describes an associative monoid A togetherwith an object M it acts upon (we think of this as an algebra together with a moduleover it) So we define LMod Lop as follows Its objects are pairs (I S) with I isin Lop

and S sube I (think of I as the set of factors wersquod like to multiply and of S as the set ofthose factors that come from M of course we canrsquot multiply more than one elementfrom M) A map (I S) (J T ) is given by a map α I J in Assoc such thatα(S) = T and for all t isin T the set αminus1(t)capS consists exactly of the maximal element ofαminus1(t) Composition is inherited from Assoc Again if we equip the 1-category LModwith the obvious map LMod Lop it becomes an infin-operad The underlying categoryLMod1 has precisely two objects a = (〈1〉 empty) and m = (〈1〉〈1〉) which we think ofcorresponding to A and M respectively We have

HomactLMod

((b1 bn) a

)=empty if bi = m for some itotal orders on 〈n〉 if bi = a for all i

This makes sense since ldquoas soon as one of the factors is from M the result will be nolonger an element of A but one of Mrdquo Similarly

HomactLMod

((b1 bn)m

)=

empty unless exactly one bi = m

total orders on 〈n〉with max element i

if bi = m

since there can be only one factor from M in any product and if there is it must bethe right-most factor

Sending the unique element lowast isin Assoc1 to a isin LMod1 defines a map Assoc LModof infin-operads which induces a map alowast AlgLMod(Cotimes) AlgAssoc(Cotimes) (ldquowe forget Mand only retain Ardquo) for any symmetric monoidal category Cotimes For any A isin AlgAssoc(Cotimes)the pullback

LModA(Cotimes) AlgLMod(Cotimes)

lowast AlgAssoc(Cotimes)

alowast

A

is called the infin-category of left modules in C over A We can convince ourselvesin examples that this terminology really makes sense If Cotimes = Settimes then an algebra

136


A isin AlgAssoc(Settimes) Mon(Set) is a monoid and elements of LModA(Settimes) are really leftA-sets Similarly if Cotimes = Abotimes then A is an associative (not necessarily commutative)ring and LModA(Abotimes) is really the 1-category of left A-modules

II56 General Properties of Module infin-Categories mdash We will now discuss (andpartially prove) a series of results to show that Assoc and LMod really do what they aresupposed to do and to see how one works with the left module infin-categories LModA(Cotimes)After that we are ready for modules over Einfin-ring spectra

First off recall the functor Cut ∆∆op Lop from Definition II15 which sends a totally

ordered set I to its set of Dedekind cuts Since the set of Dedekind cuts inherits a total orderCut lifts to a functor Cut ∆∆op Assoc over Lop Also recall the notation introduced beforeTheorem II43 For any infin-operad O and any infin-category C with finite products we denoteby OMon(C) sube Fun(O C) the full sub-infin-category of functors satisfying the Segal condition

II56a Theorem (Lurie [HA Proposition 41210]) mdash If C has finite products then theinduced functor

Cutlowast AssocMon(C) AlgAssoc(Ctimes) simminus Mon(C)is an equivalence More generally if Cotimes is any symmetric monoidal infin-category then

Cutlowast AlgAssoc(Cotimes) minus Fun(∆∆op Cotimes)timesFun(∆∆opLop) Cut

is fully faithful with essential image those functors F ∆∆op Cotimes that send inerts toinerts That is whenever α [n] [m] is inert in ∆∆ (without op) the induced morphismF (αop) F ([m]) F ([n]) is inert in Cotimes

II56b Remark mdash We wonrsquot prove Theorem II56a but let me at least give someadditional motivation The condition that F preserves inerts is just a somewhat unusualformulation of the Segal condition Indeed if we denote A = F ([1]) isin Cotimes1 C then the factthat the Segal maps ei [1] [n] induce inert maps in Cotimes shows

F([n]) (A A) isin Cotimesn Cn

(in particular F ([0]) 1C) For general α [n] [m] in ∆∆ we can think of the inducedmap F (αop) F ([m]) F ([n]) as follows Let Cut(αop)lowast (A A) (B1 Bn) be acocartesian lift of Cut(αop) 〈m〉〈n〉 A quick unravelling shows

Bi otimes

α(iminus1)ltj6α(i)

A

for i = 1 n Then Cut(αop)lowast and F (αop) induce a map Λ20 Cotimes which has an extension

to ∆2 since Cotimes Lop is a cocartesian fibration Restricting the extension to ∆12 givesmaps Bi A in C which we think of as the ldquomultiplication maps in the algebra Ardquo

Lecture 1812th Jan 2021

There is a similar description of AlgLMod(Cotimes) To this end we can extend Cut to afunctor LCut ∆∆op times [1] LMod that fits into a diagram

∆∆op AssocLop

∆∆op times [1] LMod

Cut

minustimes1 a

LCut

p

137


LCut sends elements of the form ([n] 1) to the ldquoalgebra-likerdquo element (a a) isin LModn(this guy has n repetitions of a) and elements of the form ([n] 0) to the ldquomodule-likerdquoelement (a am) isin LModn+1 (this guy also has n repetitions of a) In particular andthis might be confusing at first the structure map p LCut ∆∆op times [1] Lop is not givenby projecting to the first factor and applying Cut ∆∆op

Lop But if you think about thisit makes sense to have it that way The morphisms LCut([n] 0) LCut([n] 1) should begiven by forgetting the ldquomodule partrdquo of (a am) and only retaining the algebra part(a a)mdashwhich only works if the morphism in question runs from LModn+1 to LModn

II56c Theorem (Lurie [HA Proposition 42212]) mdash The functor

LCutlowast AlgLMod(Cotimes) minus Fun(∆∆op times [1] Cotimes

)timesFun(∆∆optimes[1]Lop) p LCut

is fully faithful with essential image those functors F ∆∆optimes[1] Cotimes that fulfill the followingconditions(a) The ldquoalgebra-likerdquo part F|∆∆optimes1 of F satisfies the condition from Theorem II56a

That is whenever α [n] [m] is inert in ∆∆ (without op) the induced morphismF (αop id1) F ([m] 1) F ([n] 1) is inert in Cotimes

(b) F ([n] 0) F ([n] 1) is always inert in Cotimes(c) If αop [n] [m] is an inert morphism in ∆∆ satisfying α(n) = m then the induced

morphism F (αop id0) F ([m] 0) F ([n] 0) is inert in Cotimes

II56d Remark mdash Again we wonrsquot prove Theorem II56c but at least explain howthe structure of an algebra and a module over it are encoded in the right-hand side LetM = F ([0] 0) and A = F ([1] 1) both are elements of Cotimes1 C by construction Bycondition (a) the part F|∆∆optimes1 encodes the algebra structure on A so in particularF ([n] 1) (A A) isin Cn as in Remark II56b Combining this with conditions (b) and(c) shows

F([n] 0

) (A AM) isin Cn+1

Given a general map α [n] [m] in ∆∆ we can think of F (αop id0) F ([m] 0) F ([n] 0)as follows Choose a cocartesian lift p(LCut(αop))lowast (A AM) (B1 Bn N) ofp(LCut(αop)) 〈m+ 1〉〈n+ 1〉 Then

Bi otimes


A and N

( otimesα(n)ltj6m

A

)otimesM

As in Remark II56b we get maps Bi A and N M which encode the multiplicationin A and the left action of A on M

In particular Theorem II56a says that any algebra A isin AlgAssoc(Cotimes) defines an elementCutlowast(A) isin FunLop(∆∆op Cotimes) which satisfies the Segal condition And then by Theorem II56cwe find that LModA(Cotimes) is a full sub-infin-category of the pullback

P FunLop(∆∆op times [1] Cotimes

)lowast FunLop(∆∆op Cotimes)

(minustimes1)lowast

Cutlowast(A)

138


Note that besides the operad map a Assoc LMod of extracting algebras there is alsoa map m Triv LMod that extracts the module object Also recall that AlgTriv(Cotimes) Cas shown in Example II45(b)

II56e Corollary mdash If Cotimes has smallfinite operadic colimits then LModA(Cotimes) hassmallfinite colimits and these are computed ldquounderlyinglyrdquo That is the functor

mlowast LModA(Cotimes) minus AlgTriv(Cotimes) C

preserves smallfinite colimits Likewise if C has smallfinite limits (which are automaticallyoperadic) then so has LModA(Cotimes) and mlowast preserves smallfinite limits In particular ifCotimes is a stable infin-operad then LModA(Cotimes) is a stable infin-category for all A isin AlgAssoc(Cotimes)

Proof sketch Let F I LModA(Cotimes) be a diagram Then Theorem II56c induces adiagram F I Fun(∆∆op times [1] Cotimes) satisfying

F (i)([n] 1

) (A A) and F

([n] 0

)(A AM(i)

)

where M I C is another diagram (the ldquomodule partrdquo of F ) We have to check that thethere are functors

F lim F colim I times∆∆op times [1] minus Cotimes

(i [n] 0) 7minus (A A)

(i [n] 1) 7minus

(A A limiisinIM(i)) for F lim

(A A colimiisinIM(i)) for F colim

Moreover we need to verify that F lim F colim constitute the limit and colimit over F takeninside the pullback P above and that both satisfy the conditions from Theorem II56c sothat they lie inside the full subcategory LModA(Cotimes) sube P We didnrsquot work this out in thelecture but in these notes I would like to give a rough sketch of what I think happens

Naively I would expect that the limit and colimit over F taken in P should be given bysome kind of pointwise limit in Fun(∆∆op times [1] Cotimes) There are two problems though Firstwe see from Remark II48a that (A A colimiisinIM(i)) doesnrsquot necessarily coincide withthe colimit colimiisinI(A AM(i)) in Cotimes even though colimits in C are operadic unless Iis weakly contractible This problem doesnrsquot occur in the limit case but there is a secondone which affects both cases We arenrsquot taking limitscolimits in Fun(∆∆op times [1] Cotimes) but insome pullback P of it And if I isnrsquot weakly contractible then the limitcolimit over theconstant A-diagram in FunLop(∆∆op Cotimes) might not be A any more Hence we canrsquot expectthat limitscolimits in the pullback P can be computed in each factor separately

Fortunately there is a trick that solves both problems at once Since C is cocomplete byassumption it has an initial object hence the diagram M I C extends canonically to adiagram M I C that sends the tip of the cone to the initial object Moreover

colimiisinI

M(i) colimiisinI

M(i)

and the right-hand side is also a colimit in Cotimes by Remark II48a since I is weaklycontractible Similarly taking the colimit over an I-shaped in FunLop(∆∆op Cotimes) which isconstant on A gives A again After some unravelling we see (I think) that this gives a way ofcomputing I-shaped colimits in P by factorwise I-shaped colimits A similar trick works forlimits Irsquoll leave it to you to make this argument precise and also to check that the conditionsfrom Theorem II56c are satisfied

139


The structure map Assoc Lop defines an infin-operad map Assoc Comm which inturn induces a map CAlg(Cotimes) AlgComm(Cotimes) AlgAssoc(Cotimes) as one would expect Inparticular the tensor unit 1C isin C also defines an associative algebra In fact one can showsimilar to LemmaDefinition II45a that 1C is also initial in AlgAssoc(Cotimes)

II56f Corollary mdash For any symmetric monoidal infin-category q Cotimes Lop the forgetfulfunctor

mlowast LMod1C (Cotimes) simminus AlgTriv(Cotimes) Cis an equivalence

Proof sketch Since the map empty (1C) in Cotimes is a q-cocartesian lift of 〈0〉〈1〉 we see thata functor F ∆∆op times [1] Cotimes fulfills the conditions from Theorem II56c iff all induced mapsin Cotimes are q-cocartesian But since ([0] 0) is initial in ∆∆op times [1] a functor F ∆∆op times [1] Cotimeswith F ([1] 1) 1C is the same thing as a natural transformation

ηF constF ([0] 0) =rArr F |∆∆optimes[1]r([0]0)

in Fun(∆∆op times [1] r ([0] 0) Cotimes) In other words F corresponds to a lift in the diagram

[0] Fun(∆∆op times [1] r ([0] 0) Cotimes

)[1] Fun

(∆∆op times [1] r ([0] 0)Lop)

constF ([0]0)

qlowast

const 〈0〉rArrpLCut

ηF

(in the bottom arrow p LMod Lop denotes the structure map of LMod)As wersquove seen above F defines an element of LMod1C (Cotimes) iff it takes all maps in ∆∆optimes [1]

to q-cocartesian ones Using [HCII Corollary IX25] this is equivalent to ηF being aqlowast-cocartesian edge Hence the space of lifts ηF and thus the space of functors F withgiven F ([0] 0) is the space of qlowast-cocartesian lifts in the above diagram Now the functormlowast LMod1C (Cotimes) C corresponds to the choice of F ([0] 0) ie to the choice of startingpoint Since qlowast is a cocartesian fibration we see that mlowast is essentially surjective and sincea map between starting points defines a contractible space of maps between qlowast-cocartesianlifts (you can make this precise using [HCII Proposition IX24]) we see that mlowast is fullyfaithful

Fabianrsquos script mentions a (stronger version of) another corollary which wersquoll need toprove Theorem II57 below I thought about whether there is an easy proof using what weknow so far but didnrsquot succeed so for now Irsquoll just refer to Lurie

II56g Corollary (Lurie [HA Proposition 4242]) mdash Let Cotimes be a symmetric monoidalinfin-category and A isin AlgAssoc(Cotimes) an algebra in it (whose underlying object we will alsodenote A isin C) Then the functor Aotimesminus C C lifts to a functor

LModA(Cotimes)

C Cmlowast

Aotimesminus

Aotimesminus

which is right-adjoint to the forgetful functor mlowast LModA(Cotimes) C In particular we seethat A isin LModA(Cotimes)

140


With that out of the way letrsquos discuss a cool application Any Einfin-ring spectrum R definesan element of AlgAssoc(Spotimes) via the canonical map CAlg AlgAssoc(Spotimes) Consequentlywe may define the infin-category of modules over R as

ModR = LModR(Spotimes)

We know from Corollary II56e that this is a stable infin-category

II57 Theorem mdash Let R be an ordinary commutative ring Then the canonical functor

D(R) LModR[0](D(R)otimes

LR) Hminus LModHR(Spotimes) ModHR

induced by Corollary II56f and Lemma II55 is an equivalence

We can interpret this theorem as follows Since S is the tensor unit in Sp (by II53) it is theinitial element in CAlg (LemmaDefinition II45a) and we have Sp ModS (Corollary II56f )Hence what Theorem II57 says is that the passage from D(R) to Sp is nothing else but theforgetful functor ModHR ModS with respect to the unique map S HR in CAlg

As an easy consequence we obtain that also

K Dgt0(R) simminus LModR(CGrp(An)otimes)

is an equivalence where CGrp(An)otimes CGrpOpinfin(Antimes) denotes the symmetric monoidalstructure on CGrp(An) obtained via Proposition II51

Proof of Theorem II57 Fabian has sketched out a proof in the lecture notes [AampHKTheorem II57] But then about a month after the final lecture Fabian has come up with amore straightforward argument which he explained to me and Irsquoll include it here

Recall that an object c in an infin-category C is called compact if HomC(cminus) commuteswith filtered colimits (which were introduced in Remark II31c) Note that a finite colimitof compact objects is compact again this follows from the fact that HomC(minusminus) transformsfinite colimits in the first variable into finite limits and finite limits commute with filteredcolimits in An This is proved in [HTT Proposition 5333] but I think it also follows fromRemark II31c after some fiddling With that out of the way the proof of Theorem II57 isbased on the following three observations(1) The objects R[i] isin D(R) are compact and the functors π0 HomD(R)(R[i]minus) D(R)

Ab for i isin Z are jointly conservative The same is true for the objects HR[i] isin ModHRand the functors π0 homModHR(HR[i]minus) ModHR Ab

(2) The functor H D(R) ModHR preserves colimits(3) D(R) is generated under colimits by the R[i] and ModHR is generated under colimits

by the HR[i]To show that HomD(R)(R[i]minus) Ωinfin homD(R)(R[i]minus) commutes with filtered colimits itsuffices to show the same for πj homD(R)(R[i]minus) using that Ωinfin and πj preserve filteredcolimits (Remark II31c) and that equivalences of spectra can be detected on homotopygroups (Lemma II23c) Wersquove seen in the proof of Lemma II55 that homD(R)(R[0]minus) H D(R) Sp Hence

πj homD(R)(R[i]minus

) πjH

(minus [minusi]

) Hi+j D(R) minus Ab

141


is the functor taking homology in degree i+ j These functors preserve fitered colimits Theabove description also immediately shows that the functors π0 HomD(R)(R[i]minus) D(R) Abfor i isin Z are jointly conservative

The corresponding assertions for HR hold more generally for any Einfin-ring spectrum T Indeed Corollary II56g implies

HomModT(T [i]minus

) HomSp

(S[i]minus

) Ωinfin+i

Ωinfin+i preserves filtered colimits by Remark II31c and the functors π0Ωinfin+i πi Sp Abfor i isin Z are jointly conservative by Lemma II23c Combine this with the fact that colimitsand equivalences in ModHR can be detected on underlying spectra by Corollary II56e andthe Segal condition from Theorem II56c respectively to finish the proof of (1)

The above arguments also show that H D(R) ModHR commutes with filtered colimitsBut H is an exact functor between stable infin-categories hence it also commutes with finitecolimits But every colimit can be obtained from coproducts (which are filtered colimits offinite coproducts) and pushouts (which are finite) proving (2)

It remains to show (3) We only show the part about HR since the part about D(R) issimilar Again we may replace HR by an arbitrary Einfin-ring spectrum T Let C sube ModTdenote the full sub-infin-category generated by the T [i] under finite colimits We claim thatthe canonical map

colimXisinCM

X simminusM

is an equivalence for allM isin ModT which will imply that ModT is generated by the T [i] undercolimits By (1) the equivalence above may be checked after applying π0 HomModT (T [i]minus)on both sides Note that the indexing infin-category CM is filtered Indeed any mapK CM from a finite simplicial set K can be extended over K by taking the colimit in CUsing that T [i] is compact we thus need to show that

colimXisinCM

π0 HomModT(T [i] X

) simminus π0 HomModT(T [i]M

)is an isomorphism of abelian groups It is surjective because each map f T [i]M on theright-hand side also defines an element of CM the image of idT [i] isin π0 HomModT (T [i] T [i])in the colimit maps to f For injectivity assume g T [i] X is mapped to 0 on theright-hand side Then X M extends over cofib(g) Note that cofib(g) is an element ofC again and that g maps to 0 in π0 HomModT (T [i] cofib(g)) Hence the image of g in thecolimit is 0 This proves (3)

Now note that homD(R)(R[0]M) simminus homHR(HRHM) is an equivalence for all M isinD(R) since both sides coincide with HM by the arguments in (1) The full sub-infin-categoryof all X isin D(R) for which homD(R)(R[0]M) simminus homHR(HRHM) is an equivalence isclearly closed under shifts and colimits hence it must be all of D(R) by (3) Thus H is fullyfaithful Its essential image is closed under colimits by (2) and contains all HR[i] Hence His also essentially surjective by (3) again

Fabian also mentioned in the lecture (and elaborates in the script see [AampHK Theo-rem II58]) that one can more or less the same argument as in the proof of Theorem II57 toshow a vast generalisation The SchwedendashShipley theorem

142


II58 Theorem (SchwedendashShipley) mdash If C is a complete stable infin-category containingan object x isin C such that homC(xminus) C Sp is conservative and commutes with colimitsthen this functor lifts to an equivalence

homC(xminus) C simminus Modend(x)op

where end(x)op denotes the opposite algebra of the canonical refinement of end(x) =homC(x x) isin Sp to an object of AlgAssoc(Spotimes)

Tensor Products over Arbitrary BasesBy Theorem II57 the symmetric monoidal structure on D(R) can be carried over to ModHRMore generally we would like to define symmetric monoidal structures ModotimesTT on ModT forany Einfin-ring spectrum T In the special case T HR it should be given by

HM otimesHR HN H(M otimesLR N)

for anyMN isin D(R) In fact constructing symmetric monoidal structures oninfin-categories ofalgebras works ridiculous generality Given a symmetric monoidal infin-category q Cotimes Lop

and an infin-operad p O Lop the infin-category of algebras AlgO(Cotimes) inherits a symmetricmonoidal structure AlgO(Cotimes)otimes

II59 Construction mdash With notation as above define AlgO(Cotimes)otimes Lop via II38aapplied to the presheaf F ∆∆Lop An which is given by

F(f [n] Lop) HomCatinfin

([n]timesO Cotimes

)timesHomCatinfin ([n]timesOLop) f times p

where f times p [n]timesO Lop denotes the composition [n]timesO Lop timesLop timesminus

Lop Letrsquoscompute the fibres

AlgO(Cotimes)otimesn AlgO(Cotimes)otimes timesLop 〈n〉 Using the method from the proof sketch of Proposition II40 we find that the Rezk nerve ofthe fibre over 〈n〉 is a simplicial anima Y given by Yk F (const 〈n〉 [k] Lop) Pluggingin the definitions we get

Yk HomCatinfin([k]timesO Cotimes

)timesHomCatinfin ([k]timesO

Lop) const 〈n〉 times p HomCatinfin

([k]Fun(O Cotimes)

)timesHomCatinfin ([k]Fun(O

Lop)) const 〈n〉 times p HomCatinfin

([k]Fun(O Cotimes)timesFun(OLop) 〈n〉 times p

) HomCatinfin

([k]FunLop(O Cotimes timesLop middot middot middot timesLop Cotimes︸︷︷︸

n times

))

For the second and third equivalence we use currying and the fact that HomCatinfin([k]minus)commutes with pullbacks For the fourth equivalence we use that maps O Cotimes over Lopwhere O is equipped with the structure map 〈n〉 times p O Lop are the same as maps fromO with its usual structure map p to the pullback of Cotimes along 〈n〉 times minus Lop

Lop Via theSegal maps said pullback is easily identified with the n-fold fibre product of Cotimes over Lop

The calculation above shows that Y Nr(FunLop(O Cotimes)n) hence the fibre wersquore inter-ested in is given by

AlgO(Cotimes)otimesn FunLop(O Cotimes)n We let AlgO(Cotimes)otimes sube AlgO(Cotimes)otimes be the full sub-infin-category spanned fibrewise by those(F1 Fn) such that each Fi is a map of infin-operads

143


II60 Proposition (Lurie [HA Proposition 3243]) mdash The map AlgO(Cotimes)otimes Lop isalways a symmetric monoidal monoidal infin-category and for every x isin O1 evaluation at xrefines to a strongly monoidal functor

evx AlgO(Cotimes)otimes minus Cotimes

II61 Observation mdash In the special case O Comm the symmetric monoidal structureon AlgComm(Cotimes) CAlg(Cotimes) is the cocartesian symmetric monoidal structure as defined inProposition II40 In particular the tensor product of two algebras is also their coproduct inCAlg(Cotimes)

Proof sketch of Observation II61 Use a generalisation of Lemma II20 which we used toshow that CMon(C) is semi-additive In fact Luriersquos version ([HA Proposition 24319]) ofthat lemma gives a criterion for a symmetric monoidal structure to be cocartesian

By Observation II61 the second half of Theorem II43 we get

CAlg(CAlg(Cotimes)otimes

) FunOpinfin

(CommCAlg(Cotimes)otimes

) Fun

(Comm1CAlg(Cotimes)

) CAlg(Cotimes)

Hence every commutative algebra A isin CAlg(Cotimes) canonically refines to a map of infin-operadsA Comm CAlg(Cotimes)otimes Now consider the following pullback diagram

LModA(Cotimes) LModA(Cotimes)otimesA AlgLMod(Cotimes)otimes

lowast Comm AlgComm(Cotimes)otimes AlgAssoc(Cotimes)otimes

alowast

〈1〉 A

II62 Theorem mdash Suppose Cotimes has all small operadic colimits Then

alowast AlgLMod(Cotimes)otimes minus AlgAssoc(Cotimes)otimes

is a cocartesian fibration In particular LModA(Cotimes)otimesA Comm defines a symmetricmonoidal structure on LModA(Cotimes) for all A isin CAlg(Cotimes) Moreover LModA(Cotimes)otimesA hasagain all operadic colimits

II62a The Bar Construction mdash The tensor product in LModA(Cotimes) can be computedby the bar construction ForMN isin LModA(Cotimes) the underlying C-object ofMotimesAN (whichwersquoll abusingly also denote that way) is given by

M otimesA N colim∆∆op

Bar(MAN)

Informally the functor Bar(MAN) ∆∆op C sends [n] isin ∆∆op to M otimesAotimesn otimesN with thetensor product being taken in C

In the script [AampHK Chapter II pp 123ndash124] Fabian explains how to define the barconstruction formally Let rev ∆∆op ∆op denote the functor that reverses the order of atotally ordered set Using Theorem II56a it induces a functor

revlowast AlgAssoc(Cotimes) minus AlgAssoc(Cotimes)

144


But one easily checks that Cut ∆∆op Lop and Cut rev ∆∆op ∆∆op are naturally

equivalent hence revlowast becomes equivalent to the identity upon composition with the canonicalmap CAlg(Cotimes) AlgAssoc(Cotimes) Of course this is nothing but the infin-categorical way ofsaying that a commutative algebra is naturally equivalent to its opposite

Using Theorem II56c we may now consider the functors

M (revtimes id[1]) N ∆∆op times [1] minus Cotimes

Letrsquos also write M (revtimes id[1]) = MA for simplicity (indicating that we now considerthe right-A module structure on M although we never defined anything of that sort)Then MA|∆∆optimes1 Arev A N|∆∆optimes1 by what we just showed In particular ifwe regard the functors MA and N as natural transformations MA|∆∆optimes0 rArr MA|∆∆optimes1and N|∆∆optimes0 rArr N|∆∆optimes1 in Fun(∆∆op Cotimes) then their endpoints coincide and we canform the pullback P isin Fun(∆∆op Cotimes) Pullbacks in functor categories are formed pointwise(Lemma I39) hence P ([n]) fits into a pullback diagram

P([n])

N([n] 0

)MA

([n] 0

)A([n])

in Cotimes By construction we have MA([n] 0) (MA A) N([n] 0) (A AN)and A([n]) (A A) where we abusingly identify algebras and modules with theirunderlying C-objects and the maps between them are the evident projections Thus usingDefinition II36(c) one easily checks

P([n]) (MA AN) isin Cotimesn+2

Now observe that P ∆∆op Cotimes factors over Cotimesact = Cotimes timesLopLop

act Indeed for a generalα [n] [m] in ∆∆ the morphism Cut(αop) 〈m〉〈n〉 is only undefined on those i isin 〈m〉for which i 6 α(0) or α(n) lt i see Definition II15 But those i with α(n) lt i are usedto encode the left-action of A on N (compare this to Remark II56d) and those i withi 6 α(0) are used to encode the right action of A on MA Hence the image of P (αop) in Lop

must be active as claimedThe unique active maps fn 〈n〉〈1〉 define a natural transformation idLop

actrArr const 〈1〉

in Fun(Lopact

Lopact) By unstraightening magic it defines a map of cocartesian fibrations from

Cotimesact to its pullback along const 〈1〉 which is C times Lopact Composing P with that map gives a

functorBar(MAN) ∆∆op minus C

which is the one wersquore looking for Since cocartesian lifts of the active maps fn+2 〈n+2〉〈1〉just tensor stuff together we see that Bar(MAN) does indeed take [n] to M otimesAotimesn otimesN

We wonrsquot explain why the bar construction computes the tensor product in LModA(Cotimes)and refer instead to [HA] again In Proposition 4428 Lurie proves that the tensor productin bimodules is given by the bar construction and in Theorem 4521 he shows that the tensorproduct on LModA(Cotimes) for A isin CAlg(Cotimes) factors over the specialisation from bimodules toleft modules

145


II62b The Tensor Unit mdash Unravelling the construction in Theorem II62 we seethat A is the tensor unit in LModA(Cotimes)otimesA as it should be As a reality check letrsquos see thatthis also comes out of the Bar construction We must show

N colim∆∆op

Bar(AAN)

This is in fact a special case of a general principle Observe that the bar constructionB = Bar(AAN) is the deacutecalage of another simplicial object in C That is there is anotherfunctor Bprime ∆∆op C such that the following diagram commutes

∆∆op ∆∆op

C

σop

B Bprime

Here σ ∆∆ ∆∆ denotes the functor taking objects [n] to [n+ 1] and morphisms α [n] [m]to σ(α) [n+ 1] [m+ 1] given by σ(α)(i) = α(i) for i 6 n and σ(α)(n+ 1) = m+ 1

Itrsquos not hard to construct Bprime in our example Basically one has to place another M indegree 0 and add those maps Aotimesn otimes N Aotimesn+1 otimes N that are given by 1C A on thefirst factor and the identity on the rest as well as those maps Aotimesn+1 otimesN Aotimesn otimesN thatmultiply the first factor with N (so we are forming some kind of cyclic bar construction) Irsquollleave the formal construction to you

In general if BBprime ∆∆op C are simplicial objects in some infin-category C and ifB dec(Bprime) is the deacutecalage of Bprime then

colim∆∆op

B colim∆∆op

constBprime0 Bprime0

The right equivalence follows from the fact that ∆∆op is weakly contractible (with [0] an initialobject) For the left-hand side observe that there are canonical transformations B rArr constBprime0and constBprime0 rArr B in Fun(∆∆op C) induced by suitable natural transformations η σop rArrconst [0] and τ const [0] rArr σop in Fun(∆∆op∆∆op) Using higher order transformationsbetween η and σ one shows that

τlowast Nat(B constx) simsim Nat(constBprime0 constx) ηlowast

are mutually inverse homotopy equivalences which is what we need

II62c Relative Tensor Products of Spectra mdash We can now define symmetricmonoidal structures on the stable infin-categories ModotimesRR for all Einfin-ring spectra R Wersquodhope the following properties to hold(a) The functor homModR Modop

R timesModR Sp has a natural refinement to a functorhomR Modop

R timesModR ModR satisfying the tensor-Hom adjunction

homR(M otimesR NL) homR

(MhomR(NL)

)

(b) If R S is a map in CAlg we get a symmetric monoidal functor

S otimesR minus ModotimesRR minus ModotimesSS

which does what it says on underlying spectra If moreover S T is another map inCAlg then T otimesS (S otimesR minus) T otimesR minus

146


(c) The functor S otimesR minus ModR ModS has a right adjoint the forgetful functor

FSR ModS minus ModR

which is the identity on underlying spectra Moreover FSR has a right adjoint itselfwhich is a refinement of homR(Sminus) ModR ModR

Proof Part (b) works in fact in much greater generality see [HA Theorem 4531] Wewill see that the rest are formal consequences

The existence of a functor homR ModopR times ModR ModR in part (a) follows from

adjoint functor theorem magic (initially we only get a tensor-Hom adjunction with the outerhomR replaced by the actual Hom functor HomModR Modop

R timesModR An but then itformally follows for homR as well) To see that homR equals homModR on underlying spectrawe have to prove (c) first

The existence of FSR follows again from Luriersquos adjoint functor theorem We knowthat S otimesR minus as an endofunctor of ModR preserves colimits by Theorem II62 hencethe same is true for S otimesR minus ModR ModS since both equivalences and colimits inModS can be detected on underlying spectra by the Segal condition from Theorem II56cand Corollary II56e respectively Hence FSR exists To see that it is the identity onunderlying spectra note that S otimesR (R otimesS minus) S otimesS minus implies FRS FSR FSS andboth FRS and FSS are just extraction of underlying spectra by Corollary II56g Thisalso implies that FSR preserves colimits since these can be detected on underlying spectraHence FSR has itself a right adjoint hSR ModR ModS Since right adjoints composeFSR hSR is a right adjoint of SotimesRminus ModR ModR hence hSR is indeed a refinementof homR(Sminus) ModR ModR This proves (c)

Finally to show that the underlying spectrum of homR(MN) is homModR(MN) itsuffices to consider the case M = R since then the general case follows by writing Mas a colimit of shifts of R (see the proof of Theorem II57 for why thatrsquos possible) ButhomR(RN) N by the tensor-Hom adjunction and homModR(RN) homSp(S N) Nby the adjunction from Corollary II56g

It remains to see why in the case of an ordinary ring R the derived tensor product onD(R) and the tensor product on ModHR coincide To this end observe that for arbitrarysymmetric monoidal infin-categories therersquos a strongly monoidal functor

Cotimes minus AlgLMod(Cotimes)otimes

which informally sends objects (c1 cn) to ((1C c1) (1C cn)) considering each ci as amodule over the trivial algebra 1C via Corollary II56f

II63 Proposition mdash The composite

D(R)otimesLR minus AlgLMod

(D(R)otimes

LR)otimesLR Hminus AlgLMod(Spotimes)otimes

induces an equivalence of symmetric monoidal infin-categories

H D(R)otimesLRsimminus ModotimesHRHR

Proof sketch We already know that H is an equivalence of underlying infin-categories Henceit suffices to show

H(R[0]

) HR and H(M otimesLR N) HM otimesHR HN

147


(since then it formally follows that H preserves cocartesian lifts of active morphisms henceall cocartesian lifts) Therersquos nothing to show for the first equivalence For the second wersquolluse the bar construction from II62a Since H being an equivalence preserves colimits andshifts it suffices to consider the case M = R[0] which follows immediately from the fact thatR is the tensor unit in D(R)otimesLR and HR is the tensor unit in ModotimesHRHR

MiscellaneaThe following stuff was only implicitly mentioned in the lecture (if at all) but Fabian gave adetailed account in his notes [AampHK Remarks II53]

II64 Graded Structures on Homotopy Groups mdash The functor π0 Sp Ab islax symmetric monoidal since it is the underlying functor of the composition

Spotimes Ωinfinminus CGrp(An)otimes π0minus Abotimes

where we denote CGrp(An)otimes CGrpOpinfin(Antimes) (which makes sense since the right-handside is symmetric monoidal by Proposition II51) and use CGrpOpinfin(Settimes) Abotimes byExample II51b(a)

This implies that π0 induces functors

π0 CAlg CAlg(Spotimes) minus CAlg(Abotimes) CRing π0 ModR LModR(Spotimes) minus LModπ0(R)(Abotimes) Modπ0(R)

for every Einfin-ring spectrum R In particular π0(R) is an ordinary ring and if M is anR-module then π0(M) is an ordinary π0(R)-module But we can do better For all i j isin Zthe multiplication micro RotimesM M induces morphisms

π0(S[minusi]otimesR

)otimes π0

(S[minusj]otimesM

)minus π0

(S[minus(i+ j)]otimesRotimesM

) microminus π0

(S[minus(i+ j)]otimesM

)

which provide a multiplication πi(R) otimes πj(M) πi+j(M) In the special case R = M itturns πlowast(R) into a graded ring and for general M we get a graded πlowast(R)-module structureon πlowast(M)

In fact πlowast(R) is graded commutative To see this we need to investigate the effect ofthe canonical equivalence S[i]otimes S[j] S[j]otimes S[i] (we drop the minus signs for convenience)on homotopy groups This equivalence corresponds to an element of

π0 HomSp(S[i+ j]S[i+ j]

) πi+j

(S[i+ j]

) Z

Now S[i] ΣinfinSi and S[j] ΣinfinSj Since Σinfin is strongly monoidal (Proposition II51) wesee that the equivalence S[i]otimes S[j] S[j]otimes S[i] corresponds to

Si and Sj (S1)andi and (S1)andj (S1)andj and (S1)andi Sj and Si

given by permutation of factors The permutation in question can be written as ij transposi-tions hence its sign is (minus1)ij Hence its action on πi+j(Si+j) Z is given by (minus1)ij as wellThis proves graded commutativity

Combining these considerations with II15c(clowast) for example we see that the algebraicK-theory Klowast(R) = πlowast(Binfink(R)) of any commutative ring R carries a graded commutativering structure called the cup product on K-theory In the next paragraph wersquoll see that alsothe cup product on singular cohomology arises in this way

148


II65 The Cup Product and the Pontryagin Product mdash IfM isin CGrp(An) thenS[M ] is an Einfin-ring spectrum as noted way back in II54 Using Proposition II60 thisimplies that S[M ] otimesHR is an Einfin-ring spectrum too for any ordinary ring R Hence byII64 and Corollary II55b there is a graded commutative ring structure on

πlowast(S[M ]otimesHR

) Hlowast(MR)

This is (our definition of) the Pontryagin product A natural question is whether the cupproduct in cohomology arises in a similar way It does The main ingredient is a constructiondue to Glasman which upgrades the Yoneda embedding Y C C Fun(CopAn) to a laxsymmetric monoidal functor

Cotimes minus Day(Copotimes Antimes)

where Copotimes denotes the induced symmetric monoidal structure on Cop (ie compose the

functor Stcocart(Cotimes) Lop Catinfin with (minus)op and observe that the Segal condition survives)In particular if A isin CAlg(Cotimes) is a commutative algebra which we abusingly identify withits underlying C-object then

HomC(minus A) Cop minus An

is an element of CAlg(Day(Copotimes Antimes)) FunOpinfin(Cop

otimes Antimes) (as in the proof of Lemma II55we donrsquot even need that this is an equivalence) hence a lax symmetric monoidal functorThus it defines a functor

HomC(minus A) CAlg(Copotimes ) minus CAlg(Antimes)

The right-hand side equals CMon(An) by Theorem II43 In particular if B isin CAlg(Copotimes ) is

a coalgebra in C then HomC(BA) has a canonical refinement in CMon(An)Moreover one can check that SpOpinfin(Day(CotimesO)) Day(Cop

otimes SpOpinfin(O)) for all infin-operads O isin Oplex

infin Hence if Cotimes is a stable infin-operad we can apply Theorem II50 to thefunctors HomC(minus A) above and obtain upgraded lax symmetric monoidal functors

homC(minus A) Copotimes minus Spotimes

homC(minus A) CAlg(Copotimes ) minus CAlg(Spotimes) CAlg

If Cotimes Ctimes is the cartesian monoidal structure from Proposition II42 then Coptimes (Cop)t is

the cocartesian monoidal structure and CAlg(Coptimes ) CAlg((Cop)t) Cop by the second half

of Theorem II43 In particular every anima X isin An is a coalgebra in the cartesian monoidalstructure Hence S[X] isin CAlg(Spop

otimes ) and then we see that homSp(S[X] E) is canonically anEinfin-ring spectrum for every Einfin-ring spectrum E Now Corollary II55b implies

πlowast homSp(S[X] HR

) Hminuslowast(XR)

whence II64 endows the cohomology of X with a graded commutative ring structure Thisis (our definition of) the cup product

II66 E1-Ring Spectra mdash Elements of AlgAssoc(Spotimes) are called E1-ring spectra Manyof our results about Einfin-ring spectra can be generalised to E1-ring spectra But let me onlymention those that wersquoll need in Chapter III

149


(a) Since S[minus] Antimes Spotimes is symmetric monoidal by Proposition II51 it induces a functor

S[minus] Mon(An) AlgAssoc(Antimes) minus AlgAssoc(Spotimes)

In particular if M isin Mon(An) is an E1-monoid then its spherical group ring S[M ] isan E1-ring spectrum

(b) Tensor products of E1-ring spectra inherit a E1-ring structure by Proposition II60(c) If A is an E1-ring spectrum and M a left module over it then πlowast(A) is a graded (but of

course not necessarily commutative) ring and πlowast(M) a graded left πlowast(A)-module

150

Chapter III

The Group CompletionTheorem and the K-Theoryof Finite Fields

IIIOur first goal in this chapter and the content of the group completion theorem is tounderstand S[Minfin-grp] in terms of S[M ] for M isin CMon(An) This turns out to be an easyto understand localisation So letrsquos understand localisations first

Localisations of Einfin-Ring SpectraRecall that π0(E) π0 homSp(S E) π0 HomSp(S E) for every spectrum E (this followsfor example from Lemma II53a) Hence every element e isin π0(E) defines up to hommotopya map e S E

III1 Definition mdash Let R isin CAlg is an Einfin-ring spectrum M isin ModR some moduleover it and S sube π0(R) some subset Then M is called S-local if for every s isin S the map

M SotimesM sotimesidMminusminusminusminus RotimesM microminusM

(ldquomultiplication by srdquo) is an equivalence

Our goal is to produce a localisation functor Letrsquos do the case of a single elements isin π0(R) first We put

M [sminus1] = colimN

(M

middotsminusM

middotsminusM

middotsminus

)

the colimit being taken in ModR (which is cocomplete by Corollary II56e this corollaryalso shows that we get a colimit on underlying spectra too) Since the tensor product onModR commutes with colimits (Theorem II62) we have M [sminus1] R[sminus1]otimesRM and sincetaking homotopy groups commutes with sequential colimits (Lemma II31b(alowast)) we getπlowast(M [sminus1]) πlowast(M)[sminus1] Also there is a functorial map M M [sminus1] given by includingM as the first element of the colimit

III2 Proposition mdash The functor minus[sminus1] ModR ModR is a Bousfield localisationonto the s-local R-modules

Proof Itrsquos clear that M [sminus1] is s-local Indeed whether s M [sminus1]M [sminus1] is an equiva-lence can be checked on homotopy groups of underlying spectra (by the Segal condition fromTheorem II56c plus Lemma II23c) where it is evidently true as πlowast(M [sminus1]) πlowast(M)[sminus1]

151

Localisations of Einfin-Ring Spectra

Therefore it suffices to check that the functorial maps ηM M M [sminus1] satisfy thecondition from Proposition I61a That is we must show that

ηM [sminus1] ηM [sminus1] M [sminus1] simminus(M [sminus1]

)[sminus1]

are equivalences This is clearly true for ηM [sminus1] because M [sminus1] is s-local hence it followsfor ηM [sminus1] by the same ldquocoordinate fliprdquo trick as in the proof of Theorem II30

This now allows us to define localisations for arbitrary S sube π0(R) If T sube S is a finitesubset say T = s1 sn put M [Tminus1] = M [(s1 sn)minus1] Since being T -local can bedetected on homotopy groups one readily checks that the T -local R-modules are preciselythe s1 sn-local R-modules hence minus[Tminus1] ModR ModR is a Bousfield localisationonto the T -local R-modules by Proposition III2 In general we put

M [Sminus1] = colimTsubeS finite

M [Tminus1]

Note that a priori there is a coherence issue in this colimit Namely the si in T = s1 snare only defined up to homotopy However we have described M [Tminus1] in terms of a universalproperty independent of any choice so we donrsquot have to worry about coherence

III3 Corollary mdash The functor minus[Sminus1] ModR ModR is a Bousfield localisation ontothe S-local objects

Proof Analogous to Proposition III2

Now whatrsquos still missing is an argument why R[Sminus1] is an Einfin-ring spectrum again andwhy the forgetful functor ModR[Sminus1] ModR (induced by II62c and the map R R[Sminus1]which we are to show is a map of Einfin-ring spectra) is an equivalence onto the S-local objectsAs we will just see all of these follow from general principles since R[Sminus1] isin ModotimesRR is atypical example of a otimes-idempotent object

III4 PropositionDefinition mdash Let Cotimes be a symmetric monoidal infin-category withtensor unit 1C isin C A map f 1C x in C is called otimes-idempotent if

idx otimes f xotimes 1C simminus xotimes x

is an equivalence In this case the following assertions hold(a) The functor minusotimes x = Lf C C is a localisation onto the full sub-infin-category Cf sube C

spanned by all y isin C that absorb f ie those y for which idy otimesf y otimes 1C simminus y otimes x isan equivalence

(b) Cotimesf is a symmetric monoidal infin-category Lf C Cf refines to a strongly monoidalfunctor Morever for any y isin Cf z isin C we have y otimes z isin Cf and Cotimesf sube Cotimes only fails tobe strongly monoidal because it doesnrsquot preserve tensor units

(c) The strongly monoidal functor Lf Cotimes Cotimesf from (b) induces a Bousfield localisationLf CAlg(Cotimes) CAlg(Cotimesf ) sube CAlg(Cotimes) onto those algebras whose underlying C-objectabsorbs f In particular x isin CAlg(Cotimes) and the induced functor

Cf LModx(Cotimesf ) simminus LModx(Cotimes)

is an equivalence

152


(d) If Cotimes has all small operadic colimits then the equivalence from (c) refines to a stronglysymmetric monoidal equivalence

Cotimesfsimminus LModx(Cotimes)otimesx

Proof For (a) wersquoll apply Proposition I61a as usual The map f 1C x induces a naturaltransformation η idC minus otimes 1C rArr minus otimes x Lf so we must show that ηLf and Lfη areequivalences Up to reordering tensor factors both amount to showing that

idy otimes idxotimesf y otimes xotimes 1C simminus y otimes xotimes x

is an equivalence for all y isin C which is clear from the condition on x Hence Lf is indeed aBousfield localisation with unit η By definition an element y isin C absorbs f iff ηy y Lf (y)is an equivalence hence Cf is indeed the essential image of Lf

For (b) we use the criterion from II37e(b) Unravelling the statement we must showthat whenever g y yprime is a morphism in C such that Lf (g) idxotimesg xotimes y simminus xotimes yprime isan equivalence then so is Lf (g otimes idz) idxotimesg otimes idz xotimes y otimes z simminus xotimes yprime otimes z for all z isin CBut thatrsquos just obvious and so are the additional two assertions from (b)

For (c) note that the adjunction Lf C Cf i descends to adjunctions

Lf CAlg(Cotimes) CAlg(Cotimesf ) i

Lf LModx(Cotimes) LModx(Cotimesf ) i

because unit and counit as well as the triangle identities are inherited (and for the loweradjunction we also need Lf (x) x) Moreover the unit being an equivalence is inherited aswell hence CAlg(Cotimesf ) CAlg(Cotimes) is fully faithful and the characterisation of its essentialimage follows by inspection The same argument shows that LModx(Cotimesf ) LModx(Cotimes) isfully faithful To see that itrsquos essentially surjective as well we must show that the underlyingC-object of every left x-module already absorbs f So let m isin LModx(Cotimes) By abuse ofnotation we identify m with its underlying C-object Then there is a multiplication mapmicro xotimesm m fitting into a diagram

m xotimesm m

xotimesm xotimes xotimesm xotimesm

fotimesidm

fotimesidm

micro

sim

fotimesidxotimes idm fotimesidmfotimesidxotimes idm idxotimesmicro

which shows that f otimes idm m xotimesm is a retract of an equivalence hence an equivalenceitself Hence LModx(Cotimesf ) simminus LModx(Cotimes) is an equivalence Finally Cotimesf LModx(Cotimesf )follows from Corollary II56f since x is the tensor unit of Cotimesf

For (d) first note that by functoriality of the relative tensor product construction fromTheorem II62 we get a lax monoidal functor

i LModx(Cotimesf )otimesx minus LModx(Cotimes)otimesx

We would like to show that itrsquos a strongly monoidal equivalence It certainly is an equivalenceon underlying infin-categories by (c) Hence it suffices to check i(x) x and i(m otimesx n) i(m) otimesx i(n) The former is obvious For the latter recall that i only fails to be stronglymonoidal by not preserving the tensor unit and hence both sides can be computed by theexact same bar construction (see II62a)

153


Now by construction and Proposition II60 there are infin-operad maps

u LModx(Cotimesf )otimesx minus AlgLMod(Cotimesf )otimes evmminusminus Cotimesf

We are done if we can show that their composition u is a strongly monoidal equivalenceIt is an equivalence on underlying infin-categories by (c) hence it suffices to show u(x) xand u(m otimesx n) u(m) otimes u(n) The former is obvious again and the latter follows fromthe fact that the bar construction computing u(m otimesx n) is constant on u(m) otimes u(n) asBn(u(m) x u(n)) u(m)otimes xotimesn otimes u(n)

III4a Corollary mdash The R-module R[Sminus1] is naturally an Einfin-ring spectrum and themap R R[Sminus1] is a map of Einfin-ring spectra The corresponding base change functor (seeII62c)

R[Sminus1]otimesR minus ModR minus ModR[Sminus1]

is a Bousfield localisation onto the S-local objects ModRS sube ModR Under this equivalencethe symmetric monoidal structures on ModR[Sminus1] (via Theorem II62) and ModRS (viaPropositionDefinition III4(b)) get identified Finally for any Einfin-ring spectrum T

HomCAlg(R[Sminus1] T

)sube HomCAlg(R T )

is the collection of path components of maps R T that take S sube π0(R) to units in π0(T )

Proof This may seem tautological after PropositionDefinition III4 but I donrsquot think itis For the first part we know from PropositionDefinition III4(c) that R R[Sminus1] is amap in CAlg(ModotimesRR ) hence itrsquos also a map of Einfin-ring spectra via the forgetful functorModotimesRR ModotimesS Spotimes Moreover II62c provides an adjunction

R[Sminus1]otimesR minus ModR ModR[Sminus1] FR[Sminus1]R

The forgetful functor FR[Sminus1]R takes values in the full sub-infin-category ModRS sube ModRof S-local objects We claim the counit R[Sminus1]otimesR FR[Sminus1]R rArr id is an equivalence Butequivalences of R[Sminus1]-modules can be detected on underlying spectra (even on homotopygroups) so it definitely suffices that R[Sminus1]otimesR N simminus N is an equivalence in ModR (ratherthan ModR[Sminus1]) for every R[Sminus1]-module N But this follows from Corollary III3 and thefact that N is S-local as was noted above

In particular ModR[Sminus1] is a full subcategory of ModR To show that it coincides withModRS consider the diagram

LModR[Sminus1](ModotimesRR

)LModR

(ModotimesRR

)LModR[Sminus1](Spotimes) LModR(Spotimes)

sim

induced by the lax symmetric monoidal functor ModotimesRR Spotimes The right vertical arrow isan equivalence by Corollary II56f and by PropositionDefinition III4(c) the top arrowcan be identified with the inclusion ModRS sube ModR Hence the bottom arrow which isfully faithful with essential image contained in ModRS also hits of ModRS In particularthe left vertical arrow is an equivalence as well and one can even show that it is stronglymonoidal It clearly preserves tensor units hence all we need to check is

colim∆∆op

Bar(MRN) colim∆∆op

Bar(MR[Sminus1] N

)

154


for all S-local R-modules M and N But we can permute colim∆∆op with the colimits definingR[Sminus1] and then M and N being S-local does the rest

The final assertion about HomCAlg can be checked on (derived) fibres over HomCAlg(R T )That is it suffices to check that once we choose a map f R T

HomRCAlg(R[Sminus1] T

)is either contractible or empty depending on whether f maps S sube π0(R) to units in π0(T )or not If it doesnrsquot then HomCAlg(R[Sminus1] T ) HomCAlg(R T ) doesnrsquot hit the pathcomponent of f hence the fibrendashderived or notmdashis empty So letrsquos assume f maps S sube π0(R)to units in π0(T ) To handle this case we need to bring out a big gun By a theorem ofLurie

RCAlg CAlg(ModotimesRR )

The proof is in [HA] See Corollary 3417 for the statement Example 33112 for why Commis coherent and 4515 for a proof that our definition of module infin-categories coincides withLuriersquos

In any case we see that f R T turns T into an element of CAlg(ModotimesRR ) By ourassumption on f T is even an element of the full sub-infin-category CAlg(ModotimesRRS) hence

HomCAlg(ModotimesRR

)

(R[Sminus1] T

) HomCAlg(ModotimesR

R)(R T )

by PropositionDefinition III4(c) The right-hand side is contractible since R is initial inCAlg(ModotimesRR ) RCAlg whence we are done


As an example of this machinery consider the unique Einfin ring map S HZ (usingthat S is initial in CAlg by LemmaDefinition II45a) This map is sometimes called theldquoHurewicz maprdquomdashif you tensor with S[X] for some X isin An and take homotopy groups youget the stable Hurewicz morphism πslowast(X) Hlowast(XZ) Localising at Z r 0 sube Z = π0Sgives an equivalence

S[pminus1 ∣∣ p prime

] simminus HQ

Indeed first of all the natural map HZ HQ factors over HZ[pminus1 | p prime] HQ bythe second part of Corollary III4a and this map must be an equivalence as one immediatelysees on homotopy groups Moreover the πi(S) are finite for i gt 0 by a theorem of Serrehence they die in the localisation so S[pminus1 | p prime] simminus HQ is an equivalence on homotopygroups as well

III5 Corollary mdash The EilenbergndashMacLane functor

H D(Q) minus Sp

defines an equivalence onto all spectra whose homotopy groups are rational (ie Q-vectorspaces)

Proof Having rational homotopy groups is equivalent to being (Z r 0)-local hence thesespectra are precisely ModS[pminus1 | p prime] ModHQ On the other hand H D(Q) simminus ModHQis an equivalence by Theorem II57

155

The Group Completion Theorem and K1(R)

So in particular H(M otimesLQ N) HM otimes HN holds for all MN isin D(Q) by Proposi-tionDefinition III4(b) We would already know this for minus otimesHQ minus instead of minus otimes minus (seeProposition II63) but as it is itrsquos really new information We also get that the ldquorationalstable Hurewicz maprdquo

πslowast(X)otimesQ πlowast(S[X]

)otimesQ simminus Hlowast(XQ)

is an equivalence for all X isin An

The Group Completion Theorem and K1(R)From Proposition II51 we get an adjunction S[minus] Antimes Spotimes Ωinfin in which the rightadjoint is lax monoidal and the left adjoint even strongly so As usual this adjunction persistsafter taking CAlg(minus) on both sides Since CAlg(Antimes) CMon(An) by Theorem II43 thenew adjunction appears as

S[minus] CMon(An) CAlg Ωinfin

In particular if R is an Einfin-ring spectrum then ΩinfinR carries a Einfin-monoid structure Infact it carries another one due to the fact that Ωinfin Sp An factors over CGrp(An)But these two structures are different The former comes from the multiplicative structureon R whereas the latter comes from the additive structure Since Ωinfin Sp CGrp(An)is an exact functor between additive categories it induces a strongly monoidal functorSpoplus CGrp(An)oplus between the cartesian monoidal structures from Proposition II42 (andalso Sp CMon(Spotimes) CGrp(An) CMon(CGrp(An)otimes) by Theorem II19) but these areobviously different from Spotimes and CGrp(An)otimes For example on discrete abelian groupsCGrp(An)otimes encodes the usual tensor product (Example II51b(a)) whereas CGrp(An)oplusencodes their product

Back to the matters at hand Let R isin CAlg and take ΩinfinR with its multiplicativeEinfin-monoid structure Let moreover M isin CMon(An) We compute

HomCAlg(S[Minfin-grp] R

) HomCMon(An)(Minfin-grpΩinfinR)sube HomCMon(An)(MΩinfinR) HomCAlg

(S[M ] R

)

and the image of the inclusion on the second line consists of all path components of mapsM ΩinfinR that take π0(M) to units in π0(ΩinfinR) Now regard π0(M) as sitting insideπ0(S[M ]) via the isomorphism

π0(S[M ]

) H0(MZ) Z

[π0(M)

]from Corollary II55b Then we have more or less just shown

III6 Theorem (Group completion theorem McDuffndashSegal) mdash For all E isin Sp andM isin CMon(An) there is a canonical equivalence(

E[M ])[π0(M)minus1] simminus E[Minfin-grp]

as S[M ]-module spectra and even S[M ]-algebra spectra if E isin CAlg In particular

Hlowast(Minfin-grpZ) Hlowast(MZ)[π0(M)minus1]

156


In Theorem III6 and henceforth we use the notation E[X] = E otimes S[X] for X isin AnAlso recall that Minfin-grp ΩBM by Corollary II21 which is how yoursquoll probably find thisresult in the literature

Proof of Theorem III6 It follows from our calculation above that


)sube HomCAlg

(S[M ] R

)is the collection of path components of those S[M ] R that send π0(M) sube π0(S[M ]) Z[π0(M)] into units in π0(R) But this shows(

S[M ])[π0(M)minus1] S[Minfin-grp]

by Corollary III4a The general case follows by tensoring with E For the statement abouthomology take E HZ and apply Corollary II55b

To warm up for our calculation of K1(R) letrsquos compute the homology of (ΩinfinS)0 the0-component of the Einfin-group ΩinfinS colimnisinN ΩnSn To really warm up Fabian decidedto give both corollaries the number III7 but Irsquom not going do that as wellIII6a Corollary mdash There is a canonical isomorphism

Hlowast((ΩinfinS)0Z

) colim

nisinNHgrplowast (SnZ)

Here Hgrplowast (SnZ) denotes the group homology of the nth symmetric group and the transition

maps Sn Sn+1 are given by mapping Sn to the permutations that fix the element n+ 1Proof First up recall the standard fact that Hgrp

lowast (SnZ) Hlowast(BSnZ) where BSn is theimage of Sn under

Grp(Set) sube Grp(An) | |minusminus An (see II6 for notation) Also recall that

πi(BSn lowast)

0 if i 6= 1Sn if i = 1

since Sn is a discrete anima and B shifts homotopy groups up (because its inverse Ω shiftsthem down)

Now let S ∐ngt0BSn denote the free commutative monoid on a point or equivalently

the symmetric monoidal groupoid (finite sets bijectionst) see page 88 In particularπ0(S) N So in order to invert π0(S) in Hlowast(SZ) Z[π0(S)] we only need to invertthe generator [1] isin π0(S) Fabian remarks that this notation is a bit awkward [1] doesnrsquotcorrespond to the unit in the ring Z[π0(S)] which is instead given by the unit 1 = [0] isin π0(S)of the monoid π0(S) N Also by the BarattndashPriddyndashQuillen theorem (Corollary II32) wehave Sinfin-grp ΩinfinS Hence Theorem III6 shows

Hlowast(ΩinfinSZ) colimN

(Hlowast(SZ) middot[1]

minusminus Hlowast(SZ) middot[1]minusminus

)

Now isolate the components corresponding to (ΩinfinS)0 (Sinfin-grp)0 on both sides In thecolimit on the right-hand side middot[1] maps the component BS0 into BS1 then into BS2 theninto BS3 etc and these maps BSn BSn+1 are precisely the transition specified in theformulation of the corollary Hence we really obtain


) colim

nisinNHlowast(BSnZ) colim


157


III6b mdash Therersquos another way of phrasing the result from Corollary III6a Considerthe ldquogroup of finitely supported permutationsrdquo Sinfin = colimnisinN Sn The canonical mapS Sinfin-grp ΩinfinS induces a map

ZtimesBSinfin colimnisinN

(Smiddot[1]minusminus S

middot[1]minusminus

)minus colim

nisinN

(ΩinfinS middot[1]

minusminus ΩinfinS middot[1]minusminus

) ΩinfinS

(where we use that ΩinfinS is already an Einfin-group so all transition maps in the second colimitare equivalences) As all components of an Einfin-group are equivalent Corollary III6a impliesthat the map ZtimesBSinfin ΩinfinS induces isomorphisms

Hlowast(ZtimesBSinfinZ) simminus Hlowast(ΩinfinSZ)

on homology But itrsquos far from being a homotopy equivalence Amusingly this alreadyfollows from Corollary III6a itself As seen in its proof

π1(ZtimesBSinfin (0 id)

) colim

nisinNπ1(BSn id) Sinfin

whereas

π1(ΩinfinS 0) H1((ΩinfinS)0Z

) colim

nisinNH1(SnZ) colim

nisinNSabn Z2Z

The first isomorphism follows from the Poincareacute lemma as π1(ΩinfinS 0) π1(S) is alreadyabelian the third isomorphism follows from the Poincareacute lemma in group homology and thefourth from the fact that the commutator [SnSn] An is the alternating group for n gt 2which has index 2

So Z times BSinfin ΩinfinS is an example where the homology Whitehead theorem fails foranima which arenrsquot simple And also it shows that the group completion theorem failsrather drastically before taking suspension spectra The exact nature of this failure will bethoroughly investigated in III11a

III7 Corollary mdash Let R be a ring Then therersquos a canonical isomorphism

Hlowast(k(R)0Z

) simminus colimnisinN

Hgrplowast(

GLn(R)Z)

In particularK1(R) = colim

nisinNGLn(R)ab GLinfin(R)ab

Proof While we donrsquot understand π0 Proj(R) to invert it in Hlowast(Proj(R)Z) it suffices toinvert the element [R] isin π0 Proj(R) Indeed for every finite projectve R-module P the class[P ] ldquodividesrdquo some power of [R] in π0 Proj(R) since there is some Q such that P oplusQ RnThus from Theorem III6 applied to k(R) Proj(R)infin-grp we find

Hlowast(k(R)Z

) colim

nisinN

(Hlowast(

Proj(R)Z) middot[R]minusminus Hlowast

(Proj(R)Z

) middot[R]minusminus

)We have Proj(R)

∐[P ]isinπ0 Proj(R)BGL(P ) by direct inspection Indeed by II6 construct-

ing an equivalence from BGLn(P ) onto the component of P is the same as constructingan equivalence BGLn(P ) simminus ΩP Proj(R) HomProj(R)(P P ) But since Proj(R) is the

158


1-groupoid of finite projective R-modules the right-hand side just is the discrete animaGLn(P ) as required

As in the proof of Corollary III6a the assertion now follows by isolating the componentof 0 on both sides In the colimit right-hand side the component of 0 is mapped to BGL1(R)then to BGL2(R) then to BGL3(R) etc so we really get

Hlowast(k(R)0Z

) colim

nisinNHlowast(BGLn(R)Z

) colim

nisinNHgrplowast(

GLn(R)Z)

In the special case lowast = 1 we apply the Poincareacute lemma in singular homology (using thatπ1(k(R) 0) is already abelian) and group homology to get the desired formula for K1(R)

III7a mdash As for Corollary III6a we could also reformulate Corollary III7 as followsThere exists a map K0(R)timesBGLinfin(R) k(R) which induces an isomorphism on homology

Fabian also mentioned in the lecture and elaborates in his notes [AampHK Chapter IIIpp 48ndash49] that for r isin quad even one can computeHlowast(gwr(Z)Z) via a similar calculationbut this requires non-trivial input about π0 Unimodr(Z)

III7b K1(R) and Determinants mdash Note that for P isin Proj(R) we get a map

Aut(P ) π1(

Proj(R) P)minus π1

(k(R) P

) K1(R)

(the last equivalence follows since all path components of the Einfin-group k(R) must be equiva-lent) If R is commutative then the determinant gives a map det K1(R) GLinfin(R)ab RtimesComposing it with the morphism Aut(P ) K1(R) from above gives a way to extend de-terminants to automorphisms of arbitrary finite projective modules Of course such anextension could also obtained by hand but it pops out for free here In general for notnecessarily commutative rings R we may think of the morphism Aut(P ) K1(R) as a goodldquogeneralised determinantrdquo

If R is commutative the kernel of det K1(R) Rtimes is usually denoted SK1(R) (theldquospecialrdquo K-group) Since det admits a splitting via Rtimes GL1(R) GLab

infin we obtain

K1(R) Rtimes oplus SK1(R)

III8 Proposition (Whiteheadrsquos lemma) mdash The commutator subgroup of GLinfin(R) isgenerated by the elementary matrices

1 0 middot middot middot 0

0 r

00 middot middot middot 0 1

= n + reij

for i j isin 1 n i 6= j and r isin R in other words by those (finite) matrices with alldiagonal entries equal to 1 and one off-diagonal entry equal to some r isin R

So SK1(R) is the obstruction group for doing row operations to get any matrix intodiagonal form In particular SK1(R) = 0 if R is a euclidean domain

Proof Omitted But Fabianrsquos notes [AampHK Proposition III8] have some details


Note that SK1(R) = 0 is not necessarily true when R is a PID so we really need R tohave a euclidean algorithm However counterexamples are rather obscure (due to the factthat SK1(OF ) = 0 whenever OF is the ring of integers in a number field) One can takeR = Z[X][Φminus1

n | n isin N] for example where Φn denotes the nth cyclotomic polynomial

159

Towards the K-Theory of Finite Fields

Towards the K-Theory of Finite FieldsQuillenrsquos computation of Klowast(Fq) proceeds in a rather roundabout way with the essentialstep being a comparison with complex K-theory This necessitates a detour into the landof (various versions of) topological K-theory But before that wersquoll take another detourand compute the rational K-groups Klowast(R)otimesQ which will already have a heavy topologicalflavour

Rational K-TheoryRecall that the rational cohomology of any anima X isin An has a comultiplication

∆ Hlowast(XQ) ∆lowastminus Hlowast(X timesXQ) sim minus Hlowast(XQ)otimesHlowast(XQ)

where the right arrow is the Kuumlnneth isomorphism It is coassociative and counital (thecounit being ε Hlowast(XQ) Hlowast(lowastQ) Q) hence makes Hlowast(XQ) into a coalgebramdashinfancy words an element of AlgAssoc((Abotimes)op) As it turns out Hlowast(XQ) is moreover gradedcommutative

If X is connected we have a canonical element 1 isin H0(XQ) given by the image of1 isin H0(XZ) Z An arbitrary element α isin Hlowast(XQ) is then called primitive if

∆(α) = 1otimes α+ αotimes 1

To acquaint myself with the comultiplication a little more I decided to put some of itsproperties (which are probably clear to you) into a lemma

III8a Lemma mdash Let X be connected(a) The primitive elements form a sub-Q-vector space of Hlowast(XQ)(b) For general α isin Hlowast(XQ) we have

∆(α) = 1otimes α+ αotimes 1 +sumi

αprimei otimes αprimeprimei

where the αprimei and αprimeprimei are elements sitting in positive degrees(c) The rational Hurewicz map h πlowast(Xx)otimesQ Hlowast(XQ) lands in the primitives

Proof Part (a) is clear since ∆ is Q-linear For (b) we consider the following diagram

Hlowast(lowast timesXQ) QotimesHlowast(XQ)

Hlowast(XQ) Hlowast(X timesXQ) Hlowast(XQ)otimesHlowast(XQ)

Hlowast(X times lowastQ) Hlowast(XQ)otimesQ

sim

1otimesminus

∆lowast

minusotimes1

pr2lowast

pr1lowast

εotimesid

idotimesε

sim

sim

The diagram shows that the part of ∆(α) not contained inoplus

ijgt1Hi(XQ)otimesHj(XQ) mustbe 1otimes α+ αotimes 1 as claimed

160


Finally for (c) take some element [f ] isin πn(Xx) represented by a map f Sn XThen h[f ] is in the image of flowast Hlowast(SnQ) Hlowast(XQ) and hence it suffices to check thatall elements of Hlowast(SnQ) are primitive But Hlowast(SnQ) is only non-zero in degrees 0 and nand ∆ respects the degree of an element hence the formula from (b) shows that all elementsmust indeed be primitive

III9 Proposition (CartanndashSerre) mdash Let X be a connected simple anima That is theaction of π1(X) on πn(X) is trivial for all n (in particular π1(X) acts trivially on itself itmust be abelian since its a well-known fact that π1(X) acts on itself via conjugation) Supposefurthermore that the rational cohomology of X is a free graded-commutative Q-algebra offinite type Then the rational Hurewicz map

πlowast(X)otimesQ minus Hlowast(XQ)

is an isomorphism onto the primitives for all lowast gt 0Proof sketch Choose free algebra generators xi isin Hni(XQ) i = 1 r By Theorem I56athese determine (up to homotopy) a map

f X minusrprodi=1

K(Q ni)

We claim() The map f induces an isomorphism on rational homotopy groups πlowast otimesQOnce we have () it suffices to check the assertion of the proposition in the special casewhere X is a rational gem (ldquogeneralised EilenbergndashMacLane spacerdquo) This can be done bya direct computation using that the rational cohomology of EilenbergndashMacLane spaces isknown

Since we also need it to show () letrsquos recall the result (a reference is [Die08 Theo-rem (2071)])

Hlowast(K(Z n)Q

)= Hlowast

(K(Q n)Q

)=Q[t2i] if n = 2iΛQ[t2i+1] if n = 2i+ 1

(since K(Z n) K(Q n) is an isomorphism on rational homotopy groups itrsquos also anisomorphism on rational cohomology by Serrersquos rational Hurewicz theorem which is in turnan application of the LerayndashSerre spectral sequence) So if n = 2i is even we get a polynomialring on a single generator in degree 2i and if n = 2i+ 1 is odd we get an alternating algebraon a single generator in degree 2i + 1 Note that ΛQ[t2i+1] = Hlowast(S2i+1Q) has only twonon-zero degrees

By assumption on X and our choice of the xi this implies that the map

flowast rotimesi=1

Hlowast(K(Q ni)Q

) Hlowast

(rprodi=1

K(Q ni)Q)simminus Hlowast(XQ)

is an isomorphism In other words f is an isomorphism in rational cohomology hence alsoin rational homology (all cohomology groups are finite-dimensional Q-vector spaces thusisomorphic to their duals) Now () follows from Serrersquos rational Hurewicz theorem

III9a Remark mdash Fabian remarks that there are versions of Proposition III9 that alsowork without the finiteness assumptions But then one has to be a bit careful to put theright topology on our graded rings and also to replace ldquofreerdquo by ldquotopologically freerdquo

161


III9b Hopf Algebras mdash IfM is a connected E1-group whose rational homology groupsHlowast(MQ) are finite-dimensional Q-vector spaces for all n gt 0 then Proposition III9 appliesto M Indeed the fact that M is simple holds more generally for all H-spaces see [Hat02Example 4A3] (but mind that Hatcher uses ldquoabelianrdquo rather than ldquosimplerdquo) To see thatHlowast(MQ) is a free graded commutative Q-algebra we need to use that A = Hlowast(MQ) is aHopf algebra over Q This means the following(a) A has a multiplication micro A otimes A A and a unit map u Q A which turn A into

an associative and unital (but not necessarily commutative) Q-algebra In our caseA = Hlowast(MQ) = πlowast(HQ[M ]) we use that HQ[M ] is an E1-ring spectrum hence itshomotopy groups form a graded ring (see II66)

(b) A has a comultiplication ∆ A A otimes A and a counit ε A Q turning A into acoassociative and counital Q-coalgebra In our case A = Hlowast(MQ) we use the structurefrom the beginning of the subsection

(c) micro and u are morphisms of Q-coalgebras or equivalently ∆ and ε are morphism ofQ-algebras Irsquoll leave it to you to verify that this is satisfied in our case

(d) A has an antipode ie a Q-linear map i A A fitting into a commutative diagram

AotimesA AotimesA

A Q A

AotimesA AotimesA

(idAi)

micro

∆

∆

ε u

(iidA)

micro

In our case we can i to be induced by (minus)minus1 M M Any graded commutative and degree-wise finite-dimensional Hopf algebra A over Q has

a free graded commutative underlying algebra (see [Hat02 Theorem 3C4] for example)In particular if the rational homology of our connected E1-group M is degree-wise finitedimensional we may apply this result to the dual Hopf algebra Hlowast(MQ) Hlowast(MQ)or(which is graded commutative since Hlowast(MQ) is graded cocommutative) and obtain thatHlowast(MQ) is a free graded commutative Q-algebra Hence Proposition III9 applies to Mand we obtain that

πlowast(M)otimesQ minus Hlowast(MQ)is an isomorphism onto the primitives

This is still true in the case where Hlowast(MQ) isnrsquot necessarily finite-dimensional in everydegree since Proposition III9 can also be extended to this case (see Remark III9a above)Following [MM65 Appendix] we can say even more Let A be a nononegatively gradedHopf algebra over Q with A0 = Q (ie A is connected) Then the primitive elementsP (A) sube A form a Lie algebra under the commutator and if A is graded cocommutativethen U(P (A)) A by a theorem of Leray (where U(g) denotes the universal envelopingalgebra of a Lie algebra g) Applying this to the graded cocommutative connected Hopfalgebra Hlowast(MQ) we get

U(πlowast(M)otimesQ

) Hlowast(MQ)

The Lie algebra structure on πlowast(M) otimes Q can be described using the Samelson product[minusminus] M andM M

162


We can still say more Let A be a nonnegatively graded Hopf algebra over Q withA0 = Q If A is both graded commutative and graded cocommutative then it must be freeon its primitive elements by another general result A reference for this result is [MM65Corollary 418] but this needs a short explanation What the reference says is that in thiscase the primitive elements of A coincide with its indecomposables ie with

indeclowastA = coker(Agt0 otimesAgt0

microminus Agt0

)

If you think about this for a moment this precisely means that A is free on its primitives asdesired Observe that if M is a connected Einfin-monoid then the Hopf algebra Hlowast(MQ) isboth graded commutative and cocommutative hence the result can be applied and Hlowast(MQ)it is free on its primitive elements πlowast(M)otimesQ In the special case M Ωinfin for some E isin Spthe discussion around Corollary III5 shows πlowast(E)otimesQ πlowast(EotimesHQ) Put the right-hand sideis Hlowast(EQ) by Corollary II55b Hence the primitive elements πlowastΩinfinE otimesQ Hlowastgt0(EQ)are given by the nonnegative part of the homology of E This shows that

Hlowast(ΩinfinEQ) SymgrQ(Hlowastgt0(EQ)

)is the free graded commutative Q-algebra on Hlowastgt0(EQ)

III10 Corollary mdash For any ring R and all i gt 1 there are a canonical isomorphisms

Ki(R)otimesQ indeciHgrplowast(

GLinfin(R)Q)

Here Hgrplowast (GLinfin(R)minus) is understood to denote colimnisinNH

grplowast (GLn(R)minus)

Proof By definition we have Ki(R)otimesQ πi(k(R)0)otimesQ which is isomorphic to the degree-i primitive elements of Hlowast(k(R)0Q) by the discussion above since k(R)0 is a connectedEinfin-group But from Corollary III7 and the universal coefficient theorems for singularhomology and group homology we get Hlowast(k(R)0Q) Hgrp

lowast (GLinfin(R)Q) Since this is agraded commutative and graded cocommutative Hopf algebra its primitive elements coincidewith its indecomposables by [MM65 Corollary 418] whence we are done

Corollary III10 reduces the computation of rational K-theory to a problem purely fromgroup homology This still isnrsquot easy by any means but known in many cases For examplewe have the following theorem (whose proof as Fabian explained is pretty crazy and waybeyond the scope of this lecture)

III11 Theorem (Borel) mdash If OF is the ring of integers in a number field F (ie afinite extension FQ) then the K-groups of OF are given by

Ki(OF )otimesQ =

Q if i = 00 if i = 1 or i gt 0 evenQr+s if i equiv 1 mod 4 and i gt 1Qs if i equiv 3 mod 4

Here r and s are the numbers of real and complex embeddings of F respectively

The Cyclic Invariance Condition and Quillenrsquos Plus ConstructionLetrsquos analyse what goes wrong with the group completion theorem before taking S[minus] aspromised after Corollary III6a

163


III11a A Subtlety mdash For simplicity we assume thatM isin CMon(An) is an Einfin-monoidfor which there is an s isin π0(M) with (π0(M))[sminus1] π0(M)grp This is always true in ourcases of interest for example wersquove seen it for M = S and M = k(R) in the proofs ofCorollaries III6a and III7 If yoursquore not willing to make this assumption yoursquoll have toiterate the constructions to come

Since CMon(An) CAlg(Antimes) by Theorem II43 we can form the module categoryLModM (Antimes) as in Example II45(e) The adjunction from Corollary II56g showsπ0 HomLModM (Antimes)(MM) π0 HomAn(lowastM) π0(M) hence multiplication with s isan M -module map (as we would expect) Now put

T (M s) colimN

(M

sminusM

sminusM

sminus

)

By Corollary II56e T (M s) is naturally an M -module again But T (M s) is usually nots-local ie the map s T (M s) T (M s) fails to be an equivalence

For example take M = S and s = [1] then T (S [1]) Z times BSinfin as seen afterCorollary III6a But multiplication with [1] does not correspond to shifting componentsIt does take n times BSinfin n + 1 times BSinfin but this map is not the identity on BSinfinand in fact not even an equivalence Instead it can be described as follows Letrsquos identifypermutations with their corresponding permutation matrices and consider the map

ϕ Sinfin minus Sinfin

A 7minus

(A 00 1

)

Then [1] n timesBSinfin n+ 1 timesBSinfin is given by Bϕ BSinfin BSinfin This canrsquot be anequivalence since itrsquos not even the induced morphism on fundamental groups which is ϕ isan isomorphism

To unwind whatrsquos going on let temporarily r isin π0(M) be another element (wersquoll soontake r = s but doing it right away would be confusing rather than illuminating) Then themultiplication r T (M s) T (M s) can be described by a big diagram

T (M s)

T (M s)

r colimN

M M M

M M M

r

s

τ r

s

τ

s

r

s s s

The homotopy τ occurring in every square comes from the symmetric monoidal structure onM so in particular τ is an equivalence and the squares commute However in the special caser = s τ doesnrsquot need to be the constant homotopy ie the identity in HomFun(MM)(s2 s2)but rather whatever equivalence the symmetric monoidal structure gives us After allcommutativity is a structure not a property

In the example of M = S the symmetry isomorphism between [n] middot [m] S S and[m] middot [n] S S is not the identity Indeed on matrix representations of permutations wemay visualise [n] middot [m] and [m] middot [n] as

[n] middot [m] A 7minus

A 0 00 m 00 0 n

and [m] middot [n] A 7minus

A 0 00 n 00 0 m

164


and in this picture the symmetry isomorphism ldquoflips the lower two blocksrdquo This looks supertautological and you really have to look twice to see why this isnrsquot the identity But it reallyisnrsquot If m = n = 1 then [1] middot [1] S S adds two elements to each finite set S isin S andthe symmetry isomorphism swaps these two new elements

Back to the general situation Since wersquove seen that s T (M s) T (M s) might not bean equivalence the argument we would normally use cannot apply So what is this ldquousualargumentrdquo If s T (M s) T (M s) denotes the map induced by the constant homotopieswe can construct an inverse t T (M s) T (M s) by means of the diagram

T (M s)

T (M s)

t colimN

M M M

M M M

s

id

s s

id

s

s

s

s

s

s

(use cofinality to see that t is really an inverse) But since s T (M s) T (M s) doesnrsquotnecessarily coincide with s the map t doesnrsquot necessarily provide an inverseIII11b Why Does It Work for Spectra Though mdash For an Einfin-ring spectrum Ra module M over it and an element s isin π0(R) we have defined the localisation M [sminus1] asT (M s) in Proposition III2 and have had no problems with M not being s-local This wasbecause we can detect on homotopy groups whether s T (M s) T (M s) is an equivalenceThe reason why this fails in general even though equivalences can still be detected onhomotopy groups of underlying anima are basepoint issues In general we must consider allpossible basepoints and there might be non-equivalent connected components whereas forspectra the canonical choice of basepoints always suffices (which was something we had toargue in the proof of Lemma II23c)III11c When Does It Work in General mdash Wersquove seen in III11a that the maps T (M s) T (M s) is an equivalence if τ = id But we can do better Actually it sufficesto have τn = id isin HomFun(MM)(sn+1 sn+1) for some n gt 1 since we can also write

T (M s) colimN

(M

snminusM

snminusM

snminus

)by cofinality Moreover we donrsquot need the 2-cell τ (or τn for that matter) to be trivial rightaway it suffices when itrsquos trivial after mapping to T (M s)

To formalise these considerations letrsquos construct natural maps Sn π1(Mmn) for allm isinM and n gt 1 Irsquom still a bit confused about how we did this in the lecture so Irsquoll givean alternative description of (hopefully) the same construction Since M isin CMon(An) itdefines a functor M (minus) Lop An sending 〈n〉 to Mn From functoriality of M (minus) we get ahomotopy-commutative diagram

Sn HomLop(〈n〉〈n〉

)HomAn(MnMn) HomAn

((m m)Mn

)Mn

lowast HomLop(〈n〉〈1〉

)HomAn(MnM) HomAn

((m m)M

)M

(fn)lowast microlowast microlowast

fn

It induces a map Sn Mn timesM micro unique up to contractible choice But upon inspectionthe composition Sn Mn from the top row sends everyone of the n points of the discreteanima Sn to the point (m m) isinMn hence we even get a map

Sn minus (m m) timesMn Mn timesM micro mn timesM mn ΩmnM

165


again unique up to contractible choice This finally provides an honestly unique mapSn π0ΩmnM π1(Mmn) With this we can formulate the following criterion

III12 Proposition (ldquoCyclic invariance criterionrdquo) mdash Let M isin CMon(An) be an Einfin-monoid with an element s isin M such that π0(M)[sminus1] π0(M)grp Then the followingconditions are equivalent(a) The map s T (M s) T (M s) is an equivalence(b) The fundamental groups of all components of T (M s) are abelian(c) The fundamental groups of all components of T (M s) are hypoabelian ie have no

perfect subgroups except the trivial group e(d) The map S3 π1(Mm3) π1(T (M s)m3) kills the permutation (123) isin S3 for all

m isinM (e) There is an n gt 2 such that Sn π1(Mmn) π1(T (M s)mn) kills the permutation

(12 n) isin Sn for all m isinM In this case T (M s) inherits a canonical Einfin-monoid structure and

Minfin-grp T (M s)

ProofLecture 2121st Jan 2021

Letrsquos prove first that (a) implies the addendum This will take us a bit longer thanin the lecture or in Fabianrsquos notes since Irsquom trying to be super precise when dealing withM -modules and their symmetric monoidal structure to unconfuse myself The reason Fabianhas to deal with this technical stuff at all is an inconspicuous but nasty problem Itrsquos not atall clear why T (M s) is an Einfin-monoid again And it doesnrsquot help us to know that CMon(An)has colimits since the maps s M M the colimit is taken over are not even maps ofEinfin-monoids

Enough of that and letrsquos get going From Theorem II62 we get a symmetric monoidalinfin-categoryMotimesM = LModM (Antimes)otimesM whose tensor unit is M and whose tensor productcommutes with colimits in either variable Wersquove already seen in III11a that T (M s) iscanonically an element ofM We claim that M T (M s) is otimesM -idempotent in the sense ofPropositionDefinition III4 Indeed since M is the tensor unit and minusotimesM T (M s) commuteswith colimits what we need to show is that

T (M s) simminus colimN

(T (M s) s

minus T (M s) sminus T (M s) s

minus )

is an equivalence But that follows from (a) Now the whole package from Proposi-tionDefinition III4 can be applied By essentially the same argument as for T (M s) we seethat an element N isinM absorbs M T (M s) iff s N simminus N is an equivalence ie iff Nis s-local Hence minusotimesM T (M s) MM is a Bousfield localisation onto the full subinfin-categoryMs subeM of s-local objects Moreover T (M s) isin CAlg(MotimesMs ) sube CAlg(MotimesM )

Now we use that

CAlg(MotimesM ) MCAlg(Antimes) MCMon(An)

(see [HA Corollary 3417] and the references in the proof of Corollary III4a) In particularT (M s) is an Einfin-monoid hence an Einfin-group since π0(T (M s)) π0(M)[sminus1] π0(M)grp isa group by assumption Hence the mapM T (M s) factors over a mapMinfin-grp T (M s)which we are to show is an equivalence By Yonedarsquos lemma it suffices to show that

HomCMon(An)(T (M s) X

) simminus HomCMon(An)(Minfin-grp X)

166


is an equivalence for all X isin CGrp(An) This can be done on fibres over HomCMon(An)(MX)That is it suffices to show that once we choose a map M X we get

HomMCMon(An)(T (M s) X

) simminus HomMCGrp(An)(Minfin-grp X)

The right-hand side is contractible since HomCMon(An)(Minfin-grp X) HomCMon(An)(MX)Writing RCMon(An) CAlg(MotimesM ) we see that X isin CAlg(MotimesM ) But X is clearlys-local since it is an Einfin-group thus it is already an element of CAlg(MotimesMs ) HencePropositionDefinition III4(c) implies

HomCAlg(MotimesM )(T (M s) X

) HomCAlg(MotimesM )(MX)

and the right-hand side is contractible as M is initial in CAlg(MotimesM ) MCMon(An) Thisfinishes our rather verbose proof of the first implication

The addendum implies (b) since the fundamental group of the unit component of anyE1-monoid is abelian by the EckmannndashHilton argument and all path components of T (M s)are equivalent since it is even an Einfin-group

The implications (b) rArr (c) (b) rArr (d) and (d) rArr (e) are all trivial For (c) rArr (e)note that A6 sube S6 is perfect hence the image of (12 6) isin A6 in the hypoabelian groupπ1(T (M s)mn) must vanish

Finally letrsquos show (e) rArr (a) Wersquove seen in III11a that s T (M s) T (M s) is anequivalence if τ = id which holds if the image of the flip (12) under S2 π1(Mm2)vanishes But as argued in III11c we really only need that some power of τ vanishes inT (M s) which replaces the vanishing of (12) under S2 π1(Mm2) with the vanishing of(12 n) under Sn π1(Mmn)

If the assumptions of Proposition III12 are not satisfied we can still describe Minfin-grp interms of T (M s) To this end let Anhypo sube An be the full sub-infin-category of hypoabeliananima ie those X for which π1(Xx) has no perfect subgroup except the trivial group efor every basepoint x isin X

III13 Proposition (Kervaire Quillen) mdash The inclusion Anhypo sube An admits a leftadjoint (minus)+ An Anhypo The natural unit map X X+ induces an equivalence

S[X] simminus S[X+]

hence an isomorphism in homology Moreover (minus)+ preserves products

As you undoubtedly have guessed already (minus)+ is called the Quillen plus constructionwhich is a bad name since the construction is actually due to Kervaire

Proof of Proposition III13 Throughout the proof we assume without restriction that X isconnected (so no basepoints of fundamental groups need to be specified) Since the proofwill be a bit lengthy we divide it into four steps(1) We construct X+ in the case where π1(X) is perfect itself

Pick generators fi S1 X i isin I of π1(X) and form the pushout∐iisinI S1 X

lowast X

167


Then an easy excision calculation shows that H2(XXZ) = ZoplusI is free Also note thatH1(XZ) = π1(X)ab = 0 since the perfect group π1(X) equals its own commutator

Hence the long exact homology sequence

minus H2(XZ) minus H2(XXZ) minus H1(XZ) = 0

shows that H2(XZ) H2(XXZ) is surjective Itrsquos also easy to check that Itrsquos easy to seethat X is simply connected for example use Seifertndashvan Kampen ([Hat02 Theorem 120])plus some fiddling to get its assumptions fulfilled Hence π2(X) = H2(XZ) by Hurewiczrsquostheorem Putting everything together we see that we may pick maps gi S2 X i isin Isuch that their images in H2(XXZ) form a basis Now put∐

iisinI S2 X

lowast X+

Again itrsquos easy to check that X+ is simply connected Also X X+ is an isomorphism inhomology To see this one can for example use the same excision calculation as before tocompute that Hi(X+ XZ) equals ZoplusI for i = 3 and vanishes in all other degrees and thenuse the long exact sequence of the triple (X+ XX) to deduce Hlowast(X+ XZ) = 0(2) Still in the case where π1(X) is perfect we use obstruction theory to verify that

HomAn(X+ Z) simminus HomAn(XZ) is an equivalence for all hypoabelian anima ZWe didnrsquot have time for this step in the lecture so herersquos my own argument It suffices

to check that

HomAn(KHomAn(X+ Z)

)minus π0 HomAn

(KHomAn(XZ)

)is a bijection for all anima K We may rewrite the left-hand side as π0 HomAn(XtimesKZ) andthe right-hand side as π0 HomAn(X+ timesKZ) By the Kuumlnneth theorem and the universalcoefficients formulas we get that X timesK X+ timesK is an isomorphism in homology andcohomology with arbitrary coefficients In particular

Hlowast(X+ timesKX timesKπn(Z z)

)= 0

(as long as the homotopy group that occurs is abelian) If Z is simply connected or at leastsimple we can apply [Hat02 Corollary 473] (mind that Hatcher calls simple spaces ldquoabelianrdquoinstead) to see that π0 HomAn(X+ timesKZ) π0 HomAn(X timesKZ) But itrsquos also injectiveThe map

(X+ timesK)times 0 1 cup (X timesK)times∆1 minus (X+ timesK)times∆1

is a homology equivalence again (use MayerndashVietoris for example) hence we may use thesame argument to lift homotopies This settles the case where Z is simply connected

For the general case the idea is to consider the universal covering of Z but the detailsget a little technical First it suffices that

HomlowastAn((X+ x) (Z z)

) simminus HomlowastAn((Xx) (Z z)

)is an equivalence for all choices of base points since equivalences of anima can be checkedon (derived) fibres Since evx F(XZ) F(x Z) = Z is a Kan fibration we may use

168


Flowast((Xx) (Z z)) = F(XZ) timesZ z and similarly Flowast((X+ x) (Z z)) = F(X+ Z) timesZ zwith pullbacks taken in sSet as explicit simplicial models for the Hom anima in lowastAnMoreover we may assume that Z = Sing Y for some actual CW complex Y In this situationwe claim that there is an actual equality of simplicial sets

Flowast((Xx)Sing(Y y)

)= Flowast

((Xx)Sing(Y y)

)

where (Y y) (Y y) is the universal covering Together with the analogous assertion forX+ this will seal the deal since we already know that HomlowastAn((Xx)Sing(Y y)) andHomlowastAn((X+ x)Sing(Y y)) are equivalent from the simply connected case

To see the claim we unwind that the set of n-simplices Flowast((Xx)Sing(Y y))n is the setof all maps Xtimes∆n Sing Y that send xtimes∆n to the basepoint y or by adjunction theset of all continuous maps f |X| times |∆n| Y with the same basepoint property But sinceπ1(|X|times |∆n|) = π1(X) is perfect whereas π1(Y y) is hypoabelian any such continuous mapf induces the trivial map flowast = const 1 π1(|X| times |∆n|) π1(Y y) on fundamental groupsHence by covering theory (see [Hat02 Propositions 133 and 134] for example) f has aunique lift

Y

|X| times |∆n| Yf

exist f

such that f sends x times |∆n| to y And so we are done(3) We construct X+ in general and show that it satisfies the required universal property

Let P sube π1(X) be the largest perfect subgroup (just consider the subgoup generated byall perfect subgroup and check itrsquos perfect) Let X X be the covering space associatedto P ([Hat02 Theorem 138]) Note that P sube π1(X) is normal (since its conjugates arealso perfect) hence π1(X) = P by more covering theory This means that we know hw toconstruct X+ and can put

X X+

X X+

in general Since X+ is simply connected by construction a Seifertndashvan Kampen argumentagain shows π1(X+) = π1(X)P which is now hypoabelian by construction Moreover

HomAn(X+ Z) HomAn(XZ)timesHomAn(XZ) HomAn(X+ Z) HomAn(XZ)

for all hypoabelian anima Z since we already know HomAn(X Z) HomAn(X+ Z)(4) We verify that S[X] simminus S[X+] is an equivalence and that (minus)+ commutes with finite

productsBoth assertions are completely formal We have HomSp(S[X] E) HomAn(XΩinfinE) and

HomSp(S[X+] E) HomAn(X+ΩinfinE) But the right-hand sides coincide because ΩinfinE ishypoabelian In fact it is an Einfin-group and thus all its path components are equivalent andhave abelian fundamental group Thus S[X] S[X+] by Yoneda

For the second assertion lowast+ lowast holds for trivial reasons To see (X times Y )+ X+ times Y +first note that the right-hand side is hypoabelian since π1 An Grp commutes with

169


products Hence it suffices to show that the composite X times Y X+ times Y X+ times Y +

induces equivalences after taking HomAn(minus Z) for Z hypoabelian But

HomAn(X+ times Y Z) HomAn(YHomAn(X+ Z)

) HomAn

(YHomAn(XZ)

) HomAn(X times YZ)

if Z is hypoabelian hence X+timesY XtimesY induces an equivalence after taking HomAn(minus Z)and for X+ times Y X+ times Y + we can use the same argument again

III14 Proposition mdash If M isin CMon(An) has an element s isin π0(M) such thatπ0(M)[sminus1] = π0(M)grp then

T (M s)+ T (M+ s) Minfin-grp

Proof First of all Proposition III12(b) and (e) imply that T (M+ s) has abelian fundamentalgroup Indeed it suffices to check that the permutation (12 6) isin S6 gets killed underthe map S6 π1(T (M+ s)mn) but this map factors through the hypoabelian groupπ1(M+mn) in which the perfect subgroup A6 sube S6 containing (12 6) already dies Thusto show T (M s)+ T (M+ s) it suffices to check that the right-hand side satisfies theuniversal property of the Quillen plus construction

HomAn(T (M+ s) Z

) lim

Nop

(

slowastminus HomAn(M+ Z) slowast

minus HomAn(M+ Z))

limNop

(

slowastminus HomAn(MZ) slowast

minus HomAn(MZ))

HomAn(T (M s)+ Z

)holds for all Z isin Anhypo so indeed T (M s)+ T (M+ s) Now consider the square

T (M s)+ Minfin-grp

T (M+ s) (M+)infin-grp

sim sim

sim

The left vertical arrow is an equivalence as we just showed and the bottom arrow is anequivalence by Proposition III12 The right vertical arrow is an isomorphism on S[minus] usingS[M ] S[M+] by Proposition III13 and Theorem III6 hence an isomorphism on homologyby Corollary II55b hence an equivalence by Whiteheadrsquos theorem since both spaces aresimple (because any H-space is see [Hat02 Example 4A3] but mind that Hatcher usesldquoabelianrdquo instead of ldquosimplerdquo)

We obtain Quillenrsquos first definition of algebraic K-theory

III14a Corollary mdash For any ring R

k(R) = K0(R)timesBGLinfin(R)+

170


Proof By Proposition III14 and the arguments from the beginning of the proof of Corol-lary III7 we have

k(R) Proj(R)infin-grp colimN

(Proj(R) middot[R]

minusminus Proj(R) middot[R]minusminus

)+

Isolating the 0-component on both sides gives k0(R) (colimnisinNBGLn(R))+ BGLinfin(R)+Now k(R) π0k(R)times k(R)0 K0(R)timesBGLinfin(R)+ as all components in an Einfin-group areequivalent

Topological K-TheoryOur next goal is to get some understanding of ku and the Adams operations on it whichwill play a prominent role in Quillenrsquos comparison with the K-theory of finite fields Thefirst thing to do is to get another description of ku as well as the other Einfin-groups ko ktopand ksph from Definition II14a

III15 Corollary mdash For V isin OUTopG put BV = colimnisinNBV(n) where in thelast two cases we define Top(n) = AutTop(Rn) equipped with the compact-open topologyand G(n) = AutlowastAn(Sn lowast) Then

ko ZtimesBO ku ZtimesBU ktop ZtimesBTop and ksph ZtimesBG

Moreover if X is a finitely dominated CW complex ie a retract of a finite one then thecomparison maps from II14 are equivalences

VectR(X)infin-grp simminus HomAn(X ko) EuclR(X)infin-grp minus HomAn(X ktop) VectC(X)infin-grp simminus HomAn(X ku) SphR(X)infin-grp simminus HomAn(X ksph)

In particular ko0(X) = π0 Hom(X ko) and ku0(X) = π0 HomAn(X ku) are the groupcompletions of the (ordinary) monoids of isomorphism classes of real and complex vectorbundles on X respectively

Proof Wersquoll only do the cases of ko and ku but Fabianrsquos script [AampHK Chapter IIIpp 21ndash24] also has intersting things to say about ksph First recall Lemma II14b whichsays

VectR ∐ngt0


BU(n)

Hence π0VectR = N and π0VectC = N with generators [R] and [C] respectively and we find

T(VectR [R]

) ZtimesBO and T

(VectC [C]

) ZtimesBU

Since B shift homotopy groups up we get π1(BO) π0(O) plusmn1 (via the determinant)and likewise π1(BU) π0(U) 1 These groups are abelian thus Proposition III12(b)shows that ZtimesBO and ZtimesBU are already the Einfin-group completions of VectR and VectCThat is they coincide with ko and ku as claimed

For the assertion about the comparison maps recall VectR(X) HomAn(XVectR) (seeII14) If X is a finite anima then HomAn(Xminus) commutes with sequential (and even filtered)colimits This clearly still holds if X is only finitely dominated Hence

HomAn(X ko) HomAn(XT (VectR [R])

) T

(VectR(X) [X times R]

)

171


Since ko is an Einfin-group so is the left-hand side hence so is the right-hand side Now usethe fact that every vector bundle on X is a direct summand of a trivial bundle to see thats = [X times R] satisfies the condition from Proposition III12 and then that proposition showsT (VectR(X) [X times R]) VectR(X)infin-grp For the additional assertion recall

π0 HomAn(XVectR) π0VectR(X)

(the VectR on the right-hand side has no curly ldquoVrdquo and denotes the groupoid of real vectorbundles on X) by the classification of principal bundles and Lemma II14c Hence

ko0(X) = π0 HomAn(X ko) = π0(

HomAn(XVectR)infin-grp) =(π0VectR(X)

)grp

as claimed The complex case works completely analogous

III15a kolowast(minus) and kulowast(minus) as Cohomology Theories mdash Recall that the tensorproduct minus otimesR minus on Proj(R) turns Binfink(R) into an Einfin-ring spectrum as explained inII54(clowast) The same argument essentially works again for the real tensor product minusotimesR minuson VectR and the complex tensor product minus otimesC minus on VectC up to some check that theyare compatible with the Kan enrichment (as defined in II14) to ensure that the coherentnerves VectR and VectC are indeed elements of CAlg(CMon(An)otimes) Up to this technicalitywe see that Binfinko and Binfinku are Einfin-ring spectra Also note that his doesnrsquot work for ktopor ksph

Using Corollary II55b we now recognize

koi(X) = πi HomAn(X ko) = πi homSp(S[X] Binfinko

)= (Binfinko)i(X)

for all i gt 0 In particular kolowast(minus) πlowastHomAn(minus ko) and kulowast(minus) πlowastHomAn(minus ku)can be canonically extended to cohomology theories in both positive and negative degreesHowever this is not the extension which topologists are usually calling ldquoK-theoryrdquo Insteadit will turn out (and this is the content of the famous Bott periodicity theorem which willappear as Theorem III16 in a moment) that in positive degrees kolowast(minus) and kulowast(minus) are8-periodic and 2-periodic respectively and the ldquousualrdquo way to extend them is to make themperiodic in all integral degrees

If yoursquove seen topological K-theory before this situation might look familiar One usuallyconstructs K0(X) as the group completion of VectR(X) or VectC(X) and then one getsK-theory in negative degrees (which correspond to positive degrees in our sign convention seethe discussion after Corollary II55b) for free since the suspension isomorphism is supposedto be satisfied But after that therersquos the problem of extending it to all integral degrees thusmaking K-theory a cohomology theory at all This is done using Bott periodicity Wersquoveseen above that Binfinko and Binfinku provide another extension to a cohomology theory knownas connective topological K-theory To emphasise the difference the ldquousualrdquo topologicalK-theory is sometimes called periodic topological K-theory

III15b Hopf Bundles mdash To formulate Bott periodicity properly we need the followingvector bundles (see [Hat02 Examples 445ndash447] for a construction)

(γR1 RP1) isin Vect1R(S1) (γH1 HP1) isin VectR(S4) and (γO1 OP1) isin Vect8

R(S8)

(where H denotes the quaternions and O the octonions) and also

(γC1 CP1) isin Vect1C(S2)

172


(which is 1-dimensional as a complex bundle hence the Vect1C is not a typo) As mentioned

in Corollary III15 we have π0VectR(Sn) = π0 HomAn(Sn ko) Hence every vector bundleξ Sn defines a homotopy class of maps ξ Sn ko If the vector bundle in question is d-dimensional the map ξ has image in dtimesBO sube ZtimesBO ko Similar to our considerationsin III11a there is a map +[R] ko ko (induced by the one we took the colimit over toconstruct ko T (VectR) [R]) in the first place) which sends i timesBO i+ 1 timesBO Ifminus[Rd] denotes the d-fold iteration of its inverse then we can postcompose ξ with minus[Rd] toobtain a map ξ minus [Rd] Sn ko which lands in the 0-component ko0 0 times BO hencedefines an element of πn(ko0)

Applying this construction to the Hopf bundles γR1 γH1 and γO1 and its complex analogue

to the Hopf bundle γC1 we obtain elements

η = γR1 minus [R] isin π1(ko0) β = γC1 minus [C] isin π1(ku0) σ = γH1 minus [R4] isin π4(ko0) ν = γO1 minus [R4] isin π8(ko0)

Since Binfinko and Binfinku are Einfin-ring spectra we see that πlowast(ko0) = πlowast(Binfinko) and πlowast(ku0) =πlowast(Binfinku) are graded commutative rings by II64 (and more generally the same is true forkolowast(X) and kulowast(X) using the cup product from II65) In fact wersquoll see in a second thatthey are even honest commutative since the odd degrees vanish or have characteristic 2III16 Theorem (Bott periodicity) mdash The elements β isin πlowast(ku0) and η σ ν isin πlowast(ko0)define isomorphisms

Z[β] simminus πlowast(ku0) and Z[η σ ν](2η η3 ησ σ2 minus 4ν) simminus πlowast(ko0)

In particular we can refine the description from II14 as follows

πn(ku) Z〈βi〉 if n = 2i0 if n = 2i+ 1

and πn(ko)

Z〈νi〉 if n = 8iZ2〈ηνi〉 if n = 8i+ 1Z2〈η2νi〉 if n = 8i+ 20 if n = 8i+ 3Z〈σνi〉 if n = 8i+ 40 if n = 8i+ 50 if n = 8i+ 60 if n = 8i+ 7

We wonrsquot prove Theorem III16 If you want to read up on this yoursquoll find proofs in anybook on topological K-theory but Fabian particularly likes Bottrsquos original proof the mostaccessible treatment of which can probably be found in [Mil63 sectsect23ndash24]

As a consequence we can now define spectra KU and KO with Ωinfinminus2iKU = ku andΩinfinminus8iKO ko for all i isin Z which are thus 2-periodic and 8-periodic respectively Anotherdescription is

KU (Binfinku)[βminus1] and KO (Binfinko)[νminus1](in Proposition III2 we only saw how to localise elements from π0 but the constructionis straightforward to generalise to elements from arbitrary degrees) In this formluationwe immediately see KOKU isin CAlg The associated cohomology theories KUlowast(minus) andKOlowast(minus) were studied by Atiyah and Hirzebruch and are known as (periodic) topologicalK-theory They coincide with kulowast(minus) and kolowast(minus) in nonnegative degrees (which correspondto nonpositive degrees in the usual sign convention)

173


III17 Adams Operations on K-Theory mdash The Adams operations are certain mapsψi ku ku Wersquoll construct them in two steps(1) Wersquoll massage the exterior power maps Λi VectC(X) VectC(X) on complex vector

bundles long enough to turn them into maps λi ku ku of anima (not respecting anyEinfin-structures though)

Letrsquos suppose X is finitely dominated at first Then ku0(X) (π0VectC(X))grp byCorollary III15 The main obstacle later on will be to bypass the finiteness assumptions onX but right now therersquos another problem We would like to extend

Λi π0VectC(X) minus ku0(X)

to all of ku0(X) However knowing that ku0(X) (π0VectC(X))grp (as X is assumed to befinitely dominated) doesnrsquot help us as Λi is not even a map of monoids However this canbe easily fixed as follows Instead of considering each Λi on its own we combine all of theminto a single map

Λ π0VectC(X) minus ku0(X)JtK

[V ] 7minussumigt0

[ΛiV ]ti

where ku0(X)JtK denotes the ring of power series over ku0(X) which is a ring itself as notedbefore Theorem III16 From the formula

Λn(V oplusW ) =oplusi+j=n

Λi(V )otimes Λj(V )

we get that Λ is a homomorphism onto the subgroup 1 + t middot ku0(X)JtK sube ku0(X)JtKtimes ofthe group of units of the ring ku0(X)JtK In the case where X is finitely dominated henceku0(X) (π0VectC(X))grp we may thus extend Λ to a map

Λ ku0(X) minus ku0(X)JtK

which is a group homomorphism into ku0(X)JtKtimes Writing Λ =sumigt0 t

iΛi we obtain naturalmaps Λi ku0(X) ku0(X) for all finitely dominated X (but note that the Λi neitherpreserve the additive nor the multiplicative structure)

We would like to extend the Λi to natural maps Λi ku0(X) ku0(X) for all animaX ie to a natural transformation from the functor ku0(minus) (πAn)op Set to itself Bydefinition we have

ku0(X) = π0 HomAn(X ku) = HomπAn(X ku)

hence ku0(minus) = HomπAn(minus ku) is represented by ku By the 1-categorical Yoneda lemmaany endotransformation of ku0(minus) thus corresponds to a map ku ku in πAn or putdifferently to an element of ku0(ku) The idea how to construct the element λi isin ku0(ku)wersquore looking for is simple We ldquoapproximaterdquo ku by suitable finite anima on which wealready know what Λi does and show that all finite stages together combine into a uniqueelement of ku0(ku) Letrsquos give a sketch of how to make this precise

First off we need to compute ku0(ku) Now both ku and KU (and more generally anyEinfin-ring spectrum E satisfying π3(E) = π5(E) = = π2n+1(E) = ) is an example of a

174


multiplicative complex oriented cohomology theory These are Einfin-ring spectra E for whichthe natural inclusion S2 CP1 sube CPinfin induces a surjection E2(CPinfin) E2(S2) A generalfact about multiplicative complex oriented cohomology theories is the formula

Elowast(BU(n)

) πlowast(E)Jc1 cnK

where the ci isin Eminus2i(BU(n)) are the Chern classes in E-cohomology The power series ringis to be understood in a graded sense That is we allow infinite sums of elements of thesame degree but not of different degrees Alternatively you can work with the conventionthat graded cohomology rings are products Elowast(X) =

prodnisinZE

n(Z) rather than direct sumsoplusnisinNE

n(X) and get a power series ring on the nose A proof of the formula can be foundin [Lur10 Lecture 4] but be warned that Luriersquos sign convention is Elowast(X) = πminuslowast(EX) Soin particular Lurie has to invert the grading on πlowast(E) to make the formula work (which hedoesnrsquot tell you) Here we have a clear instance where Fabianrsquos sign convention is superior

Apply the above formula to E ku which has πlowast(ku) = Z[β] where β in degree 2 is theBott element from Theorem III16 Using ku ZtimesBU we find

ku0(ku) =prodnisinZ

Zqβici

∣∣ i gt 1y

Now that we have computed ku0(ku) we enter phase 2 of our plan and try to ldquoapproximaterdquoku by finite anima Let Grk(Cm) be the Grassmannian parametrising k-dimensional sub-C-vector spaces of Cn and let γCkm Grk(Cm) be the tautological bundle By the classificationof vector bundles it defines a homotopy class of maps Grk(Cm) BU(k) Furthermore wehave a maps BU(k) ntimesBU sube ku for all n isin Z Letting k m and n vary thus providesa single map

ku0(ku) minusprodnisinZ

prod06k6m

ku0(Grk(Cm))

This map is injective which follows essentially from the formula for ku0(ku) above and fromthe definition of Chern classes Now the Grassmannians Grk(Cm) are finite CW complexeshence we already know what Λi ku0(Grk(Cm)) ku0(Grk(Cm)) is supposed to do Usingthe injectivity above one checks that the [ΛiγCkn] combine into a unique element λi isin ku0(ku)which is what wersquore looking for

The upshot is that we get natural maps λi ku ku (these are maps of anima but fonrsquotpreserve any Einfin-structures) For any anima X they induce maps λi ku0(X) ku0(X)which satisfy

λn(x+ y) =sumi+j=n

λi(x) middot λj(y) isin ku0(X)

for all x y isin ku0(X) Wersquove thereby reached our goal of Step (1)(2) We show how one can extract ring homomorphisms ψi ku0(X) ku0(X) from the λi

by a purely algebraic procedureThe buzzword here is that λi | i gt 0 turn ku0(X) into a λ-ring For a precise definition

consult the nLab in a nutshell it means one has a ring R together with maps λi R Rof sets that satisfy the relation above as well as a bunch of other relations for λn(xy) andλn(λm(x)) which are all satisfied for exterior powers of vector bundles and thus also in ourcase And now extracting the ψi from the λi is a purely algebraic procedure in that it canbe done for any λ-ring

175


Just as the Λi in Step (1) behave nicer once combined into a single power series the samehappens for the λi For better normalisation behaviour later one we put

λ(x) =sumigt0

(minus1)iλi(x)ti isin ku0(X)JtK

this time As for Λ one checks that λ ku0(X) 1 + t middot ku0(X)JtK sube ku0(X)JtKtimes is a grouphomomorphism into the unit group of ku0(X)JtK To get a homomorphism into the additivegroup of ku0(X)JtK instead we would like to take ldquologarithmsrdquo To this end let

`(t) =sumigt1

(minus1)iminus1ti

iisin QJtK

be the formal power series of the Taylor expansion of log(1 + x) (minus1 1) R Then thecoefficients ϕi(x) of `(λ(x) minus 1) isin (ku0(X) otimes Q)JtK are additive functions of x (note thatλminus 1 has constant coefficient 0 so we may plug it into any other formal power series) Toget integer coefficients instead of rational ones we take formal derivatives (which preservesadditivity of the coefficients) and therefore we arrive at the following definition

III17a Definition mdash For i gt 1 the ith Adams operation ψi ku0(X) ku0(X) isdefined by

minus`(λ(x)minus 1

)prime =sumigt0

ψi+1(x)ti isin ku0(X)JtK

and for i = 0 we put ψ0(x) = rk(x) middot [C] isin ku0(X) where rk ku Z times BU Z is theprojection to path components


For later use we record that the derivative in Definition III17a evaluates to

`(λ(x)minus 1

)prime = minus λ(x)prime

λ(x)= minus

sumigt0

(1minus λ(x)

)i middot λ(x)prime

III18 Proposition mdash For all X isin An we have(a) For all i gt 0 ψi ku0(X) ku0(X) is a ring homomorphism(b) For all i j gt 0 ψi ψj = ψij(c) If L isin Vect1

C(X) is a line bundle on X then ψi([L]) = [L]i = [Lotimesi] for all i gt 0(d) If p is a prime then ψp(x) equiv xp mod p That is the ψp are Frobenius lifts(e) For all i n gt 0 ψi(βn) = inβn isin ku0(S2n lowast)

We should perhaps explain the notation from Proposition III18(e) as well as why βn isan element of ku0(S2n lowast) The relative ku-group ku0(S2n lowast) is defined as

ku0(S2n lowast) = ker(ku0(S2n) ku0(lowast)

)= ker

(π0 HomAn(S2n ku) π0 HomAn(lowast ku)

)

Using π1(HomAn(lowast ku) 0) = π1(ku0) = 0 by Theorem III16 we see that ku0(S2n lowast) equalsπ2n(ku0) Z〈βn〉 Hence (e) makes indeed sense

Combining Proposition III18(a) (b) and (d) we see that the ψi | i gt 0 are uniquelydetermined by the ψp for p a prime and these guys are commuting Frobenius lifts This

176


holds in fact for all λ-rings Moreover it is a theorem of Wilkerson that conversely any choiceof commuting Frobenius lifts on a Z-torsion free ring defines a unique λ-structure Thismakes λ-rings interesting to number theorists in particular to people working in finite ormixed characteristicmdashand especially Irsquod like to add to people working in ldquocharacteristic 1rdquo

Proof sketch of Proposition III18 Parts (a) (b) and (d) work for arbitrary λ-rings and canthus be proved by horrible manipulations of power series (for (a) and (b) yoursquoll need someadditional properties of the λi that we didnrsquot specify) Wersquoll immediately see that in thecase of K-theory therersquos a dirty shortcut but before we do that let me prove (d) in generalbecause it isnrsquot so horrible at all and I like Frobenii

By Definition III17a ψp(x) is the coefficient of tpminus1 in `(λ(x)minus1)prime In our formula aboveonly those summands (1minus λ(x))i middot λ(x)prime with i 6 pminus 1 will contribute to the coefficient oftpminus1 Using this together with(

1minus λ(x))p equiv 1minus λ(x)p mod p

we get that ψp(x) modulo p is the coefficient of tpminus1 in

minuspminus1sumi=0

(1minus λ(x)

)i middot λ(x)prime equiv minus1minus

(1minus λ(x)

)p1minus

(1minus λ(x)

) middot λ(x)prime

equiv minus λ(x)p

λ(x)middot λ(x)prime

equiv minusλ(x)pminus1 middot λ(x)prime mod p

(dividing by λ(x) is fine since it is an element of ku0(X)JtKtimes as seen above) Now observethat minusλ(x)pminus1 middot λ(x)prime = minus 1

p (λ(x)p)prime and that

λ(x)p equivsumigt0

(minus1)ipλi(x)iptip mod p

Hence the coefficient of tp in λ(x)p is (minus1)pλ1(x)p + py = minusxp + py for some y Hence thecoefficient of tpminus1 in minus 1

p (λ(x)p)prime is minus 1p (minuspxp + p2y) equiv xp mod p and wersquore done

Next we prove (c) and explain how it can be used to give a quick and dirty proof of (a)(b) and (d) which only works for ku0(X) but not for general λ-rings For (c) we find thatλ([L]) = 1minus [L]t hence

minus`(λ([L])minus 1

)prime =sumigt0

[L]i+1ti

and thus ψi([L]) = [L]i for all i gt 1 For i = 0 it holds by Definition III17aTo show (a) (b) and (d) we first verify that all ψi ku0(X) ku0(X) are additive This

is easy and holds more or less by construction (we took the logarithm of a homomorphisminto the multiplicative unit group of ku0(X)JtK) To show multiplicativity as well as (b) and(d) note the these follow from additivity and (c) whenever our vector bundles [V ] isin ku0(X)can be decomposed into a direct sum V = L1 oplus middot middot middot oplusLn of line bundles This clearly doesnrsquothold in general However by the splitting principle therersquos always a map f F X suchthat flowast kulowast(X) kulowast(F ) is injective and flowast(V ) decomposes into a sum of line bundles Bynaturality of the Adams operations this allows us to reduce everything to the decomposablecase whence we are done

177


Finally we prove (e) From the 1-categorical Yoneda lemma we know that the ψi alsoexist as (homotopy classes of) maps ψi ku ku Then ψi(βn) isin ku0(S2n lowast) = π2n(ku0)corresponds to the map

S2n βn

minus ku ψi

minus ku

Using (a) we see that ψi(βn) = ψi(β)n also holds in the homotopy ring πlowast(ku0) Hence itsuffices to consider the case n = 1 In this case recall β = [γC1 ]minus [C] where [C] isin ku0(S2n)acts as the multiplicative unit and compute

ψi(β) = ψk([γC1 ]minus [C]

)= [γC1 ]i minus [C] = β

iminus1sumj=0

[γC1 ]j = iβ

Here we used (c) as well as the general fact that cup products in generalised cohomologytheories vanish on suspensions so β2 = 0 isin ku0(S2 lowast) and thus β[γC1 ]j = β holds for allj = 0 iminus 1

A Sketch of Quillenrsquos ComputationThroughout this subsection p is a prime number and q some power of p By Proposi-tion III18(a) and the 1-categorical Yoneda lemma the Adams operations induce (homotopyclasses of) maps ψi ku ku which preserve the 0-component ku0 0 timesBU

III19 Theorem (Quillen [Qui72]) mdash Any embedding ρ Ftimesp Ctimes as the groupoplus` 6=p microìnfin of all roots of unity of order coprime to p gives an equivalence

Brρ k(Fq)0simminus fib(ψq minus id BU minus BU

)

Consequently

Ki(Fq) =

Z if i = 0Z(qn minus 1) if i = 2nminus 10 else

The group microìnfin of `-power roots of unity is called the Pruumlfer `-group There are manymore ways to write it down such as Z[`minus1]Z or Q`Z` or Zìnfin Using that the group ofunits in any finite field is cyclic one easily finds Ftimesp

oplus` 6=p microìnfin But there is no canonical

embedding of this into Ctimes hence we remember ρ in the notation Brρ It should be notedthat Brρ also depends on the embedding Fq Fp into an algebraic closure which we choseto suppress in the notation

Proof sketch of Theorem III19 The computation of Klowast(Fq) = πlowast(k(Fq)0) is a consequenceof the long exact sequence of homotopy groups associated to the fibre sequence

fib(ψq minus id) minus BU ψqminusidminusminusminusminus BU

using that πlowast(BU) vanishes in odd degrees by Theorem III16 and that in even degrees themap ψqminus id π2n(BU) π2n(BU) acts as (qnminus1) Z〈βn〉 Z〈βn〉 by Proposition III18(e)

Now wersquoll sketch a proof of the first part up to the extensive group homology calculationsthat go into it To get a map

k(R)0 minus fib(ψq minus id)

178


we will first construct ldquocompatiblerdquo maps BGLn(R) fib(ψq minus id) for all n gt 1 to get amap BGLinfin(Fq) fib(ψq minus id) Then observe that the right-hand side is simple Indeeditrsquos a general fact about fibre sequences F E B that the action of π1(F f) on πlowast(F f)factors through an action of π1(E f) on πlowast(F f) Since no one (including Quillen) ever seemsto spell that out Irsquoll give the details in Lemma III19b below Anyway this shows thatfib(ψqminus id) is indeed simple since π1(BU) = 0 The fundamental group of any simple space isabelian hence the universal property from Proposition III13 together with Corollary III14aprovide an extension

k(R)0 BGLinfin(Fq)+ minus fib(ψq minus id)

If we can show that this is an isomorphism on homology then wersquore done by Whiteheadrsquostheorem since both sides are simple k(R)0 because it is an H-space and fib(ψqminus id) becausewe just checked this By Proposition III13 we may replace BGLinfin(Fq)+ by BGLinfin(Fq)and by ldquostandard argumentsrdquo it suffices to have isomorphisms on homology with coefficientsin Q Zp and Z` for all ` 6= p The latter is again something no one ever spells out sowersquoll give a proper argument in our brief discussion of completions of spectra see [TODO]

Since πlowast fib(ψq minus id) are p-torsionfree torsion abelian groups the ldquoHurewicz theoremmodulo a Serre classrdquo (see for example [Hat02 Theorem 57] in Hatcherrsquos additional chapteron spectral sequences) shows that the same must be true for Hlowast(fib(ψq minus id)Z) hence

Hlowast(

fib(ψq minus id)Q)

= 0 and Hlowast(

fib(ψq minus id)Zp)

= 0

Furthermore if G is a finite group then the group homology Hgrplowast (GA) for any coefficients

A is annihilated by G In particular it vanishes for A = Q hence

Hlowast(BGLinfin(Fq)Q

)= colim

nisinNHgrplowast(

GLn(Fq)Q)

= 0

The bulk of Quillenrsquos proof goes into comparing the homology Hlowast(BGLinfin(Fq)Z`) andHlowast(fib(ψqminus id)Z`) for ` 6= p and another significant portion into Hlowast(BGLinfin(Fq)Zp) = 0Both problems are essentially about group homology and would take us to much time sowersquoll refer to Quillenrsquos original paper [Qui72] instead

What we will do however is to construct the desired map BGLinfin(Fq) fib(ψq minus id)To give this construction some structure I decided to divide it into four steps(0) We recall some facts from the representation theory of finite groups

If you are someone like me who dodged representation theory for their entire mathematicallife Fabian recommends you put a copy of [Ser77] on your nightstand and read a page perday it will make you a better mathematician within a semester So let G be a finite groupand Rk(G) its representation ring over some field k (that is Rk(G) = K0(k[G])) Wersquoll needthe following facts(a) There is a map

rep RC(G) minus ku0(BG)

sending a representation V isin RC(G) ie a C-vector space with a G-action to thevector bundle EGtimesG V BG Alternatively it can be described as sending a grouphomomorphism r G U(n) to the map BG BU(n) n timesBU sube ku It sufficesto consider homomorphisms r G U(n) sube GLn(C) since we can always find a G-equivariant scalar product on any G-representation V (start with any scalar product〈〉prime on V and put 〈u v〉 = Gminus1sum

gisinG〈r(g)u r(g)v〉prime to make it G-equivariant)

179


(b) For any field k there are exterior power maps Λi LModk[G] LModk[G] Just as inStep (1) of III17 (minus the technical mumbo-jumbo about complex oriented cohomologytheories) these induce maps of sets λi Rk(G) Rk(G) turning the representation ringRk(G) = K0(k[G]) into a λ-ring And just as in Step (2) of III17 we can then extractAdams operations ψi Rk(G) Rk(G) from them In the case k = C it follows fromthe constructions that rep RC(G) ku0(BG) from (a) commutes with the Adamsoperations on both sides

(c) Recall that a class function on G is a function G C which is constant on conjugacyclasses One of the first result about complex representations is that the function

χ RC(G) minus class functions G C

is injective (and an isomorphism on RepC(G)otimes C) Here χ sends a representation V tothe class function χV G C given by χV (g) = tr(g V V )

(d) The map χ from (c) interacts with the Adams operations ψn from (b) as follows

χ(ψn(V )

)(g) = χ(V )(gn)

To see this fix V and g and let λ1 λd be the eigenvalues of g V V counted withalgebraic multiplicity (so that d = dimV ) Then the eigenvalues of Λmg ΛnV ΛnVare given by the products λi1 middot middot middotλim for 1 6 i1 lt middot middot middot lt im 6 m In particular tr(Λmg)is the mth elementary symmetric polynomial in the λi Hence

χ(λ(V )

)(g) =

dsumm=0

χ(ΛmV )(g)tm =dprodi=1

(1minus λit)

The derivative of the right-hand side is minussumdi=1 λi

prodj 6=i(1 minus λjt) Plugging in our

definitions together with the formula after Definition III17a thus shows

sumngt0

χ(ψn+1(V )

)(g)tn =

dsumi=1

λi1minus λit

=sumngt0

dsumi=1

λn+1i tn

Comparing coefficients plus the fact that the eigenvalues of gn are precisely λn1 λnd(for example by Jordan decomposition) proves the claim

With that out of the way letrsquos start with the actual construction(1) For any embedding ρ Ftimesp Ctimes as the group

oplus` 6=p microìnfin of all roots of unity of order

coprime to p we construct the famous Brauer lift

Brρ RFp

(GLn(Fq)

)minus RC

(GLn(Fq)

)

For any Fp-representation V of GLn(Fq) we form its Brauer character

χρV GLn(Fq) minus C

sending an element g to χρV (g) =sumλ mult(λ)ρ(λ) where the sum runs over the Fp-eigenvalues

of g V V with their algebraic multiplicities mult(λ) Since eigenvalues are preserved

180


under conjugation χρV is a class function on GLn(Fq) By a classical result of Green [Gre55Theorem 1] χρV is in the image of the injection

χ RC(GLn(Fq)) minus class functions GLn(Fq) minus C

from (c) above This gives rise to the Brauer lift Brρ RFp(GLn(Fq)) RC(GLn(Fq)) Wecan now form a diagram

RFp

(GLn(Fq)

)RC(

GLn(Fq))

RFq(

GLn(Fq))

ku0(BGLn(Fq))

Brρ

repFpotimesFqminus

Also recall that ku0(BGLn(Fq)) = π0 HomAn(BGLn(Fq) ku) Hence the canonical represen-tation GLn(Fq) y Fnq defines a homotopy class of maps BGLn(Fq) ku We have yet toshow that these maps lift to fib(ψq minus id) and that the lifts are compatible for varying n(2) As a first step in that direction we verify that the composition

RFq(

GLn(Fq))minus RFp

(GLn(Fq)

) Brρminus RC

(GLn(Fq)

)takes values in the fixed points of ψq

Since the map χ from (c) above is injective itrsquos enough to verify this after compositionwith χ Recall that Fq sube Fp is the fixed field of the Frobenius (minus)q Fp Fp Hence if λ isan eigenvalue of g Fp otimesFq V Fp otimesFq V then so are λq λq2

λq3 each of them with the

same multiplicity as λ Hence

χρFpotimesFqV

(g) =sum[micro]

mult(micro)sumλisin[micro]

ρ(λ)

where the first sum ranges over all Frobenius orbits [micro] Using assertion (d) from Step (0)shows that applying ψq only replaces g by gq But the eigenvalues of gq are the qth powersof the eigenvalues of g (for example by Jordan decomposition) and replacing each λ by λqonly permutes the summands in

sumλisin[micro] ρ(λ) hence χρ

FpotimesFqVis indeed invariant under the

action of ψqThe upshot is that we obtain a compatible sytem of maps

RFq(

GLn(Fq))minus ku0(BGLn(Fq)

)ψq = π0 HomAn(BGLn(Fq) ku

)ψq

where (minus)ψq denotes fixed points under ψq Here we also use that the canonical maprep RC(GLn(Fq)) ku0(BGLn(Fq)) from assertion (c) above is compatible with theAdams operations on both sides(3) To get a compatible system of maps BGLn(Fq) fib(ψq minus id) we show that there are

bijections

π0 HomAn(BGLn(Fq)fib(ψq minus id)

) simminus π0 HomAn(BGLn(Fq) ku

)ψq

π0 HomAn(BGLinfin(Fq)fib(ψq minus id)

) simminus limnisinN


)

181


The first map is automatically surjective Indeed the left-hand side consists of maps to thehomotopy fibre fib(ψq minus id) which given by maps to ku plus a homotopy to the identity aftercomposition with ψq ku ku On the right-hand side we forget the choice of homotopy andonly remember that there is some To show injectivity we use that HomAn(BGLn(Fq)minus)preserves fibre sequences Hence by the long exact sequence of a fibration there is asurjection from π1(BGLn(Fq) ku) = ku1(BGLn(Fq)) onto the kernel But we have thefollowing general result

III19a Theorem (AtiyahndashSegal completion theorem) mdash If G is a finite group (or acompact Lie group) then

KUi(BG) =RC(G)I if i is even0 if i is odd

where I is the kernel of rk RC(G) Z and (minus)I denotes I-adic completion

Proof of Theorem III19a See [Ati61] for the case where G is finite and [AS69] for the caseof compact Lie groups

Back to the proof of Theorem III19 The completion theorem shows ku1(BGLn(Fq)) = 0and thus the first map in (3) is indeed a bijection I suspect that there should be a lessoverkill proof that ku1 vanishes but right now Irsquom too lazy to think about this

The second map in (3) is part of a Milnor exact sequence In general the homotopygroups of a sequential limit limnisinNXn in lowastAn sit in an exact sequence

0 minus R1 limnisinN

πi+1(Xn) minus πi

(limnisinN

Xn

)minus lim

nisinNπi(Xn) minus 0

A proof of this general fact can be found in the corresponding nLab article but itrsquos also nothard to prove this yourself The Milnor sequence now shows that the second map in (3) issurjective and its kernel is given by

R1 limnisinN


)= R1 lim

nisinNku1(BGLn(Fq)

)

By the long exact homotopy group sequence and Theorem III19a ku1(BGLn(Fq)) is aquotient of RC(GLn(Fq))I Since R2limnisinN vanishes it thus suffices to show

R1 limnisinN

RC(

GLn(Fq))I

= 0

For simplicity put Rn = RC(GLn(Fq)) and In = ker(rk Rn Z) Then limmisinNRnImn

RlimmisinNRnImn because the Mittag-Leffler condition is satisfied Moreover the short exact

sequences 0 InImn RnI

mn RnIn 0 show RnI

mn = Z oplus InImn Finally the

quotient InImn is a finite abelian group by [Ati61 Proposition (613)] Putting everythingtogether we obtain

RlimnisinN

(limmisinN

RnImn

) Rlim

nisinN

(R limmisinN

RnImn

) Rlim

nisinNRnI

nn

RlimnisinN

ZoplusRlimnisinN

InInn

182


where we use that the diagonal ∆ N N times N is final (which is straightforward to verifyvia the dual of Theorem I43(b)) But R1limnisinN InI

nn vanishes because the Mittag-Leffler

condition is satisfied in any inverse system of finite abelian groups and R1limnisinN Z vanishesbecause the Mittag-Leffler condition is trivially satisfied This proves what we want andfinishes Step (3)

Therefore we have constructed a homotopy class of maps BGLinfin(Fq) fib(ψq minus id)Now extend it over the Quillen plus construction BGLinfin(Fq)+ and the computation of grouphomology may begin

We have yet to show the fibration lemma that was used to prove that fib(ψq minus id) issimple The topology gang will probably find it trivial

III19b Lemma mdash Let F Epminus B be a fibre sequence of anima Then for all f isin F

the action of π1(F f) on πlowast(F f) factors through an action of π1(E f) on πlowast(F f)

Proof Letrsquos first recall how the action of π1(F f) on πlowast(F f) works So let n denote then-cube and α (n partn) (F f) a map of pairs representing an element of πn(F f) andlet γ ∆1 F represent a loop in F with basepoint f Consider

γ middot α =(αtimes 0

)cup(

idpartn timesγ)

(n times 0

)cup(partn times∆1) minus F

Once we remember (n times 0) cup (partn times ∆1) n it provides a well-defined element[γ] middot [α] isin πn(F f) which is the action wersquore looking for

Now suppose γ ∆1 E only represents an element of π1(E f) We can still apply theconstruction γ middot α above and obtain a lifting diagram(

n times 0)cup(partn times∆1) E

n times∆1 B

γmiddotα

p

idn timesp(γ)

Restricting any lift to n times 1 gives a map n E whose restriction to partn is constanton f and whose composition with p is constant on p(f) hence an element [γ] middot [α] isin πn(F f)Itrsquos straightforward to check that the action of π1(F f) on πlowast(F f) factors through this newaction

Next we investigate how the equivalence Brρ depends on the choices of embeddings ofFq into an algebraic closure Fp and ρ Ftimesp Ctimes as the group

prodmicroinfin`

III20 Proposition mdash The computation of K-groups from Theorem III19 via Brρsatisfies the following functoriality behaviour(a) If f Fq Fqn is a field homomorphism compatible with the chosen embeddings into an

algebraic closure Fp then the following diagram commutes

K2iminus1(Fq) K2iminus1(Fqn)

Z(qi minus 1) Z(qni minus 1)

Brρ

sim

flowast

Brρ

simsumnminus1k=0

qki

183


(b) The Frobenius Frob = (minus)p Fq Fq induces the maps

pi K2iminus1(Fq) minus K2iminus1(Fq)

on K-theory

Proof For (a) note first that ψqn minus id = (ψq)n minus id = (ψq minus id)sumnminus1k=0 ψ

qk by Proposi-tion III18(b) Now per construction of the Brauer lift also the left square in

k(Fq)0 ku ku

k(Fq)0 ku ku

Brρ

flowast

ψqminusid sumnminus1k=0

ψqk

Brρ ψqnminusid

commutes From the bijectivity of



)ψq(Step (3) in the proof of Theorem III19) we then obtain that flowast agrees with the map inducedby the right-hand square on fibres Hence the claim follows from Proposition III18(e) andthe way we used it in the proof of Theorem III19

For (b) one similarly computes from the definition of the Brauer lift and Proposi-tion III18(b) that

k(Fq)0 ku ku

k(Fq)0 ku ku

Brρ

Froblowast ψp

ψqminusid

ψp

Brρ ψqminusid

commutes Then the same argument works

III21 Corollary mdash Any embedding ρ Ftimesp Ctimes induces isomorphisms

Ki(Fp)

Z if i = 0oplus

` 6=p microìnfin if i is odd0 else

Moreover if Fq Fp is any inclusion into an algebraic closure then

Klowast(Fq) simminus Klowast(Fp)Gal(FpFq)

One can actually make the computation of Klowast(Fp) canonical We already know fromCorollary III7 and Proposition III8 that K1(Fp) Ftimesp canonically In general using thatFtimesp otimesLZ Ftimesp Ftimesp [1] we get

K2iminus1(Fp) = Himinus1(Ftimesp otimesLZ middot middot middot otimesLZ Ftimesp︸︷︷︸

i times

)

and hence an Gal(FpFp) = Z-equivariant isomorphism

Klowast(Fp) = Hlowast

(FreeAlg

D(Z)otimesLZ

(Ftimesp [1]

))

Be warned however that this isomorphism only exists at the level of homology but not atthe level of Einfin-ring spectra

184


Proof of Corollary III21Lecture 2328th Jan 2021

Note that K-theory commutes with filtered colimits One wayto see this is that GLn(minus) Ring Grp(Set) commutes with filtered colimits by inspec-tion hence so does BGLn(minus) Ring An as one can check on homotopy groups (byRemark II31c) Now (minus)+ An An doesnrsquot commute with colimits in general (it onlydoes if the target is replaced with Anhypo) But in a filtered colimit colimiisinI BGLinfin(Ri)+all BGLinfin(Ri)+ have abelian fundamental groups rather than just hypoabelian ones (be-cause they are Einfin-groups) and filtered colimits of abelian groups are abelian again hencecolimiisinI BGLinfin(Ri)+ is an element of Anhypo and thus coincides with BGLinfin(colimiisinI Ri)

With that out of the way we may write Fp = colimnisinN Fpn (instead of n we could haveused any sequence converging to 0 in Z = limmisinN Zm) and get

K2i(Fp) = 0 and K2iminus1(Fp) = colimnisinN

Z((pn)i minus 1

)

The transition maps in the colimit on the right-hand side are those from Proposition III20(a)Clearly the formula shows that K2iminus1(Fp) is p-torsion free Letrsquos determine its `-powertorsion part for primes ` 6= p Note that for every r gt 1 there is anm gt 1 such that `r | pmminus1To see this one doesnrsquot need to invoke splitting fields and cyclotomic polynomials as Fabiandecided to do it suffices to note that p being coprime to ` is an element of the multiplicativegroup (Z`r)times which has finite order

In particular we obtain `r | (pn)i minus 1 for all n gt m This shows that the colimit above iscofinal in the colimit oplus

` 6=pmicroìnfin = colim

p-sZs

in which there are transition maps Zs Zt given by multiplication with ts whenever s | t

This proves K2iminus1(Fp) =oplus6=p microìnfin The additional assertion follows by inspection

III22 Completions of Spectra mdash Fabian would like to end the chapter with someremarks on the Suslin rigidity theorem But before we can do that we need to introducecompletions of spectra This took us on a short detour in the lecture I decided to makeit a slightly longer detour (so mind the asterisks) to elaborate more on the connection toderived completion in ordinary algebra and to explain one small technical step in the proofsketch of Theorem III19

Letrsquos do the spectra case first Let R be an Einfin-ring spectrum s isin π0(R) some elementand M isin ModR a module over R We define the s-completion of M as

Ms = limnisinN

MsnM

where we define MsnM as the pushout

M M

0 MsnM

sn

In the case M = R we often just write Rsn and in general MsnM M otimesR Rsn sinceminus otimesR Rsn commutes with colimits Since homR(minusM) ModR ModR (defined as in

185


II62c(a)) preserves but flips fibre sequences we see that MsnM can equivalently be writtenas homR((Rsn)[minus1]M) This also shows

Ms homR

((Rsinfin)[minus1]M

)

where we denote Rsinfin R[sminus1]R In particular the latter description shows that Ms

doesnrsquot depend on the choice of representative of s isin π0(R) Moreover it motivates thefollowing lemmadefinition

III22a LemmaDefinition mdash For an R-module spectrum M and some s isin π0(R)the following are equivalent(a) homR(R[sminus1]M) 0(b) homSp(TM) 0 for all s-local R-module spectra T

(c) The canonical morphism M simminus Ms into the s-completion is an equivalenceSuch R-module spectra are called s-complete The s-completion functor

(minus)s ModR minus Mods-compR

is a left Bousfield localisation onto the full stable sub-infin-category Mods-compR sube ModR of

s-complete R-module spectra

Proof The implication (b) rArr (a) is clear and since T T otimesR[sminus1] by Corollary III4athe reverse implication follows from the general tensor-Hom adjunction (see II62c(a))

Using the the fact that

R[sminus1]otimesR R[sminus1]R R[sminus1]R[sminus1] 0

together with the tensor-Hom adjunction we see that Ms homR((Rsinfin)[minus1]M) is indeeds-complete in the sense of (a) which shows (c) rArr (a) Conversely from the fibre sequence(Rsinfin)[minus1] R R[sminus1] we get that M is s-complete in the sense of (a) iff it equals itsown s-completion The final assertion about (minus)s being a Bousfield localisation follows fromProposition I61a as usual We know from Corollary II56e that ModR is stable and itsclear from (a) that Mods-comp

R sube ModR is closed under finite direct sums as well as fibresand cofibres hence it is indeed a full stable sub-infin-category

Completion of spectra shares many properties with the completion of ordinary rings andmodules For example

III22b Lemma mdash Let R be an Einfin-ring spectrum and s isin π0(R)

(a) For all R-module spectra M and all n gt 0 we have M otimesR Rsn Ms otimesR Rsn(b) (BeauvillendashLaszlo) The functors (minus)s ModR ModR and minus[sminus1] ModR ModR

are jointly conservative and for all R-module spectra M there is a pushoutpullbacksquare

M Ms

M [sminus1] Ms[sminus1]

186


(c) (Nakayamarsquos lemma) If M is s-complete and MotimesRRs 0 then M 0 In particularminusotimesR Rs Mods-comp

R ModR is conservativeNote that combining all three assertions from Lemma III22b implies that also the

functors minusotimesR Rs ModR ModR and minus[sminus1] ModR ModR are jointly conservative

Proof of Lemma III22b Itrsquos straightforward to show that sn M otimesRRsn M otimesRRsnis the zero morphism This implies that M otimesR Rsn is s-complete Indeed

sn homR

(R[sminus1]M otimesR Rsn

)minus homR


)is an equivalence because it is an equivalence on R[sminus1] but itrsquos also the zero map becauseit is zero on M otimesR Rsn This forces homR(R[sminus1]M otimesR Rsn) 0 whence M otimesR Rsnis s-complete by LemmaDefinition III22a(a) The same holds for Ms otimesR Rsn Now let Nbe another s-complete R-module spectrum Since (minus)s is a Bousfield localisation

homR(M otimesR Rsn N) homR

(RsnhomR(MN)

) homR

(RsnhomR(Ms N)

) homR(Ms otimesR Rsn N)

Thus the assertion of (a) follows from the Yoneda lemma in Mods-compR We have silently used

that the adjunction involving (minus)s upgrades to the internal Hom of ModR But it clearlyupgrades to the underlying spectra homModR of homR by Theorem II30 and II62c(a) andequivalences in ModR can be detected on underlying spectra by the Segal condition fromTheorem II56c

We prove (b) next Since a morphism in ModR is an equivalence iff its fibre vanishesit suffices to show that Ms 0 and M [sminus1] 0 together imply M 0 Suppose that0 Ms homR((Rsinfin)[minus1]M) Since (Rsinfin)[minus1] R R[sminus1] is a fibre sequencethis implies

homR

(R[sminus1]M

) homR(RM) M

Hence M is already s-local which implies M M [sminus1] Thus M [sminus1] 0 iff M 0 asclaimed To prove that the commutative square from (b) is really a pullback square itnow suffices to do so after applying (minus)s and minus[sminus1] But one immediately checks that thes-completion of s-local R-module spectra vanishes hence the pullback becomes trivial afters-completion just as it becomes trivial after applying minus[sminus1]

Finally letrsquos prove Nakayamarsquos lemma From the pushout diagram

R R R

0 Rsn Rsn+1

0 Rs

sn

s

s

we get a fibre sequence Rsn sminus Rsn+1 Rs for all n gt 0 (note that we donrsquot need any

non-zero divisor condition on smdashthatrsquos the blessing of working in a ldquofully derivedrdquo setting)Thus M otimesR Rs 0 inductively implies M otimesR Rsn 0 for all n gt 0 hence also

Ms limnisinN

(M otimesR Rsn) 0

187


But M Ms by assumption and LemmaDefinition III22a(c)

III22c Derived Completion over Ordinary Rings mdash Letrsquos leave the land ofspectra for the moment and consider an ordinary commutative ring R with an element s isin RFollowing [Stacks Tag 091N] we call a complex K isin D(R) derived s-complete if

RHomR

(R[sminus1]K

) 0

Combining Theorem II57 Corollary II30a and II62c we find that K is derived s-completeiffHK isin ModHR is an s-completeHR-module spectrum In particular if Ds-comp(R) sube D(R)denotes the full sub-infin-category of derived s-complete complexes then the EilenbergndashMacLanespectrum functor restricts to an equivalence

H Ds-comp(R) simminus Mods-compHR

Moreover if C denotes the chain complex (R R[sminus1]) concentrated in degrees 0 and minus1then the derived s-completion functor

(minus)s = RHomR(Cminus) D(R) minus D(R)

is a Bousfield localisation onto Ds-comp(R) Itrsquos perhaps somewhat confusing why we usethe complex C rather than (R[sminus1]R)[minus1] This is because we have to take the derivedquotient In fact one easily checks that C[1] is the mapping cone of R R[sminus1] hence itsits inside a pushout diagram

R R[sminus1]

0 C[1]

Thus C (R[sminus1]LR)[minus1] and therefore

HC (HRsinfin)[minus1]

which implies that our definitions of (minus)s in D(R) and ModHR are indeed compatibleOver an ordinary ring the Nakayama lemma can be slightly improved (although again it

doesnrsquot look like an improvement at first glance until one realises that the quotient Rs isan underived one this time)

III22d Corollary (Derived Nakayama lemma) mdash If K is a derived s-complete complexover R then K otimesLR Rs 0 implies K 0

Proof We already know from Lemma III22b(c) and Proposition II63 that KotimesLRRLs 0implies K 0 where the derived quotient RLs can be represented by the complex (R s

minus R)concentrated in degrees 0 and minus1 But the cohomology of RLs are Rs-modules henceK otimesLR Rs 0 already implies K otimesLR RLs 0

III22e Derived Completion over Noetherian Rings mdash It is not true in generalthat derived s-completion is given by

Ks RlimnisinN

(K otimesLR Rsn

)

188


(here Rlim denotes the limit in the infin-category D(R)) The problem is as usual that theordinary quotients Rsn donrsquot coincide with the derived quotients RLsn which are requiredin general to make the formula correct Neither is it true (I think but donrsquot take my wordfor it) that derived completion is the left-derived functor of

Λs = limnisinNminussnminus ModR minus ModR

in general Fabian remarks that Λs is neither left- nor right-exact in general1 but weneed a right-exact functor to form left derivatives right Wrong Grothendieck formulatedhis theory for arbitrary additive functors between abelian categories without any exactnessrequirements it just wonrsquot be true anymore that L0Λs Λs

Both of the problems above go away if we assume R is a noetherian ring (or moregenerally has bounded s-power torsion) Indeed in this case Ks RlimnisinN(K otimesL Rsn)holds by [Stacks Tag 0923] which also implies that (minus)s LΛs since it takes the correctvalues on free R-modules

III22f Derived p-Completion over Z vs p-Completion of Spectra mdash Let p isin Zbe a prime Our considerations in III22c allow us to compare derived p-completion over Zwith p-completion over HZ But p is also an element of π0(S) = Z and it turns out that wecan also describe p-completion over the sphere spectrum S (somewhat) in terms of derivedp-completion over Z

Letrsquos first analyse derived p-completion over Z a bit more Since Z Z[pminus1] is injectivethe complex C = (Z Z[pminus1]) concentrated in degrees 0 and minus1 is quasi-isomorphic to(Z[pminus1]Z)[minus1] micropinfin [minus1] Hence derived p-adic completion in D(Z) is given by

LΛp RHomZ(micropinfin [minus1]minus

) D(Z) minus Dp-comp(Z)

In particular if A is an abelian group then

L0Λp(A) = ExtZ(micropinfin A) L1Λp(A) = HomZ(micropinfin A) and LiΛp(A) = 0 for i gt 2

Itrsquos easy to determine L1Λp(A) Writing micropinfin = colimnisinN Zpn with transition maps givenby multiplication with p we obtain

L1Λp(A) = limnisinN

HomZ(Zpn A) = limnisinN

A[pn]

where A[pn] denotes the pn-torsion part of A and the transition morphisms are again givenby multiplication with p Moreover there is a short exact sequence


A[pn] minus L0Λp(A) minus Λp(A) minus 0

To see where this sequence comes from choose a projective resolution 0 P1 P0 A 0(Z has global dimension 1) note that A[pn] = ker(P1p

nP1 P0pnP0) since the right-hand

side computes TorZ1 (Zpn A) and play a bit around with exact sequencesThe upshot is that if A has bounded p-power torsion (for example if A is finitely

generated) then there is an n 0 such that pn A[p2n] A[pn] is the zero morphismHence

RlimnisinN

A[pn] 0

1 which confused me a bit since R1limnisinNMsnM = 0 holds by the Mittag-Leffler condition Howeverthis isnrsquot enough to ensure right-exactness For an inclusion N subeM the kernel of NsnN MsnM is(snM capN)snN so we would need R1limnisinN(snM capN)snN = 0 insteadmdashwhich isnrsquot true in general

189


in this case and we see that LΛp(A) coincides with the usual underived p-adic completionΛp(A)[0] placed in degree 0 These considerations can be exploited to compute homotopygroups of p-completions of spectra

III22g Lemma mdash For every spectrum E and all i isin Z there are short exact sequences

0 minus L0Λp(πi(E)

)minus πi(Ep) minus L1Λp

(πiminus1(E)

)minus 0

In particular if πlowast(E) has degreewise bounded p-power torsion then its degreewise p-completion Λpπlowast(E) coincides with πlowast(Ep) Moreover the homotopy groups of any p-completespectrum are derived p-complete

Proof The homotopy groups of Ep sit inside the Milnor exact sequence


πi+1(EpnE) minus πi(Ep) minus limnisinN

πi(EpnE) minus 0

(see the nLab article and note that Ωinfin+i Sp An commutes with limits hence theMilnor sequence works for spectra as well) The morphism pn E E clearly also inducespn πlowast(E) πlowast(E) on homotopy groups Hence the long exact sequence associated to thefibre sequence E E EpnE provides short exact sequences

0 minus πi(E)pnπi(E) minus πi(EpnE) minus πiminus1(E)[pn] minus 0

Now take limits over n We have R1limnisinN πi(E)pnπi(E) = 0 by Mittag-Leffler hence thesix-term exact sequence splits into a short exact sequence

0 minus Λp(πi(E)

)minus lim

nisinNπi(EpnE) minus L1Λp

(πiminus1(E)

)minus 0

and an isomorphism R1limnisinN πi(EpnE) R1limnisinN πiminus1(E)[pn] Summarising the infor-mation wersquove got so far and plugging in the short exact sequence that computes L0Λp(πi(E))we obtain a solid commutative diagram

0

0 R1 limnisinNop

πiminus1(E)[pn] L0Λp(πi(E)

)Λp(πi(E)

)0

0 R1 limnisinNop

πi(EpnE) πi(Ep) limnisinNop

πi(EpnE) 0

L1Λp(πiminus1(E)

)0

sim

If we can show that the dashed arrow exists then wersquoll be done Indeed the snake lemmawill then show that L0Λp(πi(E)) πi(Ep) is injective and that its cokernel is isomorphic toL1Λp(πiminus1(E)) which is what we want to show

190


To construct the dashed arrow it suffices to show that πi(Ep) is derived p-complete sincethen the canonical morphism πi(E) πi(Ep) will extend over the derived p-completionLΛp(πi(E)) and hence also over L0Λ(πi(E)) since πi(Ep) is concentrated in degree 0 Butthe solid diagram is already enough to conclude that πi(Ep) is derived p-complete Indeed weknow from [Stacks Tag 091U] that if two out of three abelian groups in a short exact sequenceare derived p-complete then so is the third Moreover this result and [Stacks Tag 091T]imply that the homology of a derived p-complete complex is derived p-complete and thatp-complete abelian groups are also derived p-complete hence L0Λp(πi(E)) Λp(πi(E)) andL1Λp(πiminus1(E)) are all derived p-complete which suffices to show the same for the rest of thediagram

Regarding the additional two assertions Wersquove already seen that πlowast(Ep) is degreewisederived p-complete If πlowast(E) has degreewise bounded p-power torsion then LΛp(πi(E)) Λpπi(E)[0] by our discussion in III22f hence πi(Ep) = Λp(πi(E)) in this case

Only the third part of the following lemma was mentioned in the lecture but the other twofit nicely and provide a proof of a small technical step in the proof sketch of Theorem III19as promised at the beginning of III22

III22h Lemma mdash (alowast) The functors minus otimes HQ Sp Sp and (minus)p Sp Sp for allprimes p are jointly conservative The same is true for (minus)p replaced by minusotimes Sp

(blowast) The functors minusotimesZ Q D(Z) D(Z) and minusotimesLZ Zp D(Z) D(Z) for all primes p arejointly conservative In particular if f X Y is a map of anima such that

flowast Hlowast(XQ) simminus Hlowast(YQ) and flowast Hlowast(XZp) simminus Hlowast(YZp)

are isomorphisms then f is also an isomorphism on homology with Z-coefficients(c) For all spectra E there is a canonical pullback square (the ldquoarithmetic fracture squarerdquo)

Eprodp

Ep

E otimesHQ

(prodp

Ep

)otimesHQ

Observe that Lemma III22h(blowast) is wildly false for underived tensor products For exampleQZ vanishes upon tensoring with Q or any Zp

Proof of Lemma III22h For (alowast) it suffices to show that E otimes HQ 0 and Ep 0 forall p imply E 0 By the same argument as in Lemma III22b(b) Ep 0 implies thatE is p-local If thatrsquos the case for all p then E E otimesHQ by Corollary III5 hence theright-hand side vanishes iff E vanishes Moreover we can replace (minus)p by minusotimes Sp becauseminusotimes Sp Spp-comp Sp is conservative by Nakayamarsquos lemma for spectra

Part (blowast) is completely analogous except that we have to use the slightly stronger derivedNakayama lemma from Corollary III22d The ldquoin particularrdquo follows as

Hlowast(minusQ) = Hlowast(Cbull(minus)otimesZ Q

)and Hlowast(minusZp) = Hlowast

(Cbull(minus)otimesLZ Zp

)

191


For (c) we may apply (a) to see that it suffices to check the pullback property after p-completion and after tensoring with HQ Itrsquos easy to check that p-completion kills all rationalspectra and all `-complete spectra for ` 6= p Hence after p-completion the lower row becomeszero whereas the upper row becomes the identity on Ep Similarly the pullback propertybecomes trivial after tensoring with HQ

As a consequence of Lemma III22h to understand the connective K-theory spectrumK(F ) = Binfink(F ) of a field F it suffices to understand its p-completions and its rationalisationSurprisingly it is the latter which causes problems in praxis even though we have a formulafor its homotopy groups Ki(F ) otimes Q = indeciHgrp

lowast (GLinfin(F )Q) by Corollary III10 Butcomputing the right-hand side is an insanely hard problem and very much depends on F asone can already see in the special case K1(F ) = Ftimes (by Proposition III8)

In contrast to that at least when F is a separably closed field the `-adic completionsK(F ) are well understood and donrsquot really depend on F

III23 Theorem (Suslinrsquos rigidity theorem) mdash For any morphism F F prime of separablyclosed fields and all primes `

K(F ) simminus K(F prime)is an equivalence The same is true for the tautological map

K(C) simminus (Binfinku) Moreover if p 6= ` is a prime Op sube Qp the absolute ring of integers over Zp and m sube Opits maximal ideal then also

K(Op) simminus K(Qp) and K(Op) simminus K(Opm)are equivalences

We wonrsquot prove Theorem III23 But note that Opm Fp Hence any pair of embeddingsQ Qp and Q C induces an equivalence K(Fp) (Binfinku) This fits with ourcalculations of their homotopy groups Since πlowast(Binfinku) = Z[β] has degreewise bounded`-power torsion Lemma III22g implies

πi((Binfinku)) =

Z` if i is even0 if i is odd

Also recall that Klowast(Fp) is 0 in even degrees andoplus

q 6=p microqinfin in odd degrees by Corollary III21But one can compute

L0Λ`

(oplusq 6=p

microqinfin

)= 0 and L1Λ`

(oplusq 6=p

microqinfin

)= Z`

hence Lemma III22g implies that `-completion shifts the homotopy groups of K(Fp) fromodd degrees into even degrees Thus

πi(K(Fp)) =


192


as Theorem III23 predicts Fabian also mentioned that even though the Adams operationsψi Binfinku Binfinku arenrsquot maps of Einfin-ring spectra they can be refined to an action of Zon the `-completion (Binfinku) through Einfin-ring spectra maps

To end this chapter we state the analogue of Theorem III19 for Hermitian K-theoryGro-thendieckndashWitt theory

III24 Theorem (Friedlander FiedorowiczndashPriddy [FP78]) mdash For odd q Brauer liftinginduces equivalences

gwminussym(Fq)0simminus fib(ψq minus id BSp minus BSp)

gwsym(Fq)0simminus fib(ψq minus id BO minus BO)

(where Sp denotes the symplectic group and not the infin-category of spectra) and in the evencase we obtain

gwsym(F2n) gweven(F2n) simminus fib(ψq minus id BSp minus BSp)gwquad(F2n) gwsym(F2n)timesBZ2

193

Chapter IV

The K-Theory of Stableinfin-Categories

IVThe Modern Definition of K-TheoryK0 of a Stable infin-CategoryIV1 Some Motivation mdash Throughout we define the connective K-theory spectrumof a ring R to be K(R) = Binfink(R) One of our goals in this chapter will be to explainQuillenrsquos fibre sequence oplus

p primeK(Fp) minus K(Z) minus K(Q)

and more generally for R a Dedekind domainoplus0 6=m prime

K(Rm) minus K(R) minus K(FracR)

The immediate problem here is that K(Fp) K(Z) and K(Rm) K(R) canrsquot possiblybe induced by by ring maps Fp Z or Rm R since there are none However there areperfectly fine exact (but not fully faithful) functors D(Fp) D(Z) and D(Rm) D(R)with essential image the p-torsion or m-torsion complexes respectively Motivated by thatwersquoll define K-theory of a ring in terms of its derived infin-category D(R) (or really its sub-infin-category Dperf(R) see below) and once wersquore there nothing can stop us from definingK-theory of arbitrary stable infin-categories

Before we move on Fabian remarks that the sequences above already contain interestinginformation in low degrees For example after taking homotopy groups in degrees 0 and 1the first sequence becomes

K1(Z) K1(Q)oplus

p primeK0(Fp) K0(Z) K0(Q)

plusmn1 Qtimesoplus

p primeZ Z Zsim

(we use Proposition III8 to get K1(Z) = Ztimes = plusmn1) The lower row yields an exactsequence

0 minus plusmn1 minus Qtimes minusoplus

p primeZ minus 0

194

The Modern Definition of K-Theory

which says precisely that Z has unique prime decomposition In a similar way the sequencefor arbitrary Dedekind domains encodes their unique prime ideal decomposition

IV1a Perfect Modules and Complexes mdash To get started we note that K0(R)admits more cycles than just projective modules Let M isin ModR be a perfect module thatis a module admitting a finite projective resolution

0 minus Pn minus Pnminus1 minus minus P0 minusM minus 0

Then we can put

[M ] =nsumi=0

(minus1)i[Pi] isin K0(R)

IV1b Exercise mdash Show that [M ] is independent of the choice of resolution

So you donrsquot need a projective module to specify an element in K0(R) all you need is aperfect one More generally let Dperf(R) sube D(R) denote the full sub-infin-category spanned bythe finite complexes of finite projective R-modules One can show that these are preciselythe compact objects in D(R)

IV2 Lemma mdash Sending a perfect complex (Pn Pnminus1 Pminusm) to its ldquoEulercharacteristicrdquo

sumni=minusm(minus1)i[Pi] defines an isomorphism(

π0 coreDperf(R)) simext

simminus K0(R)

where ldquosimextrdquo is the equivalence relation generated by B simext Aoplus C for all fibre sequencesA B C in Dperf(R)

Proof Well-definedness of the map follows from a sufficient generalisation of Exercise IV1bIf P is any finite projective module over R then also P [1] isin Dperf(R) which is mapped tominus[P ] Hence the map in question hits all inverses and is thus surjective

It remains to show injectivity Since the map in question is compatible with the additionon both sides it suffices to show that K isin Dperf(R) being mapped to 0 implies K simext 0Write K (Pn Pnminus1 Pminusm) Taking stupid truncations gives fibre sequencesPminusm[minusm] K Kgtminusm+1 which proves K simext Pminusm[minusm] oplus Kgtminusm+1 Continuinginductively we may thus replace K by the sum of its entries with all differentials being0 Next we note that Pminusm[i minus 1] 0 Pminusm[i] are fibre sequences for all i isin Z so weget Pminusm[iminus 1]oplus Pminusm[i] simext 0 Applying this first for i = minusm+ 2 and then backwards fori = minusm+ 1 gives

K noplus

i=minusmPi[i] simext Pminusm[minusm+ 2]oplus Pminusm[minusm+ 1]oplus

noplusi=minusm

Pi[i]

simext Pminusm[minusm+ 2]oplusnoplus

i=minusm+1Pi[i]

Continuing inductively we see that we may shift all even degrees into degree 0 and all odddegrees into degree 1 Hence we may assume K P0[0]oplus P1[1] Now K being mapped to0 implies [P0] = [P1] hence by construction of the Grothendieck group completion thereexists a finite projective module Q such that P0 oplusQ P1 oplusQ But by an argument wersquovealready seen QoplusQ[1] simext 0 hence K simext (P0 oplusQ)[0]oplus (P1 oplusQ)[1] This has again theform P prime[0]oplus P prime[1] hence K simext 0 and wersquore done

195


IV3 Definition mdash Given a (small) stable infin-category C put

K0(C) = π0 core(C) simext

where again ldquosimextrdquo is the equivalence relation generated by b simext aoplus c for all fibre sequencessequence a b x in C

One easily checks that taking direct sums in C descends to an abelian group structure onK0(C) For example inverses are given by minus[c] = [Σc] = [Ωc]

The functorK0(minus) Catstinfin Ab is already an interesting invariant in that it parametrises

dense sub-infin-categories We call a full sub-infin-category D sube C dense if every object in C canbe written as a retract of some d isin D ie there are maps

c minus d minus c

composing to idc For example free R-modules are dense in Proj(R)

IV4 Theorem (Thomason [Tho97 Theorem 21]) mdash If C is a stable infin-categories andD sube C a dense stable sub-infin-category then the inclusion induces an injection

K0(D) minus K0(C)

Moreover therersquos a ldquoGalois correspondencerdquo of partially ordered sets

dense stable sub-infin-categories of C simminus subgroups of K0(C)D sube C 7minus K0(D)

CH =x isin C

∣∣ [x] isin H minus [ H sube K0(C)

IV4a Remark mdash Two remarks on our ldquosub-infin-categoryrdquo terminology(a) By saying stable sub-infin-category we always implicitly assume that D C is a functor

in Catstinfin ie exact In particular being a stable sub-infin-category is slightly stronger

than being a sub-infin-category that happens to be stable(b) The left-hand side of the ldquoGalois correspondencerdquo is a bit ambiguous in that a priori

there might be equivalent but non-equal sub-infin-categories of C So depending on yourtaste you either have to identify two sub-infin-categories if their essential images agree oryou have to restrict to only those sub-infin-categories D sube C which are equal (as simplicialsets) to their essential image or equivalently those D sube C such that D C is anisofibration

IV4b Example mdash Theorem IV4 tells us that the smallest dense stable sub-infin-category

Dfree(R) sube Dperf(R)

containing R[0] corresponds to im(Z K0(R)) since the image of R[0] in K0(R) is 1Moreover any stable infin-category C has a smallest dense stable sub-infin-category

Cmin = C0 =x isin C

∣∣ [x] = 0 in K0(C)

Already the existence of such a smallest dense stable sub-infin-category is not quite obvious

196


Proof of Theorem IV4 Injectivity will be the very last thing we check so throughout theproof we work with the image

HD = im(K0(D) K0(C)

)instead We will proceed in two steps(1) In (temporary) absence of injectivity we will show the ldquoGalois correspondencerdquo with

K0(D) replaced by HDWe will do this by verifying that the map H 7 CH in the reverse direction is an inverse

So the first thing to check is that CH is indeed a dense stable sub-infin-category of C SinceC K0(C) is additive with respect to finite direct sums and fibre sequences one immediatelyverifies that CH sube C is closed under finite limits and colimits hence it is indeed a stable sub-infin-category It is also dense because every x isin C is a retract of xoplusΣx but [xoplusΣx] = 0 isin Hso xoplus Σx is always an element of CH

Next we check HCH = H Clearly im(K0(CH) K0(C)) sube H But for any h isin H thereis an x isin C with [x] = h and then x isin CH by definition hence the inclusion must be anequality This shows HCH = H

It remains to check CHD = D Unravelling what we need to check is that x isin D iff[x] isin HD which is the case iff [x] = 0 in K0(C)HD To show this we introduce anotherequivalence relation ldquosimrdquo on C Put x sim xprime iff there exist d dprime isin D with xoplus d xprime oplus dprime Itrsquoseasy to check that ldquosimrdquo is indeed an equivalence relation and also it clearly descends toπ0 core(C) Moreover we have x sim 0 iff x isin D Indeed x isin D implies x sim 0 as xoplus 0 0oplus xand 0 x isin D Conversely if xoplusd dprime for some d dprime isin D then x dprime d is a fibre sequencein C two of whose terms are in D hence the third term must be in D as well because D sube Cis a stable sub-infin-category (in the sense of Remark IV4a(a))

By our newfound knowledge we see now that to prove CHD = D it suffices to checkthat x sim xprime iff [x] minus [xprime] isin HD To show this we first observe that taking direct sumsturns π0 core(C) sim into an abelian group Indeed well-definedness associativity andcommutativity of addition is clear as is the existence of a neutral element For inversesrecall that every x isin C is a retract of some d isin D by assumption hence d x oplus xprime forxprime cofib(x d) isin C But then xprime provides an inverse of x Furthermore the mapπ0 core(C) π0 core(C) sim descends to a map

K0(C) minus π0 core(C) sim

To verify this we must check that y sim xoplus z for every fibre sequence x y z in C Butthere elements xprime zprime isin C such that xoplus xprime z oplus zprime isin D and xoplus xprime y oplus xprime oplus zprime z oplus zprime isstill a fibre sequence But now two of its terms are in D hence also y oplus xprime oplus zprime isin D becauseD sube C is a stable sub-infin-category (in the sense of Remark IV4a(a)) Now the elementsd = xoplus xprime oplus z oplus zprime isin D and dprime = y oplus xprime oplus zprime isin D satisfy y oplus d xoplus z oplus dprime thus witnessingy sim xoplus z This shows that we really get a homomorphism K0(C) π0 core(C) sim of abeliangroups

This homomorphism is surjective for trivial reasons Since we checked above that x sim 0iff x isin D the kernel must consist precisely of those [x] such that [x] = [d] for some d isin DThat is the kernel is HD which means the proof of the ldquoGalois correspondencerdquo is done(2) Finally we prove that K0(D) K0(C) is injective

Thomason does this with a bit of a trick Letrsquos denote N = ker(K0(D) K0(C)) SinceD is stable too we can also run the Galois correspondence machinery for D and obtain dense

197


stable sub-infin-category D0 sube DN sube D corresponding to the trivial subgroup of K0(D) andN sube K0(D) respectively One easily checks that D0 and DN are also dense stable sub-infin-categories of C But K0(D0) K0(C) and K0(DN ) K0(C) both have image 0 henceD0 = DN as full sub-infin-categories of C But then K0(D0) K0(D) and K0(DN ) K0(D)must have the same image which proves N = 0

The S-Construction and the Q-ConstructionLecture 242nd Feb 2021

Our next goal is to construct a K-theory functor

k Catstinfin minus CGrp(An)

As wersquove already seen in the baby case of K0(C) we now longer have to group completeInstead what we have to do is to ldquosplit all fibre sequencesrdquo A fibre sequence a b ccan be thought of as a filtered object a b of length 1 together with its filtration quotientc ba So the logical next step is to consider filtered objects of arbitrary length Thisleads us to the Segal S-construction (people keep calling it Waldhausen S-construction eventhough Waldhausen himself wrote Segal S-construction)

IV5 Construction mdash Let C be a stable infin-category For all [n] isin ∆∆op let

Sn(C) sube Fun(

Ar([n]) C)

be the full sub-infin-category of those functors F Ar([n]) C satisfying the following twoconditions(a) F (i 6 i) = 0 for all i = 0 n(b) All ldquosquaresrdquo in Ar([n]) go to pushoutspullbacks (these are the same in the stable

infin-category C) as indicated in the picture belowTo make sense of (b) recall from I28 that Ar([n]) can be pictured as follows (the colourcoding will be explained in a moment)

(0 6 0)

(0 6 1) (0 6 2) (0 6 3) (0 6 4) (0 6 nminus 1)(0 6 n)

(1 6 n)

(2 6 n)

(3 6 n)

(nminus 2 6 n)

(nminus 1 6 n)

(n 6 n)

(1 6 1)

(2 6 2)

(3 6 3)

(nminus 2 6 nminus 2)


198


Now (b) is equivalent to F being left Kan extended from the ldquoyellowrdquo part Together with(a) this implies

Sn(C) Fun([nminus 1] C

)via restriction to the ldquodottedrdquo part in the top row Hence Sn(C) is stable again MoreoverFun(Ar[n] C) is clearly functorial both in [n] and C and one checks that the full subcategoriesSn(C) are preserved under the face and degeneracy maps in ∆∆op Hence S(C) ∆∆op Catst

infinis a simplicial stable infin-category and we get a functor

S Catstinfin minus sCatst

infin

Unravelling we find that S0(C) lowast S1(C) C and S1(C) Ar(C) But we think of thetypical element of S2(C) as a diagram

0 a b

0 ba

0

rather than an arrow a b In this picture the face maps d0 d1 d2 S2(C) S1(C) Cextract the objects ba b and a respectively

Also note that we have π0 |coreS(C)| = 0 since it is a general fact that π0X0 π0 |X| issurjective for all simplicial anima X (see (lowast) in the proof of Proposition II2)

IV6 Definition mdash For a stable infin-category C we define

k(C) = Ω |coreS(C)|

Since |coreS(C)| is connected therersquos no need to specify a basepoint

IV7 Lemma mdash For all stable infin-categories C there is a canonical isomorphism

K0(C) π0(k(C)

) π1

(|core(C)|

)

Proof The second isomorphism is clear as the loop functor shifts homotopy groups downHence it suffices to show K0(C) π1(|core(C)|) This will be done using skeletal inductionWe didnrsquot talk about how this works for simplicial anima so this proof also serves as ademonstration of the technique (although we will actually do skeletal induction for semi-simplicial anima since thatrsquos somewhat easier and sufficient for our purposes)

Let ∆∆inj sube ∆∆ be the subcategory spanned by the injective maps It is wide ie containsall objects but obviously not full Using the criterion from Theorem I43(b) one easily checksthat ∆∆op

inj sube ∆∆op is cofinal So instead of the realisation |X| colim∆∆op X of some simplicialanima X we may consider the colimit over ∆∆op

inj of its restriction Somewhat abusingly wewonrsquot distinguish between X and its restriction to ∆∆op

injNext letrsquos define the skeletal filtration on X For all n gt 0 let in denote the inclusion

∆∆inj6n sube ∆∆opinj Then put

sknX = Lanin ilowastnX

199


Using the explicit fromulas from Theorem I52 one unravels

(sknX)k Xk if k 6 nempty if k gt n

The sknX assemble into a functor skX N Fun(∆∆opinjAn) with colimit X (which is clear

from the above description and the fact that colimits in functor categories are pointwise)Now consider

∆n = Hom∆∆opinj

(minus [n]

)and part∆n = sknminus1 ∆n

in Fun(∆∆opinjAn) the ldquosemi-simplicial n-simplexrdquo and its ldquoboundaryrdquo Note that ∆n and

part∆n are not the restrictions of ∆n and part∆n to ∆∆opinj However ∆n and part∆n are the left Kan

extensions of ∆n and part∆n along ∆∆opinj sube ∆∆op by Corollary I48 Hence

|∆n| |∆n| lowast and |part∆n| |part∆n| Snminus1

where we use that colim∆∆opinj

and colim∆∆op are the left Kan extension functors along ∆∆opinj lowast

and ∆∆op lowast respectively and left Kan extension composesBy direct inspection there are pushouts

part∆n+1 times constXn+1 sknX

∆n+1 times constXn+1 skn+1X

(IV71)

for all n gt 0 (just write out the definitions and everything will be trivial in each degree)Taking realisations everywhere gives pushouts

Sn timesXn+1 |sknX|

Xn+1 |skn+1X| (IV72)

for all n gt 0 But be aware of Warning IV7a belowNow apply (IV72) to X coreS(C) in the case n = 0 We know S1(C) C and hence

S0 times coreS1(C) core(C) t core(C) Thus the pushout diagram in question appears as

core(C) t core(C) |sk0 coreS(C)|

core(C) |sk1 coreS(C)|

We also know S0(C) lowast hence sk0 coreS(C) ∆0 Plugging this into the pushout aboveand playing a bit around with it then shows |sk1 coreS(C)| Σ(core(C) t lowast) Σ(core(C)+)Hence Ω |sk1 coreS(C)| ΩΣ(core(C)+) FreeGrp(core(C)) by Proposition II10 Therefore

π1 |sk1 coreS(C)| = π0 FreeGrp ( core(C))

= FreeGrp(Set) (π0 core(C))

is the free ordinary group on the set π0 core(C)

200


Next we apply (IV72) to X coreS(C) in the case n = 1 Plugging in what we knowso far the pushout in question appears as

S1 times coreS2(C) Σ(

core(C) t lowast)

coreS2(C) |sk2 coreS(C)|

This enables us to compute π1 |sk2 coreS(C)| using the Seifertndashvan Kampen theorem ([Hat02Theorem 120]) But note that coreS2(C) isnrsquot necessarily connected rather each equivalenceclass in S2(C) defines a connected component So in order to use Seifertndashvan Kampen in itsusual formulation we have to decompose coreS2(C)

∐s(coreS2(C))s into pathcomponents

and then iteratively form the pushout along S1 times (coreS2(C))s (coreS2(C))s (where theldquonumberrdquo of iterations can be any ordinal)

So choose an element s isin coreS2(C) which we may think of as a cofibre sequence

s = (a minus b minus ba)

in C and letrsquos unravel what the top row map does on fundamental groups We knowπ1Σ(core(C)tlowast) = FreeGrp(Set)(π0 core(C)) π1(S1) = Z and that π1 commutes with productsWe claim that the ensuing map

Ztimes π1(

coreS2(C) s)minus FreeGrp(Set) (π0 core(C)

)sends (1 e) 7 a middot bminus1 middot ba and (0 γ) 7 e for all γ isin π1(coreS2(C) s) where e denotes therespective neutral elements Indeed (1 e) corresponds to walking once around the perimeterof S1 |part∆1| and taking boundary maps which are given by d2(s) a d1(s) b andd0(s) ba in that order (see the end of Construction IV5) The argument for (0 γ) issimilar To make all of this really precise we would have to unravel what the top row mappart∆n+1 times constXn+1 sknX from (IV71) does but letrsquos not get into the details there

So taking the pushout

Ztimes π1(

coreS2(C) s)

FreeGrp(Set) (π0 core(C))

π1(

coreS2(C) s)

pr2

in Grp(Set) (which is what the Seifertndashvan Kampen theorem does) precisely enforces therelation [b] = [ba] middot [a] in the free group FreeGrp(Set)(π0 core(C)) Doing this for all cofibresequences in C thus provides an isomorphism

K0(C) π1(|sk2 coreS2(C)|

)(note in particular that a a oplus c c is always a fibre sequence hence [a oplus c] = [a] middot [c]and then by swapping a and c also [aoplus c] = [c] middot [a]) Thus we are almost done and it onlyremains to show π1(| sk2 coreS(C)|) = π1(|core(C)|) But it follows from (IV72) and theSeifertndashvan Kampen theorem again that the fundamental group doesnrsquot change anymore in

201


the higher skeleta (because π1(Sn) = 0 for n gt 2) Thus as π1 commutes with sequentialcolimits by Lemma II31b

π1(|core(C)|

)= π1

(colimnisinN

|skn coreS2(C)|)

= colimnisinN

π1(|sk2 coreS2(C)|

)= π1

(|sk2 coreS2(C)|

)

IV7a Warning mdash In contrast to the realisation of simplicial anima (see Proposi-tion II18) the realisation functor

| | colim∆∆op

inj

Fun(∆∆opinjAn) minus An

for semi-simplicial anima doesnrsquot commute with finite products in general For example∆0 times ∆1 ∆0 t ∆1 so |∆0 times ∆1| |∆0| t |∆1| But it does if one object is constant sinceboth |Xtimes constminus| An An and |X|timesminus An An preserve colimits and agree on lowast isin Anhence they must agree by Theorem I51

IV7b Edgewise Subdivision mdash Letrsquos give another construction of k(C) using theQuillen Q-construction from I70 To get a relation between the Q- and the S-constructionconsider

∆∆op minus ∆∆op

[n] 7minus [n] [n]op

It induces a functor (minus)esd sAn sAn called edgewise subdivision For example byLemma I69

Nr(C)esd Nr(

TwAr(Cop))

holds for all infin-categories C Moreover we have

|X| |Xesd|

for all simplicial anima X Indeed both sides are colimit preserving functors sAn Anhence it suffices to check that they coincide on simplices by Theorem I51 Clearly |∆n| lowastTo get the same for (∆n)esd we use Nr([n]) ∆n (see before TheoremDefinition I64)and thus (∆n)esd Nr(TwAr([n]op)) by the formula above But TwAr([n]op) has an initialobject hence its realisation is contractible too

Abstractly [n]op [n] = [2n+ 1] But rather than as the poset (0 lt 1 lt middot middot middot lt 2n+ 1) wethink of [n]op [n] as the poset

(0` lt 1` lt middot middot middot lt n` lt nr lt (nminus 1)r lt middot middot middot lt 0r)

where the indices (minus)` and (minus)r indicate the ldquoleftrdquo and ldquorightrdquo part corresponding to [n]and [n]op respectively

IV8 Proposition mdash The map TwAr([n]op) Ar([n][n]op) sending (i 6 j) to (i` 6 jr)

induces an equivalenceS(C)esd simminus Q(C)

for any stable infin-category C In particular k(C) Ω |Span(C)| where we put Span(C) asscat(coreQ(C)) as defined in PropositionDefinition I71

202


Proof sketch Itrsquos easy to check that TwAr([n]op) Ar([n][n]op) defines a map of simplicial

posets and that the induced map upon taking Fun(minus C) restricts to a map S(C)esd Q(C)as required

To show that this map is indeed an equivalence we draw a picture in the case n = 2 thatis a picture of Ar([2] [2]op) Ar([5])

(0` 6 0`)

(0` 6 1`) (0` 6 2`) (0` 6 2r) (0` 6 1r)(0` 6 0r)

(1` 6 0r)

(2` 6 0r)

(2r 6 0r)

(1r 6 0r)

(0r 6 0r)

(1` 6 1`)

(2` 6 2`)

(2r 6 2r)

(1r 6 1r)

We let In denote the image of TwAr([n]op) Ar([n] [n]op) This is the yellow part inthe picture Also let ∆`

n and ∆rn be the subposets spanned by (0` 6 0`) (n` 6 n`)

and (nr nr) (0r 0r) respectively In the picture these are coloured pink and purpleFinally let Jn be the subposet spanned by (i` 6 j`) (i` 6 jr) | i 6 j This is the dottedpart in the picture

By definition an element lies in S2n+1(C) iff all squares go to pushoutspullbacks inthe stable infin-category C Moreover Qn(C) sube Fun(In C) is the full sub-infin-category of thosefunctors that send all squares in In to pushoutspullbacks Now consider the composition

Fun(In C) Fun(

Ar([n] [n]op) C)

Fun(In cup∆`n C) Fun(In cup∆n cup∆r

n C) Fun(Jn cup∆rn C)

Lan

Ran Ran

Lan

(where ldquocuprdquo always means that we take the full subposet spanned by the union) That is westart with the yellow part left-Kan extend to the pink part right-Kan extend to the purplepart right-Kan extend to the dotted part and the left-Kan extend to all of Ar([n] [n]op) Allof these Kan extensions are along fully faithful embeddings of subposets hence fully faithfulby Corollary I54 Breaking up each of the Kan extensions into a sequence of Kan extensionsby adding one vertex at a time and using the formulas from Theorem I52 we verify thatthe essential image of Qn sube Fun(In C) is precisely S2n+1(C) sube Fun(Ar([n] [n]op) C) whichis what we want

For the ldquoin particularrdquo note that |coreS(C)| | coreS(C)esd| |coreQ(C)| holds by IV7band what wersquove shown so far and |coreQ(C)| |Span(C)| follows from Remark I66

203


The Resolution TheoremProposition IV8 allows us to define a map core(C) k(C) via

core(C) HomSpan(C)(0 0) minus Hom|Span(C)|(0 0) Ω |Span(C)|

(the equivalence on the left follows from (I652) applied to Span(C) asscat(coreQ(C)) andsome playing around with the pullback involved) This morphism is clearly natural in Chence it defines a natural transformation core rArr k in Fun(Catst

infinAn) But both functorspreserve finite products (because they are built from right adjoints except for the realisation| | but this also preserves finite products by Proposition II18) Hence it lifts canonically toa natural transformation

(core =rArr k) Catstinfin minus CMon(An)

by Theorem II19 and the following observation

IV9 Observation mdash Catstinfin is semi-additive (and the same is true for Cattinfin Cattimesinfin

and Catsemi-addinfin )

Proof sketch Evidently Catstinfin inherits products from Catinfin and the zero category 0 isin Catst

infinis a zero object Next recall Proposition II40 which associates to every infin-category C aninfin-operad Ct Lop If you stare at the construction yoursquoll find it is clearly functorial in Chence we get a functor

(minus)t Catinfin minus Opinfin

But if C is stable then it has finite coproducts and thus Ct Lop is a symmetric monoidalinfin-category Moreover any map F C D in Catst

infin preserves finite coproducts so its imageFt Ct Dt is strongly monoidal Hence (minus)t restricts to a functor

(minus)oplus Catstinfin minus CMon(Catinfin)

From this we can extract functors oplusC C times C C in Catstinfin which assemble into a natural

transformation oplus ∆ rArr id in Fun(CatstinfinCatst

infin) with the properties from Lemma II20Hence that lemma implies that Catst

infin is semi-additive The other cases are similar

IV9a mdash So for all stable infin-categories C we get a map of Einfin-monoids core(C) k(C)But k(C) is an Einfin-group (because π0(k(C)) = K0(C) is an ordinary abelian group) hencethis map factors over an Einfin-group map

core(C)infin-grp minus k(C)

In particular via Proj(R) sube coreDperf(R) this gives a map

k(R) Proj(R)infin-grp minus(

coreDperf(R))infin-grp

minus k(Dperf(R)

)

And this map turns out to be an equivalence Thereby wersquove reduced algebraic K-theoryof rings as constructed in Definition II12a to the more general construction from Defini-tion IV6

In fact the equivalence Proj(R)infin-grp k(Dperf(R)) can be proved in much more gener-ality but this needs a bit more terminology

204


IV10 Weight Structures mdash We didnrsquot introduce this in the lecture but herersquos whatFabian told the audience in his talk in Regensburg A weight structure on a stableinfin-categoryconsists of two full sub-infin-categories C60 Cgt0 sube C subject to the following conditions(a) Cgt0 is closed under pushouts C60 is closed under pullbacks and both are closed under

retracts(b) If x isin C60 and y isin Cgt0 then the spectrum homC(x y) is connective (ie its homotopy

groups vanish in negative degrees)(c) For all y isin C there is a fibre sequence x y z with x isin C60 and z isin Cgt1 Here we

put Cgta = (Cgt0)[a] and C6b = (C60)[b]We also put C[ab] = Cgta cap C6b and Chearts = C[00] = Cgt0 cap C60 A weight structure on C iscalled exhaustive if

C =⋃nisinNC[minusnn]

Fabian remarks that the notion of a weight structure is different from that of a t-structure(which you might know from [HA Definition 1214] wersquoll introduce it properly in IV37)For example in a t-structure the roles of ldquo6rdquo and ldquogtrdquo in (c) would be swapped You shouldthink of t-structures as generalisations of ldquosmartrdquo truncation of chain complexes whereas aweight structure behaves more like ldquostupidrdquo truncation In particular the fibre sequence inC is highly non-canonical whereas in a t-structure therersquos always a canonical choice usingthe truncation functors Moreover the heart of a t-structure is always an abelian categorywhereas it need not even be a 1-category for a weight structure

Note that (a) implies that both Cgt0 and C60 contain 0 since it is a retract of everything(and both sub-infin-categories are non-empty by (c)) and both are closed under finite directsums To acquaint myself a bit more with weight structures let me also list the followingfurther properties

IV10a Lemma mdash (a) An object x isin C lies in C60 iff homC(x y) is connective for ally isin Cgt0 Similarly y isin C lies in Cgt0 iff homC(x y) is connective for all x isin C60

(b) Both Cgt0 and C60 are closed under extensions That is if the outer two terms in a fibresequence lie in Cgt0 then the same is true for the inner term and likewise for C60

(c) Chearts is an additive infin-category and every fibre sequence x y z in C splits if all ofits terms lie in Chearts

Proof For (a) we only do the second case since the first one is analogous The ldquoonly ifrdquoholds by assumption So assume y isin C satisfies that homC(x y) is connective for all x isin C60Apply IV10(c) to y[1] to get a fibre sequence x y z with x isin C6minus1 and z isin Cgt0 Thenπ0 homC(x y) = πminus1 homC(x[1] y) = 0 by IV10(b) Hence the long exact homotopy groupssequence associated to the fibre sequence homC(z y) homC(y y) homC(x y) in spectrabecomes

minus π0 homC(z y) minus π0 homC(y y) minus 0

In particular π0 homC(z y) π0 homC(y y) is surjective Choosing a preimage of idywitnesses y as a retract of z hence y isin Cgt0 because it is closed under retracts This shows (a)Part (b) is an immediate consequence of (a) and the long exact homotopy groups sequence

Itrsquos clear that Chearts is contains 0 and is closed under finite direct sums (in fact under fibresequences by (b)) hence it inherits additivity from C If x y z is a fibre sequence in Cwhose terms lie in Chearts then homC(z x) homC(z y) homC(z z) is a fibre sequence in

205

K-Theory as the Universal Additive Invariant

spectra But all of its entries are connective hence the long exact sequence of homotopygroups reads

minus π0 homC(z x) minus π0 homC(z y) minus π0 homC(z z) minus 0

In particular π0 homC(z y) π0 homC(z z) is surjective whence we get a split z y bychoosing a preimage of idz Now consider the diagram

0 x 0

z y z

The right square is a pushout by assumption and the outer rectangle is a pushout too sincez y z is equivalent to the identity on z Hence the left square is a pushout as wellwhich establishes y xoplus z

The prime example of an exhaustive weight structure is Dperf(R) for a ring R with

Dperf(R)[ab] =C isin Dperf(R)

∣∣∣∣ C has a projective representativeconcentrated in degrees [a b]

Note that this is stronger than heaving homology concentrated in degrees [a b] The heart ofthis weight structure is Dperf(R)hearts Proj(R) With this we can formulate a general versionof the ldquoresolution theoremrdquo also often called the ldquoweight version of the theorem of the heartrdquo

IV11 Theorem (Resolution theorem GilletndashWaldhausen) mdash If C is a stable infin-categoryequipped with an exhaustive weight structure then composing core(Chearts)infin-grp core(C)infin-grp

with the map from IV9a induces an equivalence

core(Chearts)infin-grp minus k(C)

In particular k(R) Proj(R)infin-grp k(Dperf(R)) for every ring R

We wonrsquot prove Theorem IV11 But Fabian has promised he will sketch a new proofdue to Wolfgang Steimle and himself in the official notes In fact they proved an even moregeneral statement about Poincareacute categories which Fabian will also introduce in his notes

K-Theory as the Universal Additive InvariantLecture 254th Feb 2021

Our next goal is to analyse the functor k Catstinfin CGrp(An) and in particular to prove

the following result from [BGT13]

IV12 Theorem mdash The inclusion Fungrp(CatstinfinCGrp(An)) sube Funadd(Catst

infinAn) ad-mits a left adjoint (minus)grp Ω|Span(minus)(minus)| and

k coregrp

In other words K-theory is the initial grouplike functor under core Catstinfin An

The proof of Theorem IV12 has to wait till page 224 Letrsquos start with explaining all thatnew terminology in it and to do that wersquoll introduce even more

206


Verdier SequencesIV13 Definition mdash Letrsquos define things named after Verdier(a) A Verdier sequence is a sequence A B C in Catst

infin that is both a fibre and a cofibresequence It is called leftright split if both functors admit leftright adjoints and splitif it is both left and right split (so all four possible adjoints exist)

(b) A functor B C participating in a Verdier sequence as in (a) is called a Verdierprojection and similarly A B is called a Verdier injection A Verdier projection orinjection is called leftright split if the corresponding Verdier sequence has that property

(c) A Verdier square is a pullback square in Catstinfin

A B

C D

such that the vertical maps are Verdier projections A Verdier square is called (leftright)split if the Verdier projections have the respective property

Moreover a functor F Catstinfin E where E has finite limits is additive or Verdier-localising

if F (0) is a final object of E and F takes split Verdier squares or all Verdier squares topullback squares in E It is called Karoubi-localising if it is Verdier-localising and takes denseinclusions to equivalences The corresponding full sub-infin-categories of Fun(Catst

infin E) aredenoted

FunKar(Catstinfin E) sube FunVerd(Catst

infin E) sube Funadd(Catstinfin E)

Finally an additive functor F Catstinfin E is called grouplike if it factors over CGrp(E) (the

factorisation is automatically canonical see IV14(c)) The corresponding full sub-infin-categoryis denoted

Fungrp(Catstinfin E) sube Funadd(Catst

infin E)

IV14 More on Verdier Sequences and Additive Functors mdash Having introduceda plethora of new terminology in Definition IV13 some explanations are in order(a) The functor categories Fun(Catst

infin E) are usually not locally small But wersquoll ignoreset-theoretic issues in the following

(b) In Catstinfin it is really a property (rather than a structure) to be a Verdier sequence Given

A B C an extension toA B

0 C

is unique (up to contractible choice) because 0 isin Fun(A C) is initial and terminal(c) Because Catst

infin is semi-additive by Observation IV9 we obtain that

Atimes C C

A 0

207


is always a split Verdier square Moreover C C 0 is always a split Verdier sequenceand so is C Atimes C A in general Hence additive functors preserve finite productsBut then Theorem II19 implies that

Funadd(Catstinfin E) Funadd (Catst

infinCMon(E)) CMon

(Funadd(Catst

infin E))

for any infin-category E with finite limits Thus Fungrp(Catstinfin E) sube Funadd(Catst

infin E) asdefined in Definition IV13 makes sense and also

Fungrp(Catstinfin E) Funadd (Catst

infinCGrp(E)) CGrp

(Funadd(Catst

infin E))

Moreover the arguments above show that Funadd(Catstinfin E) is semi-additive and thus

the same is true for Fungrp(Catstinfin E) FunVerd(Catst

infin E) and FunKar(Catstinfin E) Finally

all of them are closed under limits in Fun(Catstinfin E)

(dlowast) If F Catstinfin An is a grouplike additive functor then F (B) F (A)times F (C) for every

split Verdier sequence A B C Indeed F (A) F (B) F (C) is a fibre sequencein An hence also in CGrp(An) Either of the fully faithful adjoints C B inducesa split F (C) F (B) so in particular π0(F (B)) π0(F (C)) is surjective Hence thefibre fib(F (B) F (C)) in Sp must be connective too whence it coincides with the fibreF (A) taken in CGrp(An) Spgt0 Now every fibre sequence in Sp with a split mustactually be split using the same argument as in the proof of Lemma IV10a(c)

(e) An example of a Verdier-localising but not grouplike functor is core Catstinfin An It

is Verdier-localising simply because it takes arbitrary pullbacks (and not just Verdiersquares) to pullbacks in An

Here we should probably mention that Catstinfin sube Catinfin is closed under pullbacks (and

in fact arbitrary finite limits) To see this first note that Catstinfin sube Catlex

infin has bothadjoints by Theorem II30 and the remark after it Hence it suffices that Catlex

infin sube Catinfinis closed under pullbacks But if we are given a pullback diagram in Catinfin such thatboth legs preserve finite limits then any finite limit in the pullback can be computedcomponentwise (and in particular exists if it exists componentwise)

For even more facts about Verdier sequences see [Fab+20II Appendix A] and IV15a below

IV14a Fibre Sequences in Catstinfin mdash We can explicitly characterise fibre sequences in

Catstinfin Given an exact functor p B C between stable infin-categories the fibre fib(p) is the

full subcategoryfib(p) b isin B | p(b) 0 in C sube B

Since p is exact itrsquos easy to show that fib(p) sube B is closed under finite direct sums andfibrecofibre sequences hence it is really a stable sub-infin-category and one immediatelychecks that fib(p) also satisfies the desired universal property

IV14b Cofibre Sequences in Catstinfin mdash Therersquos also an explicit description of cofibres

in Catstinfin Given an exact functor f A B by the essential image of f we mean the full

sub-infin-category of B spanned by all objects which are equivalent to some f(a) Since f isexact the essential image is closed under finite direct sums but not necessarily under fibresand cofibres since there might be more morphisms in B than in A But therersquos always asmallest stable sub-infin-category of B containing the essential image of f In any case we have

cofib(f) BA = B[mod-A equivalencesminus1] 208


where the mod-A equivalences are all those maps ϕ x y whose fibrecofibre is a retractof an object in the stable sub-infin-category generated by the essential image of f Asfib(ϕ) cofib(ϕ)[minus1] it really suffices to assume that either one is such a retract

In contrast to (c) itrsquos not so easy to show that cofib(f) BA is a stable infin-categoryagain and that B BA is an exact functor so wersquoll postpone the proof until Lemma IV27But assuming this is true itrsquos easy to check that BA is indeed the correct cofibre Let C beanother stable infin-category By definition (see Example I37)

HomCatstinfin

(BA C) sube HomCatstinfin

(B C)

is the collection of path components of functors that invert mod-A equivalences But notethat also

HomCatstinfin

(cofib(f) C

) HomCatst

infin(B C)timesHomCatst

infin(AC) 0 sube HomCatst

infin(B C)

is also a collection of path components This is because the zero functor is initial (andterminal) in Fun(A C) hence its path component in HomCatst

infin(A C) sube core Fun(A C) is a

contractible anima hence the inclusion of 0 into it is an equivalenceTherefore it suffices to show HomCatst

infin(BA C) and HomCatst

infin(cofib(f) C) comprise the

same path components But thatrsquos easy Any exact functor p B C inverting the mod-Aequivalences satisfies p f 0 as 0 p(a) is a mod-A equivalence for all a isin A Converselyany exact p B C with p f 0 also kills all retracts of the stable sub-infin-categorygenerated by the essential image of f Hence it inverts mod-A equivalences

IV15 Theorem mdash Let F A B be an exact functor of stable infin-categories(a) F is a Verdier projection iff it is a localisation In this case it automatically inverts

precisely the mod-fib(F ) equivalences(b) F is a leftright split Verdier projection iff it admits a fully faithful leftright adjoint

(ie iff it is a leftright Bousfield localisation)(c) F is a Verdier inclusion iff it is fully faithful and its essential image is closed under

retracts in B(d) F is a leftright split Verdier inclusion iff it is fully faithful and has a leftright adjointFurthermore if

A B Cf p

g q

or A B Cf p

gprime qprime

is a left or right split Verdier sequence with adjoints as indicated then for all b isin B thereare fibre sequences qp(b) b fg(b) or fgprime(b) b qprimep(b) respectively Moreover in thiscase the sequences

C B Aq g

p f

or A B Cqprime gprime

p f

209


are right or left split Verdier respectively Finally if

A B Cf p

g

gprime

q

qprime

is a split Verdier sequence then gqprime cofib(q rArr qprime) cofib(gprime rArr g) gprimeq[1] (wersquoll explain inthe proof how the middle two are functors C A) and for all b isin B there are pushoutpullbacksquares

b fg(b)

qprimep(b) fgqprimep(b)

andfgprimeqp(b) fgprime(b)

qp(b) b

Proof The proof is essentially trivial (assuming the description of cofibres in IV14b whichwersquoll prove in Lemma IV27) and was left as an exercise in the lecture

We begin with (a) Every Verdier projection is a localisation by IV14b hence it sufficesto prove the converse If B A[Wminus1] for some W sube π0 core Ar(A) then all morphismsϕ x y in W satisfy F (fib(ϕ)) 0 since ϕ is sent to an equivalence and F is exact HenceW is contained in the mod-fib(F ) equivalences This might well be a proper inclusion Butevery exact functor B C automatically inverts all mod-fib(F ) equivalences hence also

B A[mod-fib(F ) equivminus1]

and thus fib(F ) A B is also a cofibre sequence Finally the inverted morphismsare precisely the mod-fib(F ) equivalences Indeed if F (ϕ) is an equivalence in B for someϕ x y in A then F (fib(ϕ)) 0 hence fib(ϕ) isin fib(F )

Next we prove (b) PropositionDefinition I58 and Proposition I59 take care of mostof the statement and all thatrsquos left is to show that if F A B is a leftright Bousfieldlocalisation then fib(F ) sube A has a leftright adjoint Letrsquos only do the left case the rightcase is entirely analogous Let q B A be a left adjoint with counit c qF rArr idA Weclaim that g cofib(c qF rArr idA) defines a left adjoint g A fib(F ) of the fully faithfulinclusion fib(F ) sube A

First off F (g(a)) cofib(idF (a) F (a) F (a)) 0 since F is exact hence g takes valuesin the full sub-infin-category fib(F ) as required We must show that g is a left Bousfieldlocalisation onto fib(F ) so of course wersquoll apply our favourite criterion Proposition I61aWe get a natural transformation η idA rArr g for free since g was defined as a cofibre Itrsquoseasy to see that ηg(a) g(a) simminus g(g(a)) is an equivalence for all a isin A Indeed qF (g(a)) 0hence

g(g(a)

) cofib

(cg(a) qF (g(a)) g(a)

) g(a)

as required Proving that g(ηa) g(a) simminus g(g(a)) is an equivalence is only slightly harderWe have get a morphism of cofibre sequences

qF (a) a g(a)

qF(g(a)

)g(a) g

(g(a)

)qF (ηa) ηa

sim

g(ηa)

210


But qF (g(a)) 0 hence the square on the left is a pushout as indicated which impliesthat the induced map on cofibres must be an equivalence Thus Proposition I61a can beapplied which shows that g is a left Bousfield localisation with image in fib(F ) But η is anequivalence on all of fib(F ) hence the image of g is all of it This shows (b)

The ldquoonly ifrdquo part of (c) is clear from our description of fibres in IV14a Converselyassume F is fully faithful and its essential image is closed under retracts We must showthat A is the fibre of B BA This can be verified on homotopy categories So assumeb isin B has the property that the unique morphism 0 b in πB becomes an equivalence inthe localisation

π(BA) (πB)[mod-A equivminus1]

We must show that b lies in (the essential image of) πA Usually such a question is notoriouslydifficult but here we are lucky By assumption on B a morphism ϕ x y in B is a mod-Aequivalence iff fib(ϕ) lies in (the essential image of) A Using this itrsquos not hard to verifythat the mod-A equivalences form a left multiplicative system in πB in the sense of [StacksTag 04VB] In such a situation the localisation is easy to describe Since we assume thatthe identity idb b b and the zero morphism 0 b b coincide in π(BA) there must be acommutative diagram

b

b c b

b0

s

in πB where s is a mod-A equivalence But a simple diagram chase shows s = 0 Hencefib(s) boplus c[1] is in contained in (the essential image of) A and thus so is b by closednessunder retracts

For (d) the ldquoonly ifrdquo is trivial from (c) and the definitions So assume F is fully faithfulwith left adjoint g B A and unit η idB rArr Fg In the same way as in (b) we prove thatp fib(η idB rArr Fg) is a right Bousfield localisation p B B onto the full sub-infin-categoryim(p) sube B By the dual of PropositionDefinition I58(d) im(p) is a localisation at thosemorphisms ϕ x y for which

ϕlowast HomB(p(b) x

) simminus HomB(p(b) y

)is an equivalence for all b isin B But this is equivalent to

lowast HomB(p(b)fib(ϕ)

) Homim(p)

(p(b) p(fib(ϕ))

)

which is in turn equivalent to p(fib(ϕ)) 0 This again is equivalent to fib(ϕ) simminus Fg(fib(ϕ))being an equivalence or in other words to fib(ϕ) lying in the essential image of A Hencethe localisation p is a localisation at all those morphisms whose fibre is in the essential imageof A But then it inverts all mod-A equivalences hence im(p) BA is the localisationwersquore looking for This proves the left case of (d) The right case is analogous

Now for the additional assertions Assume that

A B Cf p

g q

211


is a left split Verdier sequence (the right case is analogous) The desired fibre sequences werealready constructed in the proof of (b) Letrsquos show that

C B Aq g

p f

is a right split Verdier sequence We already know from (d) that q is a right split Verdierinclusion so we only need to check that A is its cofibre By PropositionDefinition I58(d) Ais a localisation of B at the f -local equivalences ie at those morphisms ϕ x y for which

ϕlowast HomB(y f(a)

) simminus HomB(x f(a)

)is an equivalence for all a isin A Dualising the argument from the proof of (d) this is equivalentto cofib(ϕ) lying in the image of C hence the f -local equivalences coincide with the mod-Cequivalences (here we also use that q is fully faithful and its essential image is closed underretracts by (c)) This shows that indeed A cofib(q)

Finally suppose

A B Cf p

g

gprime

q

qprime

is a split Verdier sequence By adjunction constructing a natural transformation q rArr qprime isequivalent to constructing a natural transformation pq rArr idC But the unit η idC rArr pq isan equivalence hence we can take its inverse Now

p(

cofib(q rArr qprime)) cofib(pq rArr pqprime) cofib(id idC rArr idC) 0

hence cofib(q rArr qprime) C B factors naturally over A Now recall the natural fibre sequencesqp(b) b fg(b) plug in b qprime(c) and use pqprime idC to get cofib(q(c) qprime(c)) fgqprime(c)Since everything is natural in c this implies

gqprime cofib(q rArr qprime) in Fun(CA)

Moreover plugging b q(c) into the fibre sequences fgprime(b) b qprimep(b) and using pq idCimplies fgprimeq(c) fib(q(c) qprime(c)) cofib(q(c) qprime(c))[minus1] This is natural in c andmoreover one checks that the morphisms q(c) qprime(c) are the same as above Hence also

gqprime[1] cofib(q rArr qprime) in Fun(CA)

To construct a natural transformation gprime rArr g we apply g to the counit fgprime rArr idB and usegf idA Now gq 0 and thus cofib(gprimeq rArr gq) gprimeq[1] Similarly gprimeqprime 0 and thuscofib(gprimeqprime rArr gqprime) gqprime But we already know gprimeq[1] gqprime hence

cofib(gprimeq rArr gq) cofib(gprimeqprime rArr gqprime) in Fun(CA)

We call this functor simply cofib(gprime rArr g) C A Thereby wersquove obtained the desiredequivalences gqprime cofib(q rArr qprime) cofib(gprime rArr g) gprimeq[1]

212


It remains to construct the pushoutpullback squares Consider the following solidpushoutpullback diagram

[minus1] fgprime(b) 0

qp(b) b qprimep(b)

0 fg(b)

Plugging b qprimep(bprime) into the fibre sequence qp(b) b fg(b) and using pqprime idCshows that there is a fibre sequence qp(bprime) qprimep(bprime) fgqprimep(bprime) for all bprime isin B Uponinspection the first arrow coincides with the horizontal composition in the middle row ofthe diagram Hence fgqprimep(b) which establishes the first pullback square Moreoverfgqprimep(b)[minus1] f(gqprime[minus1])p(b) fgprimeqp(b) which establishes the second pullback squareWersquore done at last

IV15a Even More on Verdier Sequences mdash As a consequence of Theorem IV15(b)and (d) for a Verdier sequence to be leftright split it suffices that one of the two functorshas a leftright adjoint Moreover for a pullback square

A B

C D

in Catstinfin to be a leftright split Verdier square it suffices that the left vertical map B D is

a leftright split Verdier projection we donrsquot even need to assume that A C is a Verdierprojection at all Indeed if there is a fully faithful leftright adjoint D B then we alsoget a fully faithful leftright adjoint C A from the pullback property Hence A C isautomatically a leftright split Verdier projection as well

IV16 Stable Recollements mdash A split Verdier sequence

A B Cf p

g

gprime

q

qprime

is also known as a stable recollement (with French pronounciation) and gqprime or any of itsthree other equivalent descriptions from Theorem IV15 is called its classifying functor cThis is because one can check that

B Ar(A)

C A

grArrcp

t

c

213


is a pullback and in fact a split Verdier square so

A Ar(A) AidArArr0 t

s

fib(minus)

0rArridA

idArArridA

is the universal stable recollement with fibre ALetrsquos look at an example A map f R S is called a localisation of Einfin-rings if the

multiplication map micro SotimesRS simminus S is an equivalence For example we could takeHA HBfor A B a derived localisation of ordinary rings (see PropositionDefinition I63 andProposition II63 for why derived localisations give localisations of Einfin-ring spectra) orR R[sminus1] for some s isin π0(R) In any case we have the following lemma

IV16a Lemma mdash For any localisation f R S of Einfin-ring spectra there is a stablerecollement

ModS ModR ModS-torsRflowast IotimesRminus

SotimesRminus

homR(Sminus)

incl

homR(Iminus)

where flowast denotes the forgetful functor I fib(f R S) and ModS-torsR sube ModR denotes

the full stable sub-infin-category of those R-module spectra that die upon S otimesR minus

Proof Wersquove seen in II62c(c) that flowast has left adjoint SotimesRminus and right adjoint homR(Sminus)Moreover as f R S is a localisation flowast is fully faithful by PropositionDefinition I63(b)Note that this propositiondefinition only deals with derived localisations of ordinary ringsbut the proof can be copied verbatim for arbitrary localisations of Einfin-ring spectra Henceflowast is a split Verdier inclusion by Theorem IV15(d) and we only need to identify its cofibreas ModS-tors

R We know that the cofibre is a left Bousfield localisation p ModR C onto some full

sub-infin-category C sube ModR From Theorem IV15 again we get functorial fibre sequencesp(M) M S otimesR M for all M isin ModR Since I R S is a fibre sequence andM RotimesRM this implies p(M) IotimesRM as claimed Now C is spanned precisely by thoseM for which p(M) simminusM is an equivalence But this is equivalent to S otimesRM 0 by thefibre sequence above hence C ModS-tors

R as claimed It remains to show that homR(Iminus)is a right adjoint of I otimesR minus but thatrsquos clear from the tensor-Hom adjunction

IV16b Torsion Modules and Complete Modules mdash Lemma IV16a already hascool consequences in the special case where f is the localisation R R[sminus1] at a singleelement s isin π0(R) (we only considered the special case S S[pminus1] for some prime p in thelecture but everything will work in general) To this end we consider the following twostable sub-infin-categories

Mods-compR sube ModR and Mods

infin-torsR sube ModR

The left one is spanned by the s-complete R-module spectra as introduced in LemmaDefi-nition III22a The right one is spanned by the s-power torsion R-module spectra These

214


are those M for which M M otimesR (Rsinfin)[minus1] where we denote Rsinfin R[sminus1]R ThenRsinfin colimnisinNRs

n and in general

M otimesR (Rsn)[minus1] fib(sn M minusM) M [sn]

is the sn-torsion part of M Hence M is an s-power torsion R-module spectrum iff it is thecolimit of its sn-torsion parts as one would expect

Also recall that Ms homR((Rsinfin)[minus1]M) denotes the s-completion of M as definedin III22 Additionally we introduce the notation

divs(M) = homR

(R[sminus1]M

) lim

Nop

(

sminusM

sminusM

sminusM

)for the s-divisible part of M With this terminology out of the way we obtain that Theo-rem IV15 and Lemma IV16a imply a version of GreenleesndashMay duality

IV16c Corollary mdash The functors

(minus)s Modsinfin-torsR

simsim Mods-comp

R minusotimesR(Rsinfin)[minus1]

are inverse equivalences Furthermore for all R-module spectra M there are canonicalpushoutpullback squares

M Ms


anddivs(M otimesR Rsinfin) divs(M)[1]

M otimesR Rsinfin M [1]

Proof We apply Lemma IV16a in the special case S R[sminus1] Then I (Rsinfin)[minus1] andthus homR(Iminus) (minus)s is the s-completion functor Likewise homR(R[sminus1]minus) divs(minus)extracts the s-divisible part as above Finally we have

Modsinfin-torsR ModR[sminus1]-tors

R

Indeed from the fibre sequence (Rsinfin)[minus1] R R[sminus1] we get that M otimesR R[sminus1] isequivalent to M M otimesR (Rsinfin)[minus1] hence the R[sminus1]-torsion modules are precisely thes-power torsion modules as claimed

By Theorem IV15 the bottom half in Lemma IV16a is a left split Verdier sequenceApplying this in our special case S R[sminus1] together with our considerations above thisleft split Verdier sequence appears as

Modsinfin-torsR ModR ModR[sminus1]

(minus)s divs(minus)

minusotimesR(Rsinfin)[minus1] flowast

In particular the s-completion functor (minus)s is fully faithful with essential image the fibreof divs(minus) ModR ModR[sminus1] By LemmaDefinition III22a this means that (minus)sis an equivalence onto Mods-comp

R Hence its left adjoint minus otimesR (Rsinfin)[minus1] is an inverseequivalence This proves the first half of the corollary and the second half follows straightfrom Theorem IV15

215


Note that the left pullback square from Corollary IV16c already made its appearancein Lemma III22b(b) But now wersquove seen that it also follows from ridiculously generalprinciples Cool right Similarly applying Lemma IV16a to the localisation S HQ givesthe arithmetic fracture square from Lemma III22h(c)

The Additivity Theorem and a Proof of Theorem IV12Onwards to the proof of Theorem IV12 We start with a somewhat random lemmadefinitionIV17 LemmaDefinition mdash Let C be a stable infin-category For each 0 6 i 6 n thepushout square

[0] [nminus i]

[i] [n]

0

i +i

in ∆∆ gives a Verdier squareQn(C) Qi(C)

Qnminusi(C) Q0(C)

In particular for an additive functor F Catstinfin An the simplicial anima F (Q(C)) isin sAn

is a (not necessarily complete) Segal anima and we put

SpanF (C) = asscat(F (Q(C))

)

Proof Letrsquos first check that our would-be split Verdier square is a pullback square at all (inCatinfin or Catst

infin that doesnrsquot matter by IV14(e)) This follows from direct inspection asthe Quillen Q-construction satisfies the Segal condition by PropositionDefinition I71 soQn(C) can be written as the (n+ 1)-fold pullback Q1(C)timesQ0(C) middot middot middot timesQ0(C) Q1(C) A similarargument shows that once we know that the pullback square is split Verdier F (Q(C)) willindeed satisfy the Segal condition since F sends split Verdier squares to pullbacks

So it remains to show that the pullback square is split Verdier Recall from the proof ofPropositionDefinition I71 that Qn(C) Fun(Jn C) where Jn is a zigzag

︸︷︷︸n zigs and zags

In general if ι [k] [`] is the inclusion of an interval then it induces a fully faithful functorι Jk J` Since the Ji are finite and C has finite limits and colimits ιlowast Fun(J` C) Fun(Jk C) has both adjoints given by left and right Kan extension and they are fullyfaithful again by Corollary I54 Thus Q`(C) Qk(C) is a split Verdier projection byTheorem IV15(b) This clearly applies in our case whence wersquore done

IV18 Theorem (Waldhausenrsquos additivity theorem) mdash If F Catstinfin An is additive

then so is ∣∣SpanF (minus)∣∣ Catst

infin minus An In particular k Catst

infin An is additive

216


This particular general formulation of the additivity theorem or to be really precise itsfurther generalisation to Poincareacute categories first appeared in [Fab+20II Theorem 241]But of course the original formulation (with F core) is due to Waldhausen

Proof of Theorem IV18 The ldquoin particularrdquo follows as k(C) Ω |Span(C)| Ω |Spancore(C)|by Proposition IV8 core is additive by I28(d) and Ω preserves pullbacks The proof of themain statement consists of four steps(1) We prove that if p C D is a split Verdier projection then it is a bicartesian fibration

(ie both cartesian and cocartesian)This step is basically a consequence of the following simple lemma

IV19 Lemma mdash Let g B A f be an adjoint pair in Catinfin with counit c gf rArr idA(a) A morphism ϕ x y in A is f -cocartesian iff the square

gf(x) gf(y)

x y

gf(ϕ)

c c

ϕ

is a pushout in A(b) If A admits pushouts f preserves pushouts and g is fully faithful then f is a cocartesian

fibration

Proof of Lemma IV19 For (a) plug the (g f)-adjunction into the pullback square fromDefinition I24(a) For (b) let ϕprime xprime yprime be a morphism in B and let x isin A with f(x) xprimebe given Since g is fully faithful we have yprime fg(yprime) hence the morphism ϕprime f(x) yprime

induces a morphism gf(x) g(yprime) by adjunction Now form the pushout

gf(x) g(yprime)

x y

c ϕ

in A Note that f(c) fgf(x) f(x) xprime is equivalent to the identity on xprime since g is fullyfaithful Moreover f preserve pushouts Therefore applying f to the pushout square abovegives a pushout

xprime yprime

x y

ϕprime

f(ϕ)

Hence f(ϕ) ϕprime so ϕ x y is a lift of ϕprime By (a) it is even an f -cocartesian lift provingthat f is indeed a cocartesian fibration

Now we resume the proof of Theorem IV18 Applying Lemma IV19(b) to p and its fullyfaithful left adjoint (which are exact functors between stableinfin-categories so all assumptionsare verified) shows that p is a cocartesian fibration By a dual argument applied to p and itsfully faithful right adjoint p is also cartesian This finishes Step (1)

217


(2) We prove that if p C D is a split Verdier projection (thus an exact bicartesianfibration by Step (1)) then

SpanF (p) SpanF (C) minus SpanF (D)

is a bicartesian fibration as wellLecture 269th Feb 2021

Letrsquos first discuss the case F core which is of course the one of most interest to ussince it leads to K-theory but it will also be half of the argument for the general case Weclaim() A morphism x z in Span(C) ie a span x y z in C is Span(p)-cocartesian if

y x is p-cartesian and y z is p-cocartesianUnravelling what a map Λ2

0 Span(C) and a map ∆2 Span(D) look like we have to showthe following Suppose wersquore given two solid diagrams

x z v

y bull

w

(1) (2)

(3) (4)

p7minus

p(x) p(z) p(v)

p(y) u

p(w)

in C and D respectively such that the left one is mapped into the right one under p Wemust show that we can fill in all the dotted arrows (along with the missing object bull) in theleft diagram in such a way that the square in the middle becomes a pullback We can fill inthe dotted arrow labelled ldquo(1)rdquo since y x is p-cartesian Next we can choose the dottedarrow ldquo(2)rdquo to be a p-cocartesian lift of p(w) u Since w bull is now p-cocartesian we canfill in the two missing dotted arrows ldquo(3)rdquo and ldquo(4)rdquo Playing around with Definition I24(a)easily shows that the square in the middle is a pushout iff w bull is p-cocartesian Hence itis a pushout by construction and thus also a pullback since C is a stable infin-category Thisproves () Let us also remark that our argument shows that filling in the dotted arrows isunique up to contractible choice Indeed this is clear for ldquo(1)rdquo ldquo(3)rdquo and ldquo(4)rdquo and forldquo(2)rdquo wersquove just argued that we need to choose w bull as a p-cocartesian lift of p(w) u tomake the middle square a pullback

We can now conclude that Span(p) Span(C) Span(D) is a bicartesian fibrationIndeed p is bicartesian by Step (1) and given any span p(x) yprime zprime in D we can lift theleft map to a p-cartesian edge y x and then the right map to a p-cocartesian edge y zwhich proves that Span(p) is a cocartesian fibration by () Dualising the argument showsthat Span(p) is also cartesian

Now for the general case Let E sube Q1(C) Fun(J1 C) be the full sub-infin-category wherethe left edge is p-cartesian and the right edge is p-cocartesian Then the diagram

E C

Q1(D) D

p

d1

p

d1

is a pullback Informally this is because we can lift any span (p(x) yprime zprime) isin Q1(D) toa span (x y z) isin E as argued in the previous paragraph and this lift is necessarily

218


unique up to contractible choice since so is taking p-cartesian and p-cocartesian lifts For amore formal argument you would have to show that the map from E to the pullback is fullyfaithful and essentially surjective Essential surjectivity follows from the argument wersquove justgiven For fully faithfulness use Corollary I49 to compute Hom anima in Q1(C) and Q1(D)(we have TwAr(J1) Jop

2 so the formula really is not too bad) and then simplify using thepullback square from Definition I24(a) and its dual for cartesian edges Irsquoll leave workingout the details to you

Likewise we get a pullback diagram

E timesQ1(C) Q2(C) E timesC Q1(C)

Q2(D) Q1(D)timesD Q1(D)

p

(idE d1)

p

(d2d1)

where the fibre product in the upper left corner is formed using d2 Q2(C) Q1(C)Informally this square being a pullback is precisely what we proved in () Indeed EtimesCQ1(C)encodes the data of the diagram without the dotted arrows filled in Q2(D) encodes thedata of the solid diagram in D and Q1(D)timesD Q1(D) makes sure these two are compatibleunder p Wersquove seen in () that itrsquos always possible to fill in the dotted arrows so thatwe obtain an object of E timesQ1(C) Q2(C) Moreover this object is unique up to contractiblechoice as wersquove noted at the end of the proof of () This is the informal reason whyE timesQ1(C)Q2(C) is the pullback To make this argument formal you would again have to showthat the map from E timesQ1(C) Q2(C) to the pullback is fully faithful and essentially surjectiveEssential surjectivity follows from the informal argument and for fully faithfulness you haveto compute Hom anima again using Corollary I49 (and even Hom anima in Q2(C) are nottoo bad since limits over TwAr(J2) Jop

3 are very manageable)Long story short we get two pullback squares But in fact they are split Verdier squares

Indeed by IV15a we only need to check that C D and E timesC Q1(C) Q1(D)timesD Q1(D)are split Verdier projection For the first one this holds by assumption The second oneis a pullback of Q1(C) Q1(D) (using the first pullback square) which is a split Verdierprojection because the fully faithful left and right adjoints of C D extend to functorinfin-categories

Thus both diagrams stay pullback squares after applying the additive functor F Thesecond diagram then witnesses that the image of

F (E) minus F(Q1(C)

)minus HomCatinfin

([1]SpanF (C)

)consists of SpanF (p)-cocartesian edges (essentially by running the above arguments back-wards) and the first diagram gives a sufficient supply of these This shows that SpanF (p) isindeed a cocartesian fibration By a dual argument we see that it is cartesian as well Thisfinishes Step (2)

(3) We prove that the functor | | Catinfin An preserves pullback squares if one of the legsis a bicartesian fibration

This step is easy again Suppose wersquore given a pullback

A B

C D

p

219


in Catinfin such that B D and thus also A C are bicartesian fibrations We denoteby F = Stcocart(B D) D Catinfin be the cocartesian straightening of B D Sinceunstraightening turns compositions into pullbacks we obtain

Uncocart(F p) A and Uncocart(F ) B

Now consider the diagramD Catinfin An

|D|

F | |

|F |

Note that since F is a bicartesian fibration every morphism ϕ x y in D is sent to aleft-adjoint functor F (ϕ) F (x) F (y) hence to an equivalence |F (ϕ)| |F (x)| simminus |F (y)|Hence the dashed arrow exists by the universal property of localisations of infin-categories

By unstraightening of functors into An we now get a pullback diagram

Unleft (|F | |p|) Unleft (|F |)|C| |D|

|p|

So wersquore done if we can show that Unleft(|F |) and Unleft(|F | |p|) are the realisationsof Uncocart(F ) and Uncocart(F p) respectively The easiest way to see this is probablyProposition I36 We know that colim(F D Catinfin) is some localisation of Uncocart(F )hence ∣∣Uncocart(F )

∣∣ ∣∣∣ colim(D Fminus Catinfin

)∣∣∣ colim(D Fminus Catinfin

| |minus An

) colim

(|D| |F |minus An

) Unleft (|F |)

as desired The second equivalence follows from the fact that | | Catinfin An being a leftadjoint preserves colimits For the third equivalence we use that the localisation D |D| iscofinal as we noted after Definition I44 For Uncocart(F p) we can use a similar argumentThis finishes Step (3)(4) We prove that SpanF (minus) sends split Verdier squares to pullbacks in Catinfin and finish

the proof of Theorem IV18If we can show the first assertion then a combination of Step (2) and (3) seals the deal

Recall that SpanF asscat(F (Q(minus))) by LemmaDefinition IV17 So let

A B

C D

be a split Verdier square We know that Qn(minus) Catstinfin Catst

infin preserves split Verdiersquares for all n isin N (because Qn(minus) Fun(Jnminus) preserves pullbacks and the required

220


fully faithful adjoints extend to functor infin-categories) and F turns them into pullbacks inAn hence

F(Q(A)

)F(Q(B)

)F(Q(C)

)F(Q(D)

)

is a pullback in sAn In fact it is even a Segal anima by LemmaDefinition IV17 Howeverasscat(minus) sAn Catinfin doesnrsquot necessarily preserve pullbacks of incomplete Segal animaHowever the canonical (up to contractible choice) morphism

asscat(X timesY Z) minus asscat(X)timesasscat(Y ) asscat(Z)

is always fully faithful for Segal anima XY Z isin sAn as follows from (I652) So onlyessential surjectivity can go wrong Recall that core asscat(X) |Xtimes| for X Segal by (I651)We claim that

π0∣∣Xtimes timesY times Ztimes∣∣ minus π0

(|Xtimes| times|Y times| |Ztimes|

)is surjective if asscat(X) asscat(Y ) is a bicartesian fibration which will be sufficient forour case by Step (2) Every connected component on the right-hand side can be representedby points x isin X0 z isin Z0 and a path in |Y times| connecting their images To see why the0-skeleta already hit all path components take a look again at the skeletal method in theproof of Lemma IV7 and in particular at (IV72) Again by considering skeleta we seethat two points belonging to the same path component in |Y times| is already determined by Y times1 Hence there are edges yi isin Y times1 and a path w in Y0 that connect x and z via

xy1minus s1

y2 minus s2y3minus

y2n minusminus s2nw

z

Using that asscat(X) asscat(Y ) is a bicartesian fibration we can lift this to a similarsequence in X which gives a connected component in π0|Xtimes timesY times Ztimes| This shows thedesired surjectivity and the proof is finally done

Wersquore now almost in position to prove Theorem IV12 But Fabian decided to give abrief discussion of grouplike additive functors and their span infin-categories first To make theresults of that discussion more comprehensible I put them into a separate lemmaIV20 Lemma mdash Suppose F Catst

infin An is a grouplike additive functor and let C be astable infin-category Then

F(

Ar(C)) F (C)times F (C) and F

(Qn(C)

) F (C)2n+1

for all n isin N Moreover SpanF (C) BF (C) so in particular SpanF (C) is a connectedanimaProof sketch The equivalence F (Ar(C)) F (C)times F (C) follows from the stable recollementfrom IV16 together with IV14(dlowast) The equivalence F (Qn(C)) simminus F (C)2n+1 is induced byevaluation Qn(C) Fun(Jn C) C at the 2n+ 1 objects of Jn To see that it is indeed anequivalence letrsquos first consider the case n = 1 In this case we have a split Verdier sequenceC Q1(C) Ar(C) (adjoints are given by leftright Kan extension along [1] J1) whichproves F (Q1(C)) F (C)3 by IV14(dlowast) The general case follows either by induction or bythe an iterative application of the split Verdier square from LemmaDefinition IV17

Next we set out to prove that SpanF (C) is a connected anima For this we first provethe following claim (which wasnrsquot in the lecture so I hope itrsquos correct)

221


() Let F be grouplike Under the identifications F (Qn(C)) F (C)2n+1 the ldquocompositionmaprdquo

F (C)5 F(Q1(C)

)timesd1F (C)d0 F

(Q1(C)

)F(Q2(C)

) d1minus F(Q1(C)

) F (C)3(d0d2)

sim

is given by (a0 a1 a2 a3 a4) 7 (a0 a1 minus a2 + a3 a4) (of course we shouldnrsquot think ofthis map in terms of objects but rather in terms of the structure maps of the Einfin-groupF (C) however a precise statement would become utterly unreadable)

After unravelling what the face maps of the simplicial infin-category Q(C) do () reduces tothe following statement(prime) Let p Fun(Λ2

2 C) C be the map given by taking pullbacks As above we can identifyF (Fun(Λ2

2 C)) F (C)3 Then under this identification

F (p) F (C)3 minus F (C)

is given by (a1 a2 a3) 7 a1 minus a2 + a3To prove (prime) recall that pullbacks in the stable infin-category C can be computed as followsLet a1 a2 a3 be a span in C and consider the map

ϕ a1 oplus a3(+minus)minusminusminusminus a2

given by a1 a2 in the first component and the negative of a3 a2 in the secondcomponent Then fib(ϕ) is the required pullback This allows us to write p as a composition

Fun(Λ22 C) minus Ar(C) fib(minus)

minusminusminusminus C

Upon applying F and identifying F (Ar(C)) F (C)2 via source and target the first arrowinduces the map F (C)3 F (C)2 given by (a1 a2 a3) 7 (a1 + a3 a2) Thus it remains toshow that F (fib(minus)) F (Ar(C)) F (C) sends (a b) 7 a minus b Consider the map the mapq Ar(C) Ar(C) given by q(ψ a b) (pr2 fib(ψ) oplus b b) The lower half of theuniversal recollement from IV16 implies that F (q) F (Ar(C)) F (Ar(C)) is homotopic tothe identity Hence F (fib(minus)) + F (t) F (s) holds in HomAn(F (Ar(C)) F (C)) This showsthat F (fib(minus)) is indeed given by (a b) 7 aminus b whence (prime) follows

Using () and the fact that F (C) is an Einfin-group one easily verifies that

F(Q1(C)

)times F (Q1(C))

Hence SpanF (C) asscat(F (Q(C))

)is an anima (combine TheoremDefinition I64(c) with

(I652) to see that all edges are equivalences) Moreover it is connected In fact itrsquos truethat |SpanF (C)| is connected for every additive functor F To see this use Remark I66Proposition IV8 and IV7b (in that order) to obtain∣∣ SpanF (C)

∣∣ ∣∣F (Q(C))∣∣ ∣∣F (S(C)

)esd∣∣ ∣∣F (S(C))∣∣

Now π0F (S0(C)) π0 |F (S(C))| is surjective see claim (lowast) in the proof of Proposition II2But S0(C) lowast hence indeed π0 |F (S(C))| = 0

222


Back in the situation where F is grouplike we see that to prove SpanF (C) BF (C)it suffices to compute that Ω0 SpanF (C) F (C) by II6 Now Ω0 SpanF (C) coincides withHomSpanF (C)(0 0) To compute this consider

C Q1(C)

0 Q0(C)timesQ0(C)

(d1d0)

It is a pullback by inspection as there are C possibilities to fill in the missing spot in aspan 0 bull 0 It is even a split Verdier square since the right vertical arrow has bothadjoints given by left and right Kan extension which are both fully faithful (in fact bothadjoints agree and send a pair (x y) isin Q0(C)timesQ0(C) C times C to the span x xoplus y yin Q1(C) Fun(J1 C)) So the pullback property is preserved upon applying F But afterapplying F the new pullback diagram computes HomSpanF (C)(0 0) by (I652)

Fabian remarks that the special case k(Ar(C)) k(C) times k(C) of Lemma IV20 is thecontent of Waldhausenrsquos original additivity theorem Moreover wersquove seen in the proof thatF (Q1(C))times F (Q1(C)) for every grouplike additive functor F This tells us that the Segalanima F (Q(C)) is seldom complete By TheoremDefinition I64(c) we would need that

F (C) F(Q0(C)

) simminus F(Q1(C)

)times F (C)3

is an equivalence which only holds if F (C) 0

IV21 Theorem mdash If F Catstinfin An is grouplike and additive and C is a stable

infin-category C then applying the realisation functor | | Catinfin An to the pullback square

HomSpanF (C)(0 0) 0SpanF (C)

lowast SpanF (C)

0

in Catinfin gives a pullback square

F (C) lowast

lowast∣∣SpanF (C)

∣∣ 0

0

in An So in particular F (C) Ω|SpanF (C)|

First proof Wersquove checked (or at least sketched) in Lemma IV20 that

SpanF (C) ∣∣SpanF (C)

∣∣ BF (C)

But F (C) is already an Einfin-group by assumption so ΩBF (C) F (C) follows from Corol-lary II21(c) A posteriori this implies that the square in question is indeed a pullback

With that we can finally prove the theorem of BlumbergndashGepnerndashTabuada

223


Proof of Theorem IV12 First note that Ω|SpanF (minus)| is additive again for any additivefunctor F since it is the composition of the additive functor |SpanF (minus)| (Theorem IV18)with the limits-preserving functor Ω CMon(An) CGrp(An) (Corollary II21(b)) Thisalso shows that Ω|SpanF (minus)| takes values in the grouplikes

As always wersquoll use Proposition I61a to prove that

L Ω∣∣ Span(minus)(minus)

∣∣ Funadd(CatstinfinAn) minus Fungrp(Catst

infinAn)

is a left Bousfield localisation First we need a natural transformation η id rArr L Thecomputation in the proof of Lemma IV20 shows F (C) HomSpanF (C)(0 0) for all additivefunctors F (although we assumed that F is grouplike there but this isnrsquot needed in thecomputation) Composing with the natural map

HomSpanF (C)(0 0) minus Hom| SpanF (C)|(0 0) Ω∣∣SpanF (C)

∣∣gives a map F (C) Ω|SpanF (C)| which is functorial in C and F thus providing the requiredtransformation η id rArr L Now Theorem IV21 implies that ηLLη L sim=rArr L L areboth equivalences whence L is indeed a left Bousfield localisation Its essential image iscontained in the grouplike functors as checked above hence equals the grouplike functorssince Theorem IV21 precisely says that η is an equivalence on grouplikes

As the ldquofirst proofrdquo above suggests we will give a second proof of Theorem IV21 Thesecond proof wonrsquot rely on the sketchy parts of Lemma IV20 (at the price of being muchlonger) but more importantly it will introduce some tools that wersquoll need later Where thefirst proof showed F Ω|SpanF (minus)| right away and then deduced the pullback propertyafterwards the second proof will verify the pullback property directly The key to still gettingpullbacks after taking colimits is Rezkrsquos equifibrancy lemma

IV22 LemmaDefinition (Rezk) mdash Let I be any infin-category (which we think of as adigram shape) Suppose wersquore given a pullback square

A B

C D

σ τ

in Fun(IAn) such that τ us equifibred ie such that

Bi Bj

Di Dj

is a pullback for all morphisms i j in I Then also the diagram

colimiisinI

A colimiisinI

Bi

colimiisinI

Ci colimiisinI

Di

σ

τ

is a pullback in An

224


Proof We first prove the following claim to get a description of colimI B() Suppose τ is equifibred and let K be any anima Then a transformation η B rArr constK

together with a map K colimiisinI Di make the diagram

Bi K

Di colimiisinI

Di

a pullback for each i isin I iff η induces an equivalence colimiisinI Bisimminus K

To prove () consider the functors Fi Stleft(Bi Di) Di An We claim that byequifibrancy of τ the Fi assemble into a functor

F colimiisinI

Di minus An

On Silrsquos insistence Fabian has given a proper argument for this in the official notes [AampHKChapter IV p 46] which Irsquoll include here The transformation τ corresponds to a map[1] Fun(IAn) which we may curry to a map I Ar(An) That τ is equifibrant meansthat this map sends all morphisms in I to morphisms in Ar(An) which are represented bypullback squares in An On the other hand

Fun(

colimiisinI

DiAn) limiisinIop

Fun(DiAn) limiisinIop

AnDi

by straightening Note that the functor Anminus Anop Catinfin is not the usual slice categoryfunctor (whose codomain would be An rather than Anop) Indeed to a map i j inI we donrsquot associate the forgetful functor AnDi AnDj but the pullback functorminustimesDj Di AnDj AnDi Thus Anminus Anop Catinfin is represented by the cartesian() straightening of t Ar(An) An which is in fact a cartesian fibration by Exercise I25bMoreover the t-cartesian edges are precisely the pullback squares hence I Ar(An) sendsevery edge to a t-cartesian edge Therefore the induced map I I timesAn Ar(An) is a map ofcartesian fibrations over I and thus it induces a natural transformation(

const lowast =rArr AnD(minus))

Iop minus An

This transformation induces a single map lowast limiisinIop AnDi which provides the functorF colimiisinI An wersquore looking for

Note that Unleft(F ) colimiisinI Di is a left fibration over the anima colimiisinI Di henceUnleft(F ) is an anima itself Thus Unleft(F ) |Unleft(F )| and we may apply Proposition I36twice to get

Unleft(F ) colimcolimiisinI Di

F colimiisinI

colimDi

Di

colimiisinI

Unleft(Fi)

colimiisinI

Bi

This shows that K colimiisinI Bi Unleft(F ) does indeed fit into the pullback diagramfrom () Conversely every K colimiisinI Di is uniquely determined by its pullbacks to the

225


Di since straightening implies An colimiisinI Di Fun(colimiisinI DiAn) limiisinIop AnDias seen above So there is only one K that fits into the pullbacks from () whenceK colimiisinI Bi is the only possibility This shows ()

Now the actual proof may start We first check that σ is equifibred as well To this endlet i j be any morphism in I and consider the diagram

Ai Aj Bj colimiisinI

Bi

Ci Cj Dj colimiisinI

Di

τ

We must show that the left square is a pullback But the middle square is a pullback byassumption and the right one is a pullback by () Hence also the rectangle formed by themiddle and the right square is a pullback But then the same argument implies that also therectangle formed by all three squares must be a pullback By abstract pullback nonsensethis implies that the left square must be a pullback as well as claimed This shows that σ isequifibred

Now consider the diagram

Aj colimiisinI

Bi timescolimiisinI Di colimiisinI

Ci colimiisinI

Bi

Cj colimiisinI

Ci colimiisinI

Di

τ

The right square is a pullback by construction and that the outer rectangle is a pullbackwas checked in the previous step Hence again the left square is a pullback too for all j isin IBut then () applied to σ implies that colimiisinI Bi timescolimiisinI Di colimiisinI Ci colimiisinI Aiand wersquore done

Second proof of Theorem IV21 We split the proof into two major steps(1) We construct an easily understood Segal anima Null(C) ∆∆op An satisfying

asscat(F (Null(C))

) 0 SpanF (C)

Consider the functor [0]minus ∆∆op ∆∆op It induces a functor dec sAn sAn called thedeacutecalage It already appeared in II62b although our convention there was to consider minus [0]instead (but who cares) The deacutecalage functor comes equipped with natural transformations

p dec =rArr const ev0 and d0 dec =rArr id

induced by [0] sube [0] [n] and [n] sube [0] [n] for all [n] isin ∆∆op respectively One easily checksthat for every Segal anima X and all x isin X0 one has a commutative diagram

asscat(

fibx(p decX constX0))

x asscat(X)

asscat(X)

d0

sim

226


To see this one first checks that fibx(p decX constX0) is Segal again so the Hom animain its associated category can be computed via (I652) Once we understand the Hom animain it we find that asscat(fibx(p decX constX0)) has an initial element thatrsquos mappedto x under d0 hence d0 lifts canonically to x asscat(X) Then show that the lift is fullyfaithful and essentially surjective again using that Hom anima can be calculated via (I652)I leave the details to you since Irsquom too lazy to work them out

Now putNull(C) = fib0(p decQ(C) const C)

So explicitly Nulln(C) sube Qn+1(C) is the full sub-infin-category spanned by diagrams of thefollowing form (as usual the dotted part is uniquely determined by the solid part)

0

︸︷︷︸n+1 zigs and zags

Moreover since p decQ(C) const C) is a degreewise split Verdier projection (with leftrightKan extension as adjoints as usual) we have

F(

Null(C)) fib0

(p decF (Q(C)) constF (C)

)

Thus our considerations above show asscat(F (Null(C))) 0 SpanF (C) as desired(2) We show that there is a pullback diagram

constF (C) F(

Null(C))

const lowast F(Q(C)

)F (d0)

0

in which the right vertical arrow is equifibred in the sense of LemmaDefinition IV22It is easy to see that the diagram above is indeed a pullback Indeed the pullback

0 timesQn(C) Nulln(C) consists of diagrams of the form

0 0 0 0 0

x 0 0 0

x 0 0

x 0

x


227


hence 0timesQn(C) Nulln(C) C Moreover d0 Nulln(C) Qn(C) is a split Verdier projection(with leftright Kan extension as adjoints as usual) hence this pullback is preserved byF and we obtain 0 timesF (Qn(C)) F (Nulln(C)) F (C) as desired This proves that we get apullback diagram as above

If we can show that the right vertical arrow is equifibred then wersquore done Indeed ifthis is the case then LemmaDefinition IV22 says that the diagram stays a pullback afterapplying | | colim∆∆op Hence we get a pullback

|constF (C)|∣∣F (Null(C)

)∣∣|const lowast|

∣∣F (Q(C))∣∣

|F (d0)|

0

The upper left corner is F (C) which coincides with the anima HomSpanF (C)(0 0) as arguedin the proof of Lemma IV20 The lower left corner is clearly lowast Applying Remark I66 to

asscat(F (Null(C))

) 0SpanF (C) and asscat

(F (Q(C))

) SpanF (C)

shows that the upper and lower right corner coincide with |0 SpanF (C)| lowast and |SpanF (C)|hence the pullback diagram we get is precisely the one we want

It remains to show that F (d0) F (Null(C)) F (Q(C)) is equifibred That is we mustshow that

F(

Nullm(C))

F(

Nulln(C))

F(Qm(C)

)F(Qn(C)

)

is a pullback for every map [m] [n] in ∆∆op I think the way we did this in the lecture istoo complicated and I can give a simpler proof It seems almost too easy so I feel a bit likeIrsquom overlooking something but I couldnrsquot find anything Please tell me if Irsquom wrong

Wersquove seen above that Nulln(C) Qn(C) is a split Verdier projection with fibre C andso is Nullm(C) Qm(C) Hence the induced map on fibres over 0 in the diagram above is anequivalence namely the identity on F (C) But thatrsquos enough to show the pullback propertySince F is grouplike all terms in the diagram are Einfin-groups hence all fibres are empty orequivalent to the fibres over 0 But IV14(dlowast) implies F (Nulln(C)) F (Qn(C))times F (C) andlikewise F (Nullm(C)) F (Qm(C)) times F (C) so none of the fibres can be empty In fact Ibelieve the map F (Nullm(C)) F (Nulln(C)) is the product map which would make thepullback property even more apparent but in any case the argument given above shouldsuffice

IV23 Remark mdashLecture 2711th Feb 2021

Fabian started the final lecture with two not-so-random and onevery random comment(a) We know from Lemma IV20 and Theorem IV18 that BF SpanF (minus) | SpanF (minus)|

is additive again whenever F is a grouplike additive functor In fact BF is evengrouplike by Corollary II21(b) More generally it is true that |SpanF (minus)| |F (Q(minus))|is grouplike for any additive functor F since π0|F (Q(minus))| = 0 as seen in the proof ofLemma IV20 However if F isnrsquot grouplike then BF doesnrsquot even need to be additiveanymore let alone coincide with |F (Q(minus))|

228


(b) Quillen defined k(R) = K0(R)timesBGLinfin(R)+ and then showed a posteriori that it isan infinite loop space essentially1 by verifying

k(R) Ωi∣∣coreQi

(Dperf(R)

)∣∣ Here Qi Catst

infin Fun((∆∆op)iCatstinfin) denotes the i-fold Q-construction It takes values

in ldquoi-simplicialrdquo anima and subsequently the realisation | | in the formula above doesnrsquotdenote colimits over ∆∆op but over (∆∆op)i

In the script [AampHK Chapter IV p 53] Fabian explains a bit more In general ifF is an additive functor then

Ω∣∣F (Q(minus)

)∣∣ Ωi∣∣F (Qi(minus)

)∣∣ where again | | on the right-hand side denotes the colimit over (∆∆op)i If F is grouplikethen both sides are also equivalent to F To prove the equivalence above we useTheorem IV18 iteratively (along with Corollary I45 to replace nested colimits over ∆∆op

by colimits over (∆∆op)i) to see that |F (Qi(minus))| is additive By the same argument asin (a) it is also grouplike Hence Theorem IV21 implies |F (Qi(minus))| Ω|F (Qi+1(minus))|from which the formula above follows by induction

Therefore any additive functor F Catstinfin An upgrades to a prespectra-valued

(see II31) functorSpanF Catst

infin minus PSp

given by SpanF (C) (F (C)Ω|F (Q(C))|Ω2|F (Q2(C))| ) The construction of SpanFis clearly functorial in F hence induces a functor

Span(minus) Funadd(CatstinfinAn) minus Fun(Catst

infinPSp)

If we restrict to Fungrp(CatstinfinAn) sube Funadd(Catst

infinAn) then Span(minus) takes valuesin Fun(Catst

infinSp) Moreover it follows from (a) that Span(minus) coincides with Binfin ongrouplike functors Thus for a stable infin-category C we put

K(C) = Spank(C) Binfink(C)

and call this the K-theory spectrum or projective class spectrum of C(c) Now for the random comment If Sp[01] sube Sp denotes the full subcategory of spectra E

whose homotopy groups πi(E) vanish for i isin [0 1] then

Sp[01] CGrp(An)[01] CGrp(Grpd)

A symmetric monoidal 1-groupoid Gotimes on the right-hand side corresponds to a gem(ldquogeneralised EilenbergndashMacLane animardquo see Very Long Example I56) in the middle iffthe symmetry isomorphisms σxx xotimes x simminus xotimes x from II15c is the identity on xotimes xfor all x isin G

The Fibration TheoremTo get fibre sequences like in IV1 it doesnrsquot suffice to know that K-theory is additive Forexample if R S is a derived localisation of ordinary rings (PropositionDefinition I63)

1Quillen did not put Dperf(R) there but still thatrsquos basically what he proved

229


then D(S) D(R) D(R)S-tors is a split Verdier sequence by Lemma IV16a but this is nolonger true if we restrict to Dperf everywhere Instead wersquoll have to show that k Catst

infin Anis Verdier-localising (Definition IV13) which is our next goal It will turn out that evenbeing Verdier-localising isnrsquot quite enough but more on that once wersquore there

IV24 Definition mdash If f A B is an exact functor of stable infin-categories then therelative Q-construction Q(f) isin sCatst

infin is defined by the pullback

Q(f) Null(B)

Q(A) Q(B)

d0

f

By IV14(e) it doesnrsquot matter whether this pullback is taken in sCatstinfin or on sCatinfin We

may think of Qn(f) as diagrams of the following form (as usual the dotted part is uniquelydetermined by the solid part)

0 a0 a2 a2nminus2 a2n

b a1 a2nminus3 a2nminus1


Here a0 a2n isin A b isin B and of course the arrow b a0 should be interpreted as amorphism b f(a0) in B

IV25 Corollary mdash If F Catstinfin An is grouplike additive then for all exact functors

f A B there is a fibre sequence∣∣F (Q(f))∣∣ minus ∣∣F (Q(A)

)∣∣ minus ∣∣F (Q(B))∣∣

in An (or equivalently in CGrp(An))

Proof Recall the following facts from the second proof of Theorem IV21(a) d0 Nulln(B) Qn(B) is a split Verdier projection for all [n] isin ∆∆op see Step (2)(b) F (d0) F (Null(B)) F (Q(B)) is equifibred see Step (2)(c) |F (Null(B))| |0 SpanF (B)| lowast see Step (1)By (a) the pullback square defining Q(f) is a degreewise split Verdier square Hence it staysa pullback after applying F Now (b) together with LemmaDefinition IV22 ensures that westill get a pullback after taking realisations By (c) this pullback square provides the desiredfibre sequence

IV26 Corollary mdash For a grouplike additive functor F Catstinfin An the following are

equivalent

230


(a) F is Verdier-localising and sends Verdier projections to π0-surjections That is for allVerdier-projections B C the induced map π0(F (B)) π0(F (C)) is surjective

(b) For all Verdier inclusions i A B the canonical map∣∣F (Q(i))∣∣ simminus F (BA)

is an equivalence(clowast) The functor BinfinF Catst

infin Sp is Verdier-localising

In the lecture we only had (a) and (b) and we didnrsquot impose the condition that F sendsVerdier projections to π0-surjections I think it is needed though In view of () below itrsquosa natural question whether one can drop this condition (along with (clowast)) and in turn onlyask for |F (Q(i))| F (BA) to be an inclusion of path components However I believe thisbreaks the argument why F sends Verdier squares to pullbacks Please tell me if itrsquos trueafter all

Also note that K-theory does send Verdier projections to π0-surjections which is clearfrom Definition IV3 and the fact that B BA is essentially surjective

Proof of Corollary IV26 Letrsquos first explain where the canonical map comes from So forthe moment F is only grouplike additive Theorem IV21 implies Ω|F (Q(A))| F (A) andlikewise for B Thus rotating the fibre sequence from Corollary IV25 gives a fibre sequence

F (A) minus F (B) minus∣∣F (Q(i)

)∣∣in CGrp(An) We claim() The induced map π0(F (B)) π0 |F (Q(i))| is surjective and thus the sequence above is

also a fibrecofibre sequence in SpIf C denotes the cofibre in Sp then by Lemma II23c it suffices to show that the canonical mapC |F (Q(i))| induces isomorphisms on all homotopy groups Observe that C is connectiveso this holds for all negative homotopy groups and the usual five lemma arguments showthe same for all positive homotopy groups It remains to check that π0(C) π0 |F (Q(i))|is an isomorphism For that we must show that π0(F (B)) π0 |F (Q(i))| is surjective ByLemma IV20 the fibre sequence from Corollary IV25 can be rewritten as∣∣F (Q(i)

)∣∣ minus BF (A) minus BF (B)

Hence π1(BF (B)) π0 |F (Q(i))| π0(BF (A)) is exact But π1(BF (B)) = π0(F (B)) andπ0(BF (A)) = 0 since B shifts homotopy groups up Thus π0(F (B)) π0 |F (Q(i))| is indeedsurjective and wersquove proved ()

Now for the actual proof Since F (A) F (B) F (BA) always composes to 0() provides the desired natural map |F (Q(i))| F (BA) Assume (a) holds ThenF (A) F (B) F (BA) is a fibre sequence too By the usual five lemma arguments|F (Q(i))| F (BA) is an isomorphism on all homotopy groups except possibly π0 Butsince π0(F (B)) π0(F (BA)) is surjective by assumption and π0(F (B)) π0 |F (Q(i))| issurjective by () the five lemma also works in π0 This shows (a) rArr (b)

Conversely assume that |F (Q(i))| simminus F (BA) Then F sends Verdier sequences to fibresequences and Verdier projections to π0-surjections It remains to show that F transformsVerdier squares into pullbacks But whether a square in CGrp(An) is a pullback square

231


can be tested on fibres Since F sends Verdier projections to π0-surjections all fibres arenon-empty and thus all fibres are equivalent to the respective fibres over 0 But the inducedmap on fibres over 0 is an equivalence since we know what F does on Verdier sequencesThis shows (b) rArr (a)

Finally for (a)hArr (clowast) it suffices to observe that a fibre sequenceX Y Z in CGrp(An)stays a fibre sequence under Binfin CGrp(An) Sp iff π0(Y ) π0(Z) is surjective whichfollows from the argument in ()

We still owe a proof that cofibres in Catstinfin can be described as in IV14b The following

takes care of the essential case

IV27 Lemma mdash Let C sube D be a stable sub-infin-category (in the sense of Remark IV4a)and let DC be the localisation of D at the mod-C equivalences Then for all x y isin D

HomDC(x y) colimcisinCy

HomD(x cofib(c y)

)

In particular DC is stable and D DC is an exact functor

Lemma IV27 only takes care of the fully faithful case but it follows at once that thedescription from IV14b also holds for arbitrary exact functors Moreover note that theslice category Cy = C timesD Dy is filtered Indeed given any map K Cy from a finitesimplicial set K we can find an extension K Cy by taking the colimit in C which mapscanonically to y since C sube D preserves finite colimits Moreover recall that filtered colimitsin An commute with finite limits This is proved in [HTT Proposition 5333] but I thinkit also follows from Remark II31c after some fiddling

Proof of Lemma IV27 Let W sube π0 core Ar(D) be the collection of morphisms whose fibreis in C In the definition of mod-C equivalences we also allowed morphisms whose fibre is onlya retract of an object of C but once wersquove shown that D[Wminus1] is stable and p D D[Wminus1]is exact it will be clear that D[Wminus1] coincides with the localisation DC at all mod-Cequivalences

Recall from Lemma I60 that HomD[Wminus1](xminus) D[Wminus1] An is left-Kan extendedfrom HomD(xminus) D An Moreover left Kan extension p sits in an adjunction

p Fun(DAn) Fun(D[Wminus1]An

)plowast

in which plowast is fully faithful Hence plowastp Fun(DAn) Fun(DAn) defines a left Bousfieldlocalisation onto the full sub-infin-category of functors that invert W

The idea is now to guess an explicit description of plowastp and verify its correctness usingProposition I61a Consider the functor L Fun(DAn) Fun(DAn) defined by

LF (x) colimcisinCx

F(

cofib(c x))

The functorial maps x cofib(c x) define a map

F (x) colimcisinCx

F (x) minus colimcisinCx

F(

cofib(c x))

where the equivalence on the left follows from the fact that any filtered infin-category is weaklycontractible One checks that these maps are functorial in x and F whence they assembleinto a natural transformation η idrArr L We claim

232


(1) For all F the functor LF inverts W (2) If F already inverts W then η F LF is an equivalenceClaims (1) and (2) together easily imply that the conditions of Proposition I61a are met soindeed L plowastp This also proves that Hom anima in D[Wminus1] are given by

HomD[Wminus1](x y) colimcisinCy

HomD(x cofib(c y)

)

Using this formula together with Corollary I50 and the fact that filtered colimits commutewith finite limits shows that D D[Wminus1] preserves finite colimits Using the same argumentfor Dop Dop[(W op)minus1] shows that D D[Wminus1] also preserves finite limits Moreoverwe can get that D[Wminus1] has finite limits out of this We already have a terminal objecthence it suffices to have pullbacks But every span x y z in D[Wminus1] can be representedby a span x cofib(c y) z in D for some c isin Cy since the colimit in the formulaabove is filtered Now y cofib(c y) is in W hence an equivalence in D[Wminus1] thus itsuffices to take the pullback in D to get the desired pullback in D[Wminus1] In the same wayone shows that D[Wminus1] has finite colimits Moreover our argument also shows that pullbackand pushout squares coincide whence D[Wminus1] is stable by PropositionDefinition II27(c)This proves everything we want safe for the two claims above

We didnrsquot prove these claims in the lecture but herersquos what I think should work Letrsquosfirst check (1) Suppose ϕ x y has fibre in C Then also cofib(x y) isin C Now for anyc isin Cy consider the following diagram

ϕlowastc x 0

c y cofib(x y)

0 cofib(c y)

ϕ

From the upper two squares we obtain that ϕlowastc c cofib(x y) is a fibre sequence in Dtwo of whose terms are in C Hence also ϕlowastc isin C Thus taking pullbacks along ϕ defines afunctor ϕlowast Cy Cx which one readily checks to be right-adjoint to the forgetful functorϕ Cx Cy Recall that right adjoints are cofinal see the discussion after Definition I44Moreover the left two squares in the diagram above show cofib(ϕlowastc x) cofib(c y)Hence ϕlowast defines an equivalence

ϕlowast LF (y) colimcisinCy

F(

cofib(ϕlowastc x)) simminus colim

cprimeisinCxF(

cofib(cprime x)) LF (x)

One checks that ϕlowast is an inverse to the induced map LF (ϕ) LF (x) LF (y) so that LFindeed inverts W This shows (1)

It remains to show (2) which is easy x cofib(c x) lies in W because its fibre is cHence F (x) simminus F (cofib(c x)) is an equivalence which proves that F (x) simminus LF (x) is anequivalence as well for all x isin D

To prove that K-theory is Verdier-localising we invoke a somewhat non-standard formu-lation of Waldhausenrsquos fibration theorem

233


IV28 Theorem (Waldhausenrsquos fibration theorem) mdash Let A sube B be a stable sub-infin-category let K be a finite infin-category and let FunA(KB) sube Fun(KB) be the widesub-infin-category spanned by all objects but only the pointwise mod-A equivalences Then∣∣FunA(KB)

∣∣ minus Fun(KBA)

is faithful ie it induces inclusions of path components on Hom anima Furthermore(a) If A sube B is a Verdier inclusion (ie closed under retracts) then the map above is an

equivalence onto core Fun(KBA)(b) If A sube B is dense then |FunA(KB)| is discrete (since BA 0) In fact it is the

discrete group K0 Fun(KB)K0 Fun(KA)

Proof Lorem ipsum dolor sit amet consectetuer adipiscing elitLorem ipsum dolor sit amet consectetuer adipiscing elit Ut purus elit vestibulumUt purus elit vestibulumut placerat ac adispiscing vitae felisut placerat ac adispiscing vitae felis Curabitur dictum gravida mauris Nam arcu liberoCurabitur dictum gravida mauris Nam arcu liberononummy eget consectetuer id vulputate a magnanonummy eget consectetuer id vulputate a magna Donec vehicula augue eu neque Pel-Donec vehicula augue eu neque Pel-lentesque habitant morbi tristiquelentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestassenectus et netus et malesuada fames ac turpis egestasMauris ut leo Cras viverra metus rhoncus semMauris ut leo Cras viverra metus rhoncus sem Nulla et lectus vestibulum urna fringillaNulla et lectus vestibulum urna fringillaultrices Phasellus eu tellus sit amet tortor gravida placeratultrices Phasellus eu tellus sit amet tortor gravida placerat Integer sapien est iaculis inInteger sapien est iaculis inpretium quis viverra ac nuncpretium quis viverra ac nunc Praesent eget sem vel leo ultrices bibendum Aenean fauci-Praesent eget sem vel leo ultrices bibendum Aenean fauci-bus Morbi dolor nulla malesuada eu pulvinar at mollis ac nullabus Morbi dolor nulla malesuada eu pulvinar at mollis ac nulla Curabitur auctor sem-Curabitur auctor sem-per nulla Donec varius orci eget risusper nulla Donec varius orci eget risus Duis nibh mi congue eu accumsan eleifend saggitisDuis nibh mi congue eu accumsan eleifend saggitisquis diam Duis eget orci sit amet orci dignissim rutrumquis diam Duis eget orci sit amet orci dignissim rutrum

This proof is part of Andrearsquos Masterrsquos ThesisPlease select a purchase option

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Learn about K-Theory PREMIUM Already have an account Sign in

IV29 Corollary mdash The K-Theory functor k Catstinfin An is Verdier-localising The

same is true for its spectra-valued variant K Binfink Catstinfin Sp

Proof sketch Let i A B be a Verdier inclusion Consider the simplicial stableinfin-category

Fun(minusB) ∆∆op minus Catstinfin

sending [n] 7 Fun([n]B) We claim that there is a degreewise fully faithful inclusion

Q(i) sube Fun(minusB)esd

(where (minus)esd denotes the edgewise subdivision see IV7b) with degreewise essential imagethose functors in Fun([n]B) that land in the mod-A equivalences To see this recall thatthe elements of Qn(i) can be pictured as diagrams


b0 a1 a2nminus3 a2nminus1

b1

b2nminus1

b2n

where a0 a2n isin A b0 isin B and the b1 bn as well as the rest of the diagram areuniquely determined by the pullback conditions To such a diagram we can associate thesequence

fib(b0 a0) minus minus fib(b2n a2n) minus b2n minus minus b0

234


which evidently is an element of Fun([2n+ 1]B) (Fun(minusB)esd)n and one easily checksthat all maps are actually mod-A equivalences Conversely the diagram can be uniquelyreconstructed from the sequence To make this precise one should argue similar to Proposi-tion IV8 but wersquoll leave the details to you

We conclude that QnQm(i) consists of functors Jntimes [2m+1] B that take all morphisms(j k) (j `) to mod-A equivalences Thus QnQm(i) Fun([2m + 1]FunA(JnB)) andtherefore

coreQnQ(i) (

Nr FunA(JnB))esd

Taking realisations and using Theorem IV28(a) we obtain

|coreQnQ(i)| ∣∣FunA(JnB)

∣∣ core Fun(JnBA)

Combining this with Lemma IV20 shows∣∣Bk(Q(i))∣∣ ∣∣coreQ

(Q(i)

)∣∣ |coreQ(BA)| Bk(BA)

(the second realisation is actually a colimit over (∆∆op)2 similar to Remark IV23(b)) ByCorollary IV26 this shows that Bk Catst

infin CGrp(An) and Binfin+1k Catstinfin Sp are

Verdier-localising Hence so are k ΩBk and K Binfink (Binfin+1k)[minus1]

IV30 Corollary mdash If A sube B is a dense stable sub-infin-category then Ki(A) simminus Ki(B)is an isomorphism for all i gt 1

Proof Throughout the proof let i A B denote the inclusion and let A = K0(B)K0(A)Recall from the proof of Corollary IV26 that there is a fibre sequence

Bk(A) minus Bk(B) minus∣∣Bk(Q(i)

)∣∣ Since B shifts homotopy groups up the statement of the corollary is equivalent to |Bk(Q(i))|being the EilenbergndashMacLane anima K(A 1) As in the proof of Corollary IV29 but usingTheorem IV28(b) instead we get


∣∣ K0 Fun(JnB)K0 Fun(JnA)

Now Lemma IV20 implies K0 Fun(JnB)K0 Fun(JnA) = A2n+1 and thus∣∣Bk(Q(i))∣∣ ∣∣coreQ

(Q(i)

)∣∣ colim[n]isin∆∆op

A2n+1

Be aware that the colimit on the right-hand side is taken in An regarding the abelian groupsA2n+1 as discrete anima and thus the simplicial abelian group A2bull+1 ∆∆op Ab as adiscrete simplicial anima Letrsquos analyse the face maps in A2bull+1 Elements of Fun(JnB) aregiven by zigzags

b0 b2 b4 b2nminus2 b2n

b1 b3 b2nminus1

for some b0 b2n isin B The outer face maps d0 dn Fun(JnB) Fun(Jnminus1B) simplychop of b0 b1 or b2nminus1 b2n respectively The inner face maps dj Fun(JnB) Fun(Jnminus1B)for 1 6 j lt n remove b2jminus1 b2j b2j+1 and replace them by the pullback b2jminus1 timesb2j b2j+1 in

235


B Once we note that [b2jminus1 timesb2j b2j+1] = [b2jminus1] + [b2j+1]minus [b2j ] in K0(B) it follows thatthe face maps dj A2n+1 A2nminus1 in A2bull+1 are given thusly

dj(a0 a2n) =

(a2 a2n) if j = 0(a0 a2jminus2 a2jminus1 + a2j+1 minus a2j a2j+2 a2n) if 1 6 j lt n

(a0 a2nminus2) if j = n

The simplicial abelian group A2bull+1 is a Kan complex by Example I6(c) Hence Remark I66implies that colim[n]isin∆∆op A2n+1 is just given by A2bull+1 itself considered as an anima ByTheorem I16 the homotopy groups of A2bull+1 are the homology groups of its normalisedchain complex Nlowast(A2bull+1) where Nn(A2bull+1) =

⋂16j6n ker(dj) sube A2n+1 and the differentials

are given by d0 One easily checks N0(A2bull+1) = A N0(A2bull+1) = A2 and Nn(A2bull+1) = 0 forn gt 2 which upon inspection of the differentials shows that A2bull+1 is indeed the EilenbergndashMacLane anima K(A 1)

Localisation and DeacutevissageIV31 mdash Two final things are left to do(a) If R S is a derived localisation of rings (we will soon see how to generalise this to

an arbitrary localisation of Einfin-ring spectra) we wish to show that under a soon to bespecified mild technical condition (R S having perfectly generated fibre) there is afibre sequence

k(Dperf(R)S-tors) minus k

(Dperf(R)

)minus k

(Dperf(S)

)

This is not yet trivial since Corollary IV29 isnrsquot applicable right away Even thoughD(R)S-tors D(R) D(S) is a Verdier sequence by Lemma IV16a and Theorem IV15the same need not be true anymore after restricting to the full sub-infin-categories ofperfect complexes everywhere For example S otimesR minus Dperf(R) Dperf(S) may fail tobe essentially surjective This can already be the case for S otimesR minus Proj(R) Proj(S)However its essential image is always a dense subcategory since every finite free S-module is hit and every finite projective S-module is a retract of a finite free one Thisobservation leads us on the right track as wersquoll see in a moment

(b) If R and S are nice enough we wish to show that the K-theory of Dperf(R)S-tors canbe computed in terms of the K-theory of suitable quotients of R For example wersquodlike to have oplus

p primeK(Fp) K

(Dperf(Z)Q-tors)

More generally a similar formula should hold for arbitrary Dedekind domains Thiswill be the content of Quillenrsquos deacutevissage

Before we can tackle (a) we must take two brief detours One into Karoubi-land and oneinto the land of inductive completions

IV32 Karoubi Things mdash A Karoubi equivalence is an inclusion of a dense sub-infin-category We put

Catstinfin = Catst

infin[Karoubi equivminus1]

236


A Karoubi sequence is a sequence in Catstinfin that becomes both a fibre and a cofibre sequence

in Catstinfin Moreover we define Karoubi inclusions and Karoubi projections in the obvious

wayWe didnrsquot have time to discuss the following lemma in the lecture but it appears in

Fabianrsquos notes [AampHK LemmaDefinition IV34] and should be mentioned here

IV32a Lemma mdash The localisation Catstinfin Catst

infin is both a left and a right Bousfieldlocalisation A left adjoint is given by C 7 Cmin where Cmin denotes the minimalisation fromExample IV4b and a right adjoint is given by C 7 C where C denotes the idempotentcompletion of C

Proof Let CatstK0=0infin sube Catst

infin be the full sub-infin-category spanned by those stable infin-categories C that have no dense stable sub-infin-category or equivalently (by Theorem IV4)those C with K0(C) = 0 We claim that C 7 Cmin is a right adjoint of this inclusion (andyes we really want a right adjoint here wersquoll explain below where the leftright switchcomes from) First note that (minus)min Catst

infin Catstinfin is indeed a functor and comes with

a pointwise fully faithful transformation (minus)min rArr id Just consider a suitable simplicialsubset of Uncocart(Catst

infin Catinfin) Now Theorem IV4 implies that (minus)min rArr id satisfiesthe (dual) conditions of Proposition I61a whence CatstK0=0

infin is indeed a right Bousfieldlocalisation of Catst

infinItrsquos clear that (minus)min inverts dense inclusions Hence it descends to a functor

(minus)min Catstinfin

simminus CatstK0=0infin

which we claim is an equivalence To see this simply note that CatstK0=0infin sube Catst

infin Catstinfin

is an inverse Together with the adjunction that was verified above this now implies that(minus)min Catst

infinsimminus CatstK0=0

infin sube Catstinfin is indeed a left adjoint of Catst

infin Catstinfin as

claimed Moreover we see where the leftright switch comes from Whether C 7 Cmin isa left or right adjoint depends on which side of the adjunction contains the equivalence(minus)min Catst


infin The case of idempotent completions is similar Let Catst

infin sube Catstinfin be spanned by the

idempotent complete stable infin-categories (see [HTT Corollary 44515]) Then this inclusionhas a left adjoint (minus) This really doesnrsquot have anything to do with stable infin-categoriesas already Catinfin sube Catinfin has a left adjoint (minus) (see [HTT Proposition 54216]) and weonly need to check that C is stable again if C is (and that exactness of functors is preservedbut that will also fall out from our argument) Lurie constructs C sube P(C) as the retractclosure of C sube P(C) Since HomC(minus c) Cop An is left-exact for all c isin C one easilychecks that the retract closure is contained in Funlex(CopAn) Funex(CopSp) where theequivalence in follows from Theorem II30 Now Funex(CopSp) is stable (in fact it is a stablesub-infin-category of Fun(CopSp)) hence stability of C follows from Lemma IV32b below

The functor (minus) Catstinfin Catst

infin takes dense inclusions to equivalences This isactually not entirely clear from the construction so we refer to [HTT Definition 5141and Proposition 5149] Once this is known it follows as above that (minus) descends to anequivalence

(minus) Catstinfin

simminus Catstinfin

and thus (minus) Catstinfin

simminus Catstinfin sube Catst

infin is a right adjoint to Catstinfin Catst

infin (and we getthe same leftright switch as above)

237


IV32b Lemma mdash Let C sube D be a stable sub-infin-category (in the sense of Re-mark IV4a) and let Cprime sube D be the retract closure of C Then Cprime is stable again

Proof It suffices to show that Cprime is closed under shifts finite direct sums and fibres Thefirst two are trivial so let α cprime dprime be a morphism in Cprime By definition cprime and dprime are retractsof objects c d isin C But all retracts in a stable infin-category are split (by a similar argumentas in the proof of Lemma IV10a(c)) hence there are cprimeprime and dprimeprime such that c cprime oplus cprimeprimeand dprime doplus dprimeprime Consider the map α oplus 0 c d Then fib(α oplus 0) is an object of C againHowever fib(αoplus 0) fib(α)oplus cprimeprime oplus dprimeprime[minus1] which proves that fib(α) is again a retract of anobject in C

The upshot is that there are two ways to embed Catstinfin into Catst

infin As the full sub-infin-category where K0 vanishes and as the full sub-infin-category of idempotent complete stableinfin-categories Moreover Lemma IV32a implies that a functor f A B in Catst

infin becomesan equivalence under Catst

infin Catstinfin iff it is a Karoubi equivalence ie a dense inclusion

The ldquoifrdquo is trivial so letrsquos suppose the converse Then

f A simminus B and fmin Amin simminus Bmin

are equivalences Since A sube A and B sube B are fully faithful f being an equivalence showsthat f is fully faithful And fmin being an equivalence shows that f is dense

Moreover Lemma IV32a allows us to characterise Karoubi sequences Wersquoll see an evencooler characterisation in Theorem IV34 below

IV32c Lemma mdash For a nullcomposite sequence A B C in Catstinfin the following

are equivalent(a) It is a Karoubi sequence(b) The induced maps A fib(B C) and BA C are Karoubi equivalences(c) The induced maps A simminus fib(B C) and BminAmin simminus Cmin are equivalences

Proof Since Catstinfin Catst

infin has both adjoints by Lemma IV32a it preserves fibres andcofibres Hence (a) hArr (b)

Moreover wersquove seen in the proof above that (minus) Catstinfin

simminus Catstinfin is an equivalence

and that Catstinfin sube Catst

infin is a right adjoint Hence fib(B C) is also the fibre in Catstinfin

Thus A fib(B C) and BA C is a Karoubi equivalence iff A simminus fib(B C) is anactual equivalence Together with a similar argument for (minus)min this implies (b) hArr (c)

IV33 Inductive Completions mdash Let C be an infin-category with finite colimits Thenwe define its inductive completion to be the full sub-infin-category Ind(C) sube P(C) of functorsF Cop An that preserve finite limits Note that if C is stable then

Ind(C) Funlex(CopAn) Funex(CopSp)

by Theorem II30 This implies that Ind(C) is stable againOne can also define Ind(C) if C doesnrsquot necessarily have finite colimits but then the

correct definition is a bit different see [HTT Definition 5351] In any case Ind(C) is theldquofree wayrdquo of adding filtered colimits to C as the following lemma shows

238


IV33a Lemma mdash If D has filtered colimits then restriction along the Yoneda embeddingY Cω C Ind(C) induces an equivalence

Y Clowastω Funfilt-colim ( Ind(C)D) simminus Fun(CD)

where Funfilt-colim sube Fun is the full sub-infin-category of filtered colimits preserving functors

Proof Wersquoll only do the case where C has finite colimits since that was our assumptionabove (but the general case isnrsquot really more difficult) First note that since filtered colimitscommute with finite limits in An Ind(C) sube P(C) is closed under filtered colimits Nextobserve that every functor f C D admits a left Kan extension LanY Cω f Ind(C) D Tosee this it suffices to show that D admits Y Cω F -shaped colimits for every F isin Ind(C) sincethen the formulas from Theorem I52 provide the desired Kan extension So we must provethat Y Cω F is filtered

Let K be a finite simplicial set and let K Y Cω F be a diagram which we may think ofas objects ck isin C together with maps ck F in Ind(C) By Yonedarsquos lemma such mapscorrepond to points in F (ck) Since C admits finite colimits c = colimkisinK ck exists andsatisfies F (c) limkisinKop F (ck) since c is given by a finite limit in Cop and F preserves finitelimits Hence the given maps ck F define a contractably unique map c F and thusan extension K Y Cω F This shows that Y Cω F is filtered and thus there exists a Kanextension functor LanY Cω Fun(CD) Fun(Ind(C)D)

Next observe that any such Kan extension LanY Cω f Ind(C) D preserves filteredcolimits To see this choose a colimit-preserving embedding D sube Dprime into a cocompleteinfin-category Dprime (for example one could take Dprime Fun(DopAnop)) and consider the left Kanextension LanY C f P(C) Dprime along the usual Yoneda embedding Y C C P(C) LanY C fpreserves colimits by Theorem I51 and its restriction to Ind(C) is LanY Cω f by Corollary I54Since Ind(C) sube P(C) preserves filtered colimits this implies that LanY Cω f Ind(C) Dpreserves filtered colimits as well

The upshot is that Y Clowastω Funfilt-colim(Ind(C)D) Fun(CD) has a left adjoint LanY Cω One easily checks that unit and counit are equivalences using that the same is true byTheorem I51 for the adjunction LanY C Fun(CD) Funcolim(P(C)D) Y Clowast

IV33b Compact and Tiny Objects mdash Let D be an infin-category Recall that anobject d isin D is called compact if HomD(dminus) commutes with filtered colimits and tiny ifHomD(dminus) commutes with all colimits Moreover we say that a set S of objects of D is aset of generators if the functors HomD(sminus) for s isin S are jointly conservative

If D P(C) for some small infin-category C then the essential image of the Yonedaembedding Y C C P(C) is a set of tiny generators as follows easily from Yonedarsquos lemmaIn fact a cocomplete D is of the form D P(C) if and only if it has a set of tiny generatorsin which case D P(Dtiny) where Dtiny sube D is the full sub-infin-category spanned by the tinyobjects To see this we apply Criterion () from the proof of Lemma II38 to i Dtiny Dto see that the colimit-preserving functor P(Dtiny) D induced by Theorem I51 is anequivalence Conditions (a) and (b) are trivially satisfied To check condition (c) ie thatSingi D P(Dtiny) is conservative recall that Singi(d) HomD(i(minus) d) (Dtiny)op AnSo conservativity of Singi follows immediately from the fact that D has a set of tiny generators

The exact same thing holds for inductive completions and compact objects

IV33c Lemma mdash Let D be an infin-category with finite colimits Then D is of the formD Ind(C) for some small infin-category C iff D admits filtered colimits and has a set of

239


compact generators In this case automatically D Ind(Dω) where Dω sube D is the fullsub-infin-category of compact objects

Proof The proof is completely analogous to Criterion () in the proof of Lemma II38except for two things that should perhaps be explained First Dω sube D is closed under finitecolimits since filtered colimits commute with finite limits in An So in particular Dω hasfinite colimits whence we can define Ind(Dω) as above and the inclusion iDω sube D extendsto a functor Ind(Dω) D by Lemma IV33a

Second we have to explain why this functor has a left adjoint Choose a colimit-preservingembedding D sube Dprime into a cocomplete infin-category Then i Dω Dprime induces a colimit-preserving functor | |i P(Dω) Dprime with right adjoint Singi Dprime P(Dω) On objectsd isin Dprime Singi is given by Singi(d) HomDprime(i(minus) d) (Dω)op An For d isin D this presheafcoincides with HomD(minus d) (Dω)op An which preserves finite limits Hence Singi restrictsto a right adjoint D Ind(Dω) of Ind(Dω) D

An infin-category D as in Lemma IV33c is called compactly generated For example wersquovechecked in the proof of Theorem II57 that ModR is compactly generated for any Einfin-ringspectrum R

IV34 Theorem (ThomasonndashNeeman localisation theorem) mdash A nullcomposite sequenceA B C in Catst

infin is a Karoubi sequence iff

Ind(A) minus Ind(B) minus Ind(C)

is a Verdier sequence

Proof Throughout the proof wersquoll use that Ind(D) Funex(DopSp) holds for all stableinfin-categories D Under this identification the functor Ind(D) Ind(E) induced by an exactfunctor i D E corresponds to Lani Funex(DopSp) Funex(EopSp) as follows fromthe proof of Lemma IV33a We proceed in three steps(1) We show that a map D E in Catst

infin is a Karoubi equivalence iff Ind(D) simminus Ind(E) isan equivalence

Letrsquos first assume that i D E is a Karoubi equivalence As seen above we have toshow that Lani Funex(DopSp) Funex(EopSp) is an equivalence For this it suffices thatits right adjoint

ilowast Funex(EopSp) simminus Fun(DopSp)is an equivalence Equivalences of infin-categories can be detected on core(minus) and core Ar(minus)To check that ilowast core Funex(EopSp) simminus core Funex(DopSp) is an equivalence we useLemma IV32a to get

HomCatstinfin

(DopSp) HomCatstinfin


(DopSp)

Here we also used Sp Sp since Sp is already idempotent complete (it is even cocomplete)Similarly

HomCatstinfin

(EopSp) HomCatstinfin

(EopSp)

But now the left-hand sides agree since D sube E is a Karoubi equivalence hence also theright-hand sides agree Thus

HomCatstinfin


(EopSp)

240


as desired To show that ilowast core Ar(Funex(EopSp)) simminus core Ar(Funex(DopSp)) is anequivalence simply repeat the argument with Ar(Sp) instead of Sp

Conversely assume that Ind(D) simminus Ind(E) is an equivalence Since D sube Ind(D) andE sube Ind(E) are fully faithful it follows that D E is fully faithful as well It remainsto show that all objects of E are retracts But every e isin E can be written as a filteredcolimit e colimiisinI di over elements from D Since e is compact in Ind(E) the isomorphisme simminus colimiisinI di already factors over some dj colimiisinI di But then

e minus dj minus colimiisinI

disimminus e

witnesses e as a retract of dj as required This finishes Step (1)(2) We show that if A B C is a Karoubi sequence then Ind(A) Ind(B) Ind(C) is

a Verdier sequenceAs seen above both arrows in Ind(A) Ind(B) Ind(C) have right adjoints given by

the usual restriction functors

Funex(CopSp) minus Funex(BopSp) minus Funex(AopSp)

By Theorem IV15 it suffices to show that this is a Verdier sequence Note that A B isfully faithful (for example by Lemma IV32c(b)) hence so is the left Kan extension functorFunex(AopSp) Funex(BopSp) Therefore Funex(BopSp) Funex(AopSp) is a left splitVerdier projection by Theorem IV15(b) and thus all thatrsquos left to do is to identify its fibrewith Funex(CopSp)

By Lemma IV32c(b) and Step (1) we have Funex(CopSp) Funex((BA)opSp) so wemay show instead that

Funex((BA)opSp) minus Funex(BopSp) minus Funex(AopSp)

is a fibre sequence in Catstinfin (or equivalently in Catinfin see IV14(e)) But this is obvious

Funex((BA)opSp) Funex(BopSp) is fully faithful with essential image those exactfunctors that invert mod-Aop equivalences And an exact functor Bop Sp inverts mod-Aop

equivalences iff the composition Aop Bop Sp vanishes(3) We show that if Ind(A) Ind(B) Ind(C) is a Verdier sequence then A B C is

a Karoubi sequenceLet Aprime = fib(B C) and Cprime = BAprime Then clearly Aprime B is fully faithful and closed

under retracts hence Theorem IV15 shows that Aprime B Cprime is a Verdier sequence Inparticular Step (2) implies that Ind(Aprime) Ind(B) Ind(Cprime) is a Verdier sequence tooWersquore done if we can show that A Aprime and Cprime C are Karoubi equivalences Indeed theformer implies that BA BAprime thus also BA C is a Karoubi equivalence whence bothconditions from Lemma IV32c(b) are satisfied

By construction we get fully faithful inclusions Ind(A) sube Ind(Aprime) sube Ind(B) But Ind(A)is also the fibre of Ind(B) Ind(C) so Aprime sube Ind(A) Since Ind(A) is already closed underfiltered colimits this shows that Ind(A) simminus Ind(Aprime) must be an equivalence Hence A Aprimeis a Karoubi equivalence by Step (1) Now both

Ind(A) minus Ind(B) minus Ind(C) and Ind(Aprime) minus Ind(B) minus Ind(Cprime)

are Verdier sequences and we already know Ind(A) Ind(Aprime) so Ind(Cprime) Ind(C) holds aswell By Step (1) again this implies that Cprime C is a Karoubi equivalence Wersquore done

241


IV35 Quillenrsquos Localisation Sequence mdash If R is an Einfin-ring spectrum we put

k(R) = k(ModωR) and K(R) = K(ModωR)

for short In the case where R is an ordinary ring this is compatible with our previousdefinitions since one easily checks that Dperf(R) consists of precisely the compact objects inD(R) ModHR

Let R S be a localisation of Einfin-ring spectra (see IV16) We say that R S hasperfectly generated fibre if I = fib(R S) lies in the smallest stable sub-infin-category ofModS-tors

R generated by ModS-torsωR = ModS-tors

R capModωR under colimits This is always trueif S R[sminus1] is a localisation at some element s isin π0(R) since then

I (Rsinfin)[minus1] colimnisinN

(Rsn)[minus1]

and every Rsn cofib(sn R R) is an element of ModS-torsωR Essentially the same

argument shows that localisations R R[Sminus1] at arbitrary subsets S sube π0(R) have perfectlygenerated fibre Moreover it is a theorem of ThomasonndashTrobaugh that R S has perfectlygenerated fibre whenever R and S are ordinary rings such that SpecS SpecR is an openembedding of affine schemes (which is always a derived localisation see Example I63a(c))

The following lemma explains why having perfectly generated fibre is a very naturaltechnical condition to impose

IV35a Lemma mdash The smallest stable sub-infin-category of ModS-torsR generated by

ModS-torsωR is equivalent to Ind(ModS-torsω

R ) Moreover the following are equivalent(a) R S has perfectly generated fibre(b) The canonical functor Ind(ModS-torsω

R ) simminus ModS-torsR is an equivalence

(c) ModS-torsωR ModωR ModωS is a Karoubi sequence

Proof We know that ModR Ind(ModωR) from Lemma IV33c Hence the canonicalfunctor

Ind(ModS-torsωR ) minus Ind(ModωR) ModR

induced by Lemma IV33a is fully faithful and its essential image is contained in ModS-torsR

since being S-torsion is preserved under arbitrary colimits Moreover the essential imageis closed under filtered colimits by construction and closed under finite colimits since it isa stable sub-infin-category But every colimit can be built from coproducts (which can berewritten as filtered colimits) and coequalisers (which are finite) hence the essential image isclosed under all colimits Finally Ind(ModS-torsω

R ) is generated by ModS-torsωR under colimits

(even under filtered ones) hence it is indeed the smallest sub-infin-category wersquore looking forThe implication (b)rArr (a) is clear since the fibre I fib(R S) is contained in ModS-tors

R Conversely assume I can be written as a colimit I colimiisinI Ci with Ci isin ModS-torsω

R By the above we may even assume that I is filtered Let M be any S-torsion R-modulespectrum Since ModR is compactly generated M can be written as a filtered colimitM colimjisinJ Dj with Dj isin ModωR But then M M otimesR I since M is S-torsion whichimplies

M colimjisinJ

Dj otimesR I colim(ij)isinItimesJ

Ci otimesR Dj

using that minusotimesR I commutes with colimits along with Proposition I42 Itrsquos straightforwardto check that the Ci otimesR Dj are compact again and S-torsion whence M isin Ind(ModS-torsω

R )

242


This shows (a) rArr (b) Finally (b) hArr (c) follows from Theorem IV34 Lemma IV16a andthe fact that ModR Ind(ModωR)

Combining everything we did on the last few pages we finally obtain Quillenrsquos fibresequence

IV35b Corollary mdash If R S is a localisation of Einfin-ring spectra with perfectlygenerated fibre then therersquos a fibre sequence

k(ModS-torsωR ) minus k(R) minus k(S)

in An (or in CGrp(An)) In particular if R S is a derived localisation of ordinary ringswith perfectly generated fibre then therersquos a fibre sequence

k(Dperf(R)S-tors) minus k(R) minus k(S)

Fabian also mentioned that the perfectly generated fibre condition was recently removedby Alexander Efimov but his result would leave the scope of this lecture

Proof of Corollary IV35b We know from Lemma IV35a(c) that ModS-torsωR ModωR

ModωS is a Karoubi sequence Itrsquos straightforward to check that ModS-torsωR ModωR is fully

faithful and closed under retracts hence a Verdier inclusion by Theorem IV15(c) Thus if Cdenotes the Verier quotient ModωRModS-torsω

R then Corollary IV29 implies that

k(ModS-torsωR ) minus k(ModωR) minus k(C)

is a fibre sequence in An Moreover C ModωS is a Karoubi equivalence by Lemma IV32c(b)whence k(C) k(ModωS) is an inclusion of path components by Corollary IV30 and Theo-rem IV4 This shows that

k(ModS-torsωR ) minus k(ModωR) minus k(ModωS)

is a fibre sequence in An as well as required

Despite Corollary IV35b itrsquos not necessarily true that the K-theory spectra arrange intoa fibre sequence

K(ModS-torsωR ) minus K(R) minus K(S)

if R S is a localisation with perfectly generated fibre However the only thing that canfail is that K0(R) K0(S) isnrsquot surjective In the case R = Z S = Q and more generallywhenever R is an integral domain and S = FracR its fraction field K0(R) K0(S) is clearlysurjective Hence we do get a fibre sequence

K(Dperf(Z)Q-tors) minus K

(Dperf(Z)

)minus K

(Dperf(Q)

)in Sp Wersquove thus completed thing (a) of the two final things from IV31 and only thing (b)is left to do Again this will lead us on two detours First we have to introduce the classicaldefinition of K-theory of abelianexact categories Itrsquos a bit unsatisfying that we canrsquot stayon the stableinfin-categorical high road especially after we have been avoiding to leave it for solong but as far as Fabian knows a formulation of Quillenrsquos deacutevissage in stable infin-frameworkhas yet to be found

243


IV36 Algebraic K-Theory of AbelianExact Categories mdash Let A be an abeliancategory We define a category Span(A) as follows Its objects the objects of A Its morphismsets HomSpan(A)(a aprime) are equivalence classes of span diagrams

aminus aprimeprime minus aprime

in A where two such diagrams are equivalent iff there is an isomorphism between themwhich restricts to the identity on a and aprime Composition is given by taking pullbacks as wewould expect from a span category The same definition can be given if A is only an exactcategory In a nutshell these are additive categories together with a given supply of exactsequences subject to a bunch of axioms which Irsquom not going to recall In the exact case thearrows in a aprimeprime aprime have to be admissible epi- and monomorphisms respectively Herewe call an epi- or monomorphism admissible means if it is part of one of the specified exactsequences

Two caveats are in order First Span(A) is not the infin-category obtained by applyingPropositionDefinition I71 to A since we only allow specific spans here One should probablysignify that by adding a suitable decoration but I couldnrsquot come up with a good one I hopethat doesnrsquot cause confusion Second what we call Span(A) here is usually denoted Q(A)in the literature since thatrsquos how Quillenrsquos Q-construction was originally defined Howevercalling it Span(A) is more consistent with the notation in the rest of these notes

In any case we define the algebraic K-theory anima of A and its spectrum variant as

k(A) = Ω |Span(A)| and K(A) = Binfink(A)

where |Span(A)| denotes the infin-categorical localisation of the 1-category Span(A) at all itsmorphisms

IV37 t-Structures mdash Let C be a stable infin-category A t-structure on C consists of twofull sub-infin-categories Cgt0 C60 sube C subject to the following conditions(a) Cgt0 is closed under pushouts C60 is closed under pullbacks and both under retracts(b) If x isin Cgt0 and y isin C60 the spectrum homC(x y) is coconnective ie its homotopy

groups vanish in positive degrees(c) Every y isin C admits a fibre sequence x y z with x isin Cgt1 and z isin C60 Here we

write Cgta = Cgt0[a] and similarly C6b = C60[b]As for weight structures (see IV10) we put C[ab] = Cgta cap C6b and Chearts = C[00] Cgt0 cap C60A t-structure on C is called exhaustive (which is nonstandard terminology Lurie uses boundedinstead) if


As already mentioned in IV10 t-structures behave a lot differently than weight structuresThe inclusions Cgtn sube C have right adjoints τgtn Cgtn C and the inclusions C6n sube C haveleft adjoints τ6n C6n C With this terminology it turns out that the fibre sequence from(c) is necessarily of the form τgt1y y τ60y hence it is unique and functorial Finallyand most importantly for our purposes the heart Chearts is always an abelian 1-category Proofsof all of this can be found in [HA Subsection 121]

The standard example of a stable infin-category with a t-structure is D(R) for some ring Rtogether with the full sub-infin-categories Dgt0(R) and D60(R) of complexes whose homology

244


vanishes in negative or positive degrees respectively Note however that this usually doesnrsquotrestrict to a t-structure on Dperf(R) since truncations of a perfect complex need not beperfect anymore Wersquoll see in IV39 below how this can be fixed But first letrsquos see howK-theory of abelian categories relates to t-structures

IV38 Theorem mdash Let C be a stable infin-category equipped with an exhaustive t-structure(a) There is a canonical equivalence k(Chearts) simminus k(C) where the left-hand side is defined as

in IV36 using that Chearts is abelian(b) Suppose D is another stableinfin-category with an exhaustive t-structure and let F C D

be an exact and t-exact functor (ie F (Cgt0) sube Dgt0 and F (C60) sube D60) Supposefurthermore that the induced functor F Chearts Dhearts is fully faithful its essential imageis closed under subobjects and quotients and that every object in the abelian categoryb isin Dhearts has a finite filtration

0 = b0 sube b1 sube middot middot middot sube bn = b

whose subquotients bi+1bi are in the essential image of F Then F induces an equiva-lence

F k(C) simminus k(D)

Proof sketch of Theorem IV38 Part (a) is Clark Barwickrsquos theorem of the heart and werefer to his original proof in [Bar15] Assuming (a) we can give a surprisingly simple proofof (b) which is taken from Charles Weibelrsquos K-book [Wei13 Chapter V sect4] Using (a) itsuffices to show k(Chearts) k(Dhearts) and for this it suffices to show that the nerves of Span(A)and Span(B) are weakly equivalent as simplicial sets where we write A = Chearts and B = Dheartssince the hearts are getting on my nerves

Wersquoll show something stronger i Span(A) Span(B) is final By the dual of Theo-rem I432 we must show that the slice categories ib = Span(A) timesSpan(B) Span(B)b areweakly contractible for all b isin B Since A B is fully faithful by assumption the sameis true for Span(A) Span(B) by construction and the fact that A sube B is closed undersubobjects Hence the slice categories ia for a isin A are weakly contractible since they havea terminal object Now a general object b isin B has a finite filtration with subquotients in Ahence it suffices to show that ibprime ib is a weak equivalence for all subobjects bprime sube b whosequotient bbprime lies in A

By definition the objects of ib are given by pairs (a ϕ a b) where a isin A and ϕ is amorphism in Q(B) given by a span a c b in B Letting cprime = ker(c a) such a spanmay be equivalently described as a chain of monomorphisms cprime c b such that a = ccprime

is contained in A Now let J sube ib be the full subcategory of objects (cprime c b) wherecprime sube bprime holds as subobjects of b Then ibprime ib can be factored into a solid sequence

ibprime J ib r

s

If we can show that the dashed adjoints exist then wersquore done since adjoint functors induceinverse homotopy equivalences Define r and s explicitly as

r(cprime c b) = (cprime c cap bprime bprime) and s(cprime c b) = (cprime cap bprime c b) 2I somewhat suspect that this was in fact the original purpose of Quillenrsquos theorem A

245


Note that (c cap b)cprime is contained in A since it is a subobject of ccprime = a Similarly c(cprime cap bprime)is a subobject of bbprime and thus also contained in A So r and s are well-defined One easilychecks that they indeed provide the desired adjoints

IV39 Almost Perfect Complexes and G-Theory mdash As explained above the stan-dard t-structure on D(R) doesnrsquot restrict to a t-structure on Dperf(R) at least not withoutsome extra conditions on R Instead we say a complex D isin D(R) is almost perfect (orpseudocoherent in Stacks Project language) if it satisfies the following equivalent conditions(a) For all n isin Z the truncation τ6nD coincides with τ6nCn for some perfect complex

Cn isin Dperf(R)(b) D is quasi-isomorphic to a (not necessarily bounded above) complex of degreewise finite

projective R-modulesEquivalence is proved in [Stacks Tag 064U] (beware that The Stacks Project uses cohomolog-ical indexing) Upon replacing ldquoperfectrdquo by ldquocompactrdquo condition (a) can also be generalisedin a straightforward way to modules over an Einfin-ring spectrum

Itrsquos clear from condition (a) that the full sub-infin-category of almost perfect complexesis stable again and inherits a t-structure Moreover if we further restrict to the full sub-infin-category Daperf(R) sube D(R) of bounded almost perfect complexes then that t-structurebecomes exhaustive We define the G-theory spectrum of R to be

G(R) = K(Daperf(R)

)

If R is noetherian then a complex D isin D(R) is almost perfect iff its homology Hlowast(D) is afinitely generated R-module in each degree see [Stacks Tag 066E] In particular the heartof the t-structure on Daperf(R) is the abelian category of finite R-modules If R is moreoverregular and of finite Krull dimension then it has finite global dimension which means thatevery finite R-module has a ldquofinite finiterdquo projective resolution (ie a resolution of finitelength consisting of finite projective R-modules) This easily implies that every boundedalmost perfect complex is in fact perfect so

G(R) K(R)

holds for all regular noetherian rings of finite Krull dimension With this out of the way wecan finally prove Quillenrsquos fibre sequence from IV1

IV40 Corollary mdash Let R be a Dedekind domain Thenoplus

mDperf(Rm) Dperf(R)torswhere m runs over all maximalnonzero prime ideals of R induces an equivalenceoplus

m

K(Rm) simminus K(Dperf(R)tors)

In particular there is a fibre sequenceoplusm


Proof Since R is regular noetherian therersquos a natural t-structure on Dperf(R) whichpersists to Dperf(R)tors The functors Dperf(Rm) Dperf(R)tors are clearly t-exact andinduce fully faithful functors on hearts namely the inclusion

Modfgm-torsR sube Modfgtors

R

246


of finitely generated m-torsion R-modules into all finitely generated torsion R-modules Letme remark however that Dperf(Rm) Dperf(R)tors itself isnrsquot fully faithful since Ext1

Rm

vanishes (as Rm is a field) but Ext1R doesnrsquot This is the reason why we had to introduce

all this stuff about t-structures and K-theory of abelian categoriesNow

oplusmDperf(Rm) inherits a t-structure and by the Chinese remainder theorem its

heart can be described as the full subcategory A sube ModfgtorsR of finitely generated torsion

modules whose annihilator AnnR(M) sube R is a squarefree ideal ie the factors mi in theprime ideal decomposition AnnR(M) = m1 middot middot middotmn are distinct Note that every finitelygenerated torsion R-module has a finite filtration with subquotients in A In fact everyfinitely generated module over an arbitrary noetherian ring has a finite filtration withsubquotients of the form Rpi for some pi isin SpecR see [Stacks Tag 00L0] Thus thefunctor

oplusmDperf(Rm) Dperf(R)tors satisfies the assumptions from Theorem IV38(b)

whence we get an equivalence

K

(oplusm

Dperf(Rm))simminus K

(Dperf(R)tors)

It remains to identify the left-hand side withoplus

mK(Dperf(Rm)) oplus

mK(Rm) which is astraightforward check The additional assertion about the fibre sequence follows immediatelyfrom Corollary IV35b

IV41 Nonconnective K-Theory mdash As introduced in Definition IV13 let FunKar

denote infin-categories of Karoubi-localising functors It turns out that

Ωinfin FunKar(CatstinfinSp) simminus FunKar(Catst

infinAn)

is an equivalence Now C 7 k(C) is a Karoubi-localising functor hence an element onthe right We let K Catst

infin Sp denote a preimage and call this non-connective K-theoryFabian warns you that in general the preimage of a Karoubi-localising functor F Catst

infin Anmay not be given by BinfinF Catst

infin Sp as this guy might not be Karoubi-localising at all(but of course this only fails on π0) In particular if C is a stable infin-category then K(C)usually doesnrsquot coincide with Binfink(C) and it is usually not a connective spectrummdashhencethe name

This is where Fabianrsquos final lecture ends But needless to say that Fabian has at least adozen pages worth of additional cool stuff to say in his official notes

247

Bibliography[AampHK] Fabian Hebestreit Algebraic and Hermitian K-Theory Lecture held at the

University of Bonn Winter term 202021 url httpswwwmathuni-bonndepeoplefhebestrKtheory

[AS69] Michael F Atiyah and Graeme B Segal ldquoEquivariant K-theory and comple-tionrdquo In Journal of Differential Geometry 3 (1969) pp 1ndash18 url httpprojecteuclidorgeuclidjdg1214428815

[Ati61] Michael F Atiyah ldquoCharacters and cohomology of finite groupsrdquo In Institutdes Hautes Eacutetudes Scientifiques Publications Matheacutematiques 9 (1961) pp 23ndash64 url httpwwwnumdamorgitemid=PMIHES_1961__9__23_0

[Bar15] Clark Barwick ldquoOn exact infin-categories and the theorem of the heartrdquo InCompositio Mathematica 15111 (2015) pp 2160ndash2186 url httpsdoiorg101112S0010437X15007447

[BGT13] Andrew J Blumberg David Gepner and Gonccedilalo Tabuada ldquoA universalcharacterization of higher algebraic K-theoryrdquo In Geometry amp Topology 172(Apr 2013) pp 733ndash838 doi 102140gt201317733

[Die08] Tammo tom Dieck Algebraic topology EMS Textbooks in Mathematics Eu-ropean Mathematical Society (EMS) Zuumlrich 2008 pp xii+567 url httpswwwmathsedacuk~v1ranickpapersdiecktoppdf

[Fab+20I] Baptiste Calmegraves et al Hermitian K-theory for stable infin-categories I Founda-tions 2020 arXiv 200907223 [mathKT]

[Fab+20II] Baptiste Calmegraves et al Hermitian K-theory for stableinfin-categories II Cobordismcategories and additivity 2020 arXiv 200907224 [mathKT]

[Fab+20III] Baptiste Calmegraves et al Hermitian K-theory for stable infin-categories III Grothen-dieck-Witt groups of rings 2020 arXiv 200907225 [mathKT]

[FP78] Zbigniew Fiedorowicz and Stewart Priddy Homology of classical groups overfinite fields and their associated infinite loop spaces Vol 674 Lecture Notes inMathematics Springer Berlin 1978 pp vi+434

[Gla16] Saul Glasman Day convolution for infinity-categories 2016 arXiv 13084940[mathCT]

[Gre55] James Alexander Green ldquoThe characters of the finite general linear groupsrdquoIn Transactions of the American Mathematical Society 80 (1955) pp 402ndash447url httpsdoiorg1023071992997

[HA] Jacob Lurie Higher Algebra 2017 url httppeoplemathharvardedu~luriepapersHApdf

[Hat02] Allen Hatcher Algebraic topology Cambridge University Press Cambridge 2002pp xii+544 url httpspimathcornelledu~hatcherATATpdf

248

Bibliography

[Hau20] Rune Haugseng A fibrational mate correspondence for infin-categories 2020arXiv 201108808 [mathCT]

[HCI] Fabian Hebestreit Higher Categories I Lecture held at the University of BonnWinter term 201920 url https wwwmath uni- bonn depeople fhebestrSkript20HigherCatsI

[HCII] Fabian Hebestreit Higher Categories II Lecture held at the University ofBonn Summer term 2020 url httpswwwmathuni-bonndepeoplefhebestrSkript20HigherCatsII

[HLN20] Fabian Hebestreit Sil Linskens and Joost Nuiten Orthofibrations and monoidaladjunctions 2020 arXiv 201111042 [mathCT]

[HTT] Jacob Lurie Higher Topos Theory Vol 170 Annals of Mathematics StudiesPrinceton University Press Princeton NJ 2009 pp xviii+925 url httppeoplemathharvardedu~luriepapersHTTpdf

[JT07] Andreacute Joyal and Myles Tierney ldquoQuasi-categories vs Segal spacesrdquo In Cate-gories in algebra geometry and mathematical physics Vol 431 Contemp MathAmer Math Soc Providence RI 2007 pp 277ndash326 arXiv math0607820[mathAT]

[LT19] Markus Land and Georg Tamme ldquoOn the K-theory of pullbacksrdquo In Annalsof Mathematics 1903 (2019) p 877 doi 104007annals201919034

[Lur09] Jacob Lurie (infin 2)-Categories and the Goodwillie Calculus I 2009 arXiv09050462 [mathCT]

[Lur10] Jacob Lurie Chromatic Homotopy Theory (Math 252x) Lecture held at HarvardUniversity Spring 2010 url httpswwwmathiasedu~lurie252xhtml

[Mil63] John W Milnor Morse theory Based on lecture notes by M Spivak and RWells Annals of Mathematics Studies No 51 Princeton University PressPrinceton NJ 1963 pp vi+153 url httpswwwmathsedacuk~v1ranickpapersmilnmorspdf

[MM65] John W Milnor and John C Moore ldquoOn the structure of Hopf algebrasrdquo InAnnals of Mathematics Second Series 81 (1965) pp 211ndash264 url httpsdoiorg1023071970615

[Nik16] Thomas Nikolaus Stable infin-Operads and the multiplicative Yoneda lemma2016 arXiv 160802901 [mathAT]

[Qui72] Daniel Quillen ldquoOn the cohomology and K-theory of the general linear groupsover a finite fieldrdquo In Annals of Mathematics Second Series 96 (1972) pp 552ndash586 url httpsdoiorg1023071970825

[Ser77] Jean-Pierre Serre Linear representations of finite groups Translated from thesecond French edition by Leonard L Scott Graduate Texts in MathematicsVol 42 Springer-Verlag New York-Heidelberg 1977 pp x+170

[Spa88] Nicolas Spaltenstein ldquoResolutions of unbounded complexesrdquo In CompositioMathematica 652 (1988) pp 121ndash154 url httpwwwnumdamorgitemid=CM_1988__65_2_121_0

249

Bibliography

[Stacks] The Stacks Project Authors The Stacks Project 2020 url httpstacksmathcolumbiaedu

[Tho97] Robert W Thomason ldquoThe classification of triangulated subcategoriesrdquo InCompositio Mathematica 1051 (1997) pp 1ndash27 url httpsdoiorg101023A1017932514274

[Wei13] Charles A Weibel The K-book Vol 145 Graduate Studies in Mathemat-ics An introduction to algebraic K-theory American Mathematical SocietyProvidence RI 2013 pp xii+618 url httpssitesmathrutgersedu~weibelKbookhtml

250

List of Lectures

Introduction

Organisational Stuff

A Fairytale


Recollections and Preliminaries

Infinity-Categories and Anima

CocartesianLeft Fibrations and Luries Straightening Equivalence

Yonedas Lemma Limits and Colimits in Infinity-Categories

Yonedas Lemma and Adjunctions

First Properties and Examples of (Co)Limits

Colimits and the Yoneda embedding

Kan Extensions

Bousfield Localisations


Symmetric Monoidal and Stable infty-Categories


Einfty-Monoids and Einfty-Groups

Spectra and Stable infinity-Categories

An Interlude on Infinity-Operads

Infinity-Operads and Symmetric Monoidal Infinity-Categories

The Cocartesian and Cartesian Symmetric Monoidal Structures

Algebras over Infinity-Operads

Day Convolution

Stabilisation of Infinity-Operads and the Tensor Product on Spectra


E-infinity-Ring Spectra

Infinity-Categories of Modules

Tensor Products over Arbitrary Bases

Miscellanea

The Group Completion Theorem and the K-Theory of Finite Fields

Localisations of E-infinity-Ring Spectra



Rational K-Theory

The Cyclic Invariance Condition and Quillens Plus Construction

Topological K-Theory

A Sketch of Quillens Computation

The K-Theory of Stable Infinity-Categories


K0 of a Stable Infinity-Category

The S-Construction and the Q-Construction

The Resolution Theorem


Verdier Sequences

The Additivity Theorem and a Proof of Theorem IV12

The Fibration Theorem

Localisation and Deacutevissage

Bibliography

This text consists of unofficial notes on the lecture Advanced Topics in Algebra ndash Algebraicand Hermitian K-Theory taught at the University of Bonn by Fabian Hebestreit in thewinter term (Wintersemester) 202021

Some additions have been made by the author To distinguish them from the lecturersquosactual contents they are labelled with an asterisk So any Proof or Lemma etc that thereader might encounter are wholly the authorrsquos responsibility

Please report errors typos etc through the Issues feature of GitHub or just tell me beforeor after the lecture

ii











iii

Contents





Bibliography 248

iv











v

List of Lectures













vi

List of Lectures








vii

Chapter 0









1

A Fairytale






D(f)s =

















2

A Fairytale








χ(XE) =dsumi=0


)







3

A Fairytale




K0(X) K0(Y )


f


flowast

















(R[Sminus1]

)


4





Ki(R) = πi N(






sube

sube sube

(minus)grp

sube

(minus)infin-grp








5







6

Chapter I






Y lowast FunL(





| |F sSet C SingF


∆∆ C

sSet

F

Y LanY F=| |F



)= HomC

(F ([n]) X

)

7







Λni C

∆n




0

1

2

Λ20

0

1

2

Λ21

0

1

2

Λ22



8






)


F(∆n X) = F(

N([n])N(C))

= N Fun([n] C)













9




∆0 C times C

(st)(ab)








core C C

N(coreπC) N(πC)







10













An


Cat(2)1

subesube

sube

sube sube





simpbull (∆n) D

)




11








(Hom(CD)

)for all i gt 0









| | Z[minus]

forget

| |Cbull

SingCbull




τgtiC =(




)

12





)[minus1]

)











| | An Nc(CW) Sing




13




Un(F ) lowastAn

C An

F











14




x z

y

g

f

exist()

p7minus

p(x) p(z)

p(y)

p(g)

p(f)σ



HomE(y z) HomE(x z)

HomC(p(y) p(z)

)HomC

(p(x) p(z)

)flowast

plowast plowast

p(f)lowast





p-Cocart

P Ar(C)

E C

simq

s

p

s

p



15












HomP

(q(f) q(g)


(p(f) p(g)

)


16







x y

xprime yprime

f σ g



x x

xprime yprime

f




17





z

x y

zprime

xprime yprime

h

f

g









18





C[Cop times C] Kan


Exinfin






Ar(C) C times C

C

(st)

t pr2




19




Ar(C) C times C

C

(st)

s pr1




TwAr(C) lowastAn

Cop times C An

HomC



x xprime

y yprime

f

ms

g

mt


∆0

mlowastsmtlowast

f g


HomTwAr(C)(f g) g


h


20






evidx Nat(




21








minus HomC(yRminus)







L CR minus D












π0 | |

core

22




HomC(

colimI

F x) Nat(F constx)




colimI

F (

hocolimC[I]

F)f

and limIF

(holimC[I]

F)c







23





Γ(p) Fun(SE)

idS Fun(S S)

plowast



colimI


limIF Γcocart

(Uncocart(F )

)


colimI

F ∣∣Un(F )

∣∣ and limIF Γ

(Un(F )

)


Fun(ICatinfin)

Catinfin

Cocart(I)

sim

St

const



) HomCatinfin(C X)


24







I I

pr2 p












25



W C

|W | C[Wminus1]

F

p



)minus Fun(CD)


(C[Wminus1]





∆0 ∆1

∆1

1

0






Λ20

D Dprime


26



















Fun(IFun(D C)

)Fun(D C)

Fun(DFun(I C)

)const

const lowast

colimlowast

limlowast

27




colimI


F



colimE

F colimC

G





Fun(CP(D)

)Fun

(E P(D)

)Fun

(cP(D)

)Fun

(E|cP(D)

)plowast

clowast ilowast

plowast|c


28




EF




Cprime Dprime

C D

q

g f

p


Right(C) Right(D)

qlowast

qlowast

glowast

plowast

flowast

plowast



lowast






29








in this casecolimI

F α colimJ

F


jα Ar(J )

j times I J times J

(st)(inclα)





|I| colimI

const lowast colimJ

const lowast |J |


30



C


st const

αlowast

colimJ colimI







colimJ

F colimI

H

if either exists


colimJ

F colimUncocart(G)

F







31



E E prime

C D

p pprime

f










)=rArr HomCatinfinC

(E flowast(minus)

)


32





C D

P(C) P(D)

f

Y C Y D

f




Cx Df(x)

C Df


∆0

Cx Df(x)

x f(x)

f




HomD(F (x) G(y)

)



33












HomC(F (i) x





HomC(F (i) x)




34





) ηlowast









) simminus Fun(CD)







F (c)



35





fE Ar(E)

C E

s

f





C fE C

Ef

r

t

s

flArr=η


f s f s

f s t

η η

η


36




colim(ccE)isinY CE


is an equivalence





U xC

Y CE C

s



) E(x)





J Y CP(C) C D

I P(C)

t

s F

Θ LanY C F

37



LanY C F(

colimiisinI

Ei


(Y CE minus C F

minus D)


colimiisinI


minus D)



J C

P(C)Θ P(C)

Y C

s

andY CE C

P(C)E P(C)

Y C

s


Stcart(J C) =rArr E


I An

P(C)

Θ

evxΘ

HomP(C)

(Y C(x)minus




)

38




colimnisinN


empty = empty





)Y Clowast







39




on objects d isin D


HomD(


(minus R(d)












40




)= π0 HomDgt0(Z)

(Cbull(Si lowast) C

)= π0 HomDgt0(Z)

(Z[i] C

)= Hi(C)


πi HomAn(XK(An)

)= πi HomDgt0(Z)

(Cbull(X) A[n]

)= HiminusnC





HomlowastAn(XK(An)

) Hn(XA) HomAb

(Hn(X) A

)



πi(τ6nX) =

0 if i gt n

πi(X) if i 6 n


41




) HomAb

(Hn+1(F ) πn+1X

)


The diagram





C infinoplusi=0


Hi(C)[i]


X infinprodi=0

K(πiX i)


τgt0(C[minus1]

)0

0 C




42




0 Cbull(Xx)



) HomDgt0(Z)

(Cbull(X) D


HomAn(XK(D)



)(all pullbacks are taken in An as announced on Page 35) The complex Cbull(Xx) computesreduced homology For example if Si denotes the i-sphere then Cbull(Si lowast) cofib(Z[0] Z[0]oplus Z[i]) Z[i] as one would expect



| |F sAn D SingF




incl

| |1F

Sing1F






43



Nrn(C) HomCatinfin

([n] C

) core Fun

([n] C

)










HomC(yRd) HomC(xRd)


flowast

sim sim

Lflowast




HomC(RdRx)

clowastd

Rsim

44



evidd Nat(





cLxminusminus Lx











C[Wminus1]op

pop

HomC(minusc)


45



C D

C[Wminus1]

p

Fp

F





) G(pc) Nat










colimI

F L(

colimI

RF)

and limIF L

(limIRF)


46



HomD(L(

colimI

RF)minus) HomC

(colimI

RFRminus) Nat

(RF const(Rminus)



) Nat(F constminus)




HomC(minus xj)



(limIRF)


HomD(minus L

(limIRF)) HomC

(Rminus RL

(limIRF)) HomC

(Rminus lim

IRF)

Nat(

const(Rminus) RF)

Nat(constminus F )









)

47










π0

sube

| |

core




]









48













D(R)ϕ

ϕlowastD(S)

ϕlowast

ϕlowastD(R)



R S

0 SLR

ϕ


49










S S otimesLR S

0 S otimesLR (SLR)

SotimesLRϕ






Nrm

([n]) HomCatinfin

([m] [n]

) core Fun

([m] [n]

)


50







X3 X1 timesX1

s

∆

(ss)(d02d13)




Pwz X1

lowast X0 timesX0

(d1d0)(wz)



(d0d2)sim







51





0

1

2

3

id

id








52




lowast X0 timesX0

(d1d0)

(xy)

(I652)



core asscat(


0(C)






Nr1(C) core Ar(C)

lowast lowast

Nr0(C)timesNr


sim

(xy)(xy)

sim



colim∆∆op

T (

hocolimC[∆∆op]

T)f T f Sing |T |


| asscat(T )| |T |

53





Nrminus CSAn







s




πminus Cat(2)

1






54




Nrn

(TwAr(C)

) HomCatinfin

([n]op [n] C

) HomsAn

((∆n)op ∆nNr(C)

)



TwAr(C) lowastAn

Cop times C An

HomC


Nrn

(TwAr(C)

) HomCatinfin

([n]TwAr(C)

)(1) HomCatinfin

([n] Cop times C


([n] lowastAn

)(2) HomCatinfin

([n] Cop times C


(3) HomCatinfin

([n] Cop times C



)(4)(

HomCatinfin([n]op C

)timesHomCatinfin

([n] C


(5)(

HomCatinfin([n]op C

)timesHomCatinfin

([n] C


(6) core Fun

([n]op t0 [1] t0 [n] C

)(7) HomCatinfin

([n]op [n] C

)(8) HomsAn

((∆n)op ∆nNr(C)

)




)core(lowastAn)

HomCatinfin([n]An

)core(An)

ev0

ev0


55



core Fun([n+ 1]An

) core Fun

([1]An


([1 n+ 1]An

)



) core Funlowast

([1]An


([1 n+ 1]An

)





([n]op C

)








Qn(C) sube Fun(

TwAr([n])op C)



56











0 6 0 1 6 1 2 6 2 3 6 3

0 6 1 1 6 2 2 6 3

0 6 2 1 6 3

0 6 3

J3




57







s

∆

(ss)(d02d13)




Q0(C) P

Q3(C)s



F (0 6 0) F (1 6 1) F (2 6 2) F (3 6 3)

F (0 6 1) F (1 6 2) F (2 6 3)

F (0 6 2) F (1 6 3)

F (0 6 3)

(1) (2)(4) (4)

(2) (1)

(1) (3) (3) (1)

(1) (1)

58





F (0 6 3) F (1 6 3)

F (0 6 1) F (1 6 1)

sim

sim


59

Chapter II








Xnsimminus

nprodi=1

X1





60










Yn Xn

lowast Xn+10

(xx)




| |X sAn minus Anop


∆([n])

= ∆n and ∆0([n])

=n∐i=0

∆0






61



Xn+10

∣∣∣∣∣n∐i=0

∆0

∣∣∣∣∣X












colim∆∆op

π0X = Coeq(π0X1

d1

d0π0X0

)



62



HomlowastCSAn(


((T 1T ) (Xx)




) HomsAn









comp(

decomp(Xx)) (Xx)






T decomp(compT )

63








(lowastCatinfin)gt1

sube sim








Mon(An) Grp(An)



sim

supe

sim

sim

supe

sim

supe


64













ΩkK lowast

lowast K

k

k

simminus

HomLK(k k) Kk

lowast K

k



65














(ΩBMΩxXx

) Hom(lowastAn)gt1

(BMXx

) HomlowastAn

(BMX

)



66




FreeMon(K)1 ∐ngt0

Kn













int[n]Xk colim


Xk1







∐kgt0X


(LaniX)n ∐


1


∐ngt0K

n as claimed

67



Set Mon(Set)

An Mon(An)

Free

FreeMon






X lowast

lowast ΣX


colimI

Fun(IAn) An const


colimI




68



Grp(An) An

(lowastAn)gt1

B

ev1

Ω









prodni=1 S1







69




N N

lowast N2

middot2


Z Z

lowast N2

middot2


S1 S1

D2 RP2

middot2






obj(C)n



)=

(α f)



70




i=1


)




)











1)sube Mon(An)

71






)


K0(R) = π0(


)grp =


grp




(m 7 (r otimes 1)m








Nm XZ2 minus XZ2

[x] 7minus x+ τ(x)



72





Unimodr(PM) =









minus C


b(p p) = Nm(q(p)







73



VectR =


EuclR =


VectC =


SphR =










R



VectR ∐ngt0


BU(n)


Grp(CW) minus Grp(

Nc(CW)) Grp(An)





74







simminus Grp(An)











75




and πn(ko)



πn(

fib(ko ktop)) Θn


πn(

fib(ktop ksph)) Lq

n(Z)








Cut(αop)(i) =



76





ρi(j) =

1 if i = j

undefined else



(lr)


X1 timesX1

X1

X1 timesX1

flip factors

micro

micro









Cotimesmon Cotimes

∆∆op Lop

potimesmon

potimes

Cut

77



CMon(Cat(2)

1)

Mon(Cat(2)

1)


1

Cutlowast

sim sim

forget




idx 00 idy














78




HomC

(noplusi=1

xi

noplusi=1

yi

)

nprodij=1

HomC(xi yj)





Fun(CAn) minus Fun(


(CGrp(C)Fun(LopAn)

)



)







79





Xi



61[n] ([n]∆∆61)op


0 010 11

01 1

1 1

0 0

0





80


12

n

empty empty









Proof We compute



(Mn timesNm)


(Mn times colim

[m]isin∆∆opNm

)(


Mn

)times(


Nm

) |M | times |N |



81








B CMon(Mon(An)



CMon(Mon(C)

)CMon(C)

Mon(CMon(C)

)

sim

forgetlowastsim

simforget

(II18b2)



CMon(An) CMon(An)

Mon(An) lowastAn

B

CutlowastΩ

ev1

B

Ω


82


















83



πi((BX)1

)



Dgt0(Z) CGrp(An)

Dgt0(Z) CGrp(An)

[1]=Σ

K

BΩ

K

Ω












CMon(CMon(C)

)CMon(C)

simforgetlowast

simforget



84





CMon(C) limnisinNop

Mon(n)(C)










microxtimesy





85










simtimes

idsim

(I) (III)


sim







times

times

sim

times

86










)HomC


)


(V)

microa (VI)microa





sim

sim

microctimesd





LopXn


87







)









XnhSn


FhG colimBG

F


88





∐ngt0X

nhSn

As an example



)= S







(

Ωminus CGrp(An) Ω



limNop

(

Ωminus CGrp(An)gt2

Ωminus CGrp(An)gt1

Ωminus CGrp(An)

)


Sp(C) = limNop

(

Ωminus lowastC Ω

minus lowastC)


89



Sp(CGrp(C)

) simminus Sp(C)

is an equivalence


limNop

(



(Lop limNop

(

Ωminus lowastC Ω

minus lowastC))




) ev1minus Sp(C)








)Grp

(Fun(Dop lowastAn)


ev1

sim

Ω

Ω


90

















91







)C 7minus




Sp(Dgt0(Z)

)Sp(An)

D(Z) Sp

Sp(K)

H


D(Z) Sp


H

Ωinfinminusi




Nop

(


minus (lowastAn)op)

limNop

(

Σminus lowastAn Σ



92



Cbull Sp D(Z) H



HomD(Z)(Cbull(X) D

) limiisinNop


) limiisinNop


) limiisinNop





)= colim



)






93





Ωa Ωc 0

Ωb x g 0

0 f a b

0 c



F =

a b

c

7minus Ωminus1

Ωa Ωb

Ωc x




Nat

a b

c

x x

x

Nat

a b

c d

x x

x x

HomC(d x)



94














infin




limI










95



C Cprime












Sp


HomC

homC

of the HomC functor





)and homK(R)(CD) H

(HomR(CD)

)


ΩinfinH(RHomR(CD)

) K

(RHomR(CD)

) HomAn

(lowastK(RHomR(CD))

) HomD(Z)

(Z[0] RHomR(CD)

) HomD(R)

(Z[0]otimesLZ CD

) HomD(R)(CD)

96














97





PSp(C) limnisinNop

Fun(N26(nn) lowastC

)

Sp(C) limnisinNop

lowastC

sube

sim

η=lim ηn





lowast lowast

lowast lowast


︸︷︷︸n+1

︸︷︷

︸n+1N2

6(nn)






HomlowastC(Xn Yn)




98




ΩnXn+i














Fun(N26(nn) lowastC

)

99




lowast lowast

lowast lowast


︸︷︷︸n+1

︸︷︷

︸

n+1N26(nn)





ΩnΣn(Xx)




minus Sp




πi+n(Sn lowast)





Hlowast(S[X] A


(S[X] A

)= Hminuslowast(XA)

100



CbullS colimiisinN


colimnisinNgti




) colim



colimnisinN

colimiisinN6n


colimnisinN

Hlowast+n(Sn A)












ΩXspi+1 Ω colim



nisinNΩnXn+i Xsp

i

101






limiisinNop

(colimnisinN

ΩnXn+i Yi

) limiisinNop

HomlowastC(

colimnisinNgti


limiisinNop

limnisinNop

gti


) limnisinNop

limjisinN6n


) limnisinNop

HomlowastC(Xn Yn)

HomPSp(C)(XY )






102




(F (S0) F (S1)

)


evS0 Sp(Sp(C)


) simminus Sp(C)



X otimes d = colimX


const d



colimx

const d colimX

d

)




F sp colimnisinN

ΩnF (Σnminus)



103



A C lowast

B D Q lowast

lowast P ΣA ΣB

lowast ΣC ΣD








(colimX

E)


(limXE)





)[minusi]



104












HomO(x y)nprodi=1

HomO(x yi)


) nprodi=1


)

(ρ1ρn)

p p

(ρ1ρn)






Lopint minus


Lop





105





HomO1

((αlowastx)i yi

)

HomαO(x y)

nprodi=1

HomρiαO (x y)

sim

sim sim











(F0F1)





106







C0 C1Cyl[(α)

0 1(∆1)

subeC0 C1

C0

Cyl[(α)

C0 times (∆1)]

0 1(∆1)

subeC0 C1

C0

Cyl[(α)


0 1(∆1)

supeC0 C1

C0


0 1(∆1)



HomactO((a1 an) b

)HomO

((a1 an) b

)lowast HomLop

(〈n〉〈1〉

)

fn

107








)is an equivalence







Opinfin dOpinfin

Catinfin Catinfin

sim

(minus)1 (minus)1

(minus)op






108

















109






) Homact

Cotimes((d1 dn) d



)


)




) Homα





)





) Homα




) HomC


)



) HomD


)


110














111








HomP(E)X(

HomE(eminus) F)

HomP(E)(

HomE(eminus) F)

F (e)

lowast HomP(E)(

HomE(eminus) X)

X(e)

sim

sim





112











(∆3(∆02 cup∆13) X

)






([n]timesLop

Lopt C






HomsAn(∆nNr(Lop)

)HomsAn

(InNr(Lop)

)sim

113







Lopt C

) simminusnprodi=1


Lopt C

)


Lopt in



t )





sAn

sim






[k] Lop

pr1

const 〈n〉

114



Yk FCt(


Lopt C

) HomCatinfin

([k]times 〈n〉 C

) HomCatinfin

([k] Cn







([1]timesLop

Lopt C

)lowast HomCatinfin

([0]timesLop

Lopt C

)timesHomCatinfin

([0]timesLop

Lopt C

) (d1d0)

(xy)



Lopt C



) HomsAnNr(Lop)



([1] Ct


Lop) α




Lopt C


[0]timesLopLopt C

[1]timesLopLopt

d1




Lopt C to be




)

115







) Homfn

Ct((a1 an)minus

)




)

nprodi=1

HomC(aiminus)








([n]timesLop

Loptimes C

)




)


prodiisinS F (i)


infin-operad)

116








lowast Fun(OLop)

pprimelowast

p


prodni=1 F (ai)















117





α





(b) If O is Lopint


FunLop(Lop

int Cotimes) FunLop

int


Lopint)



) Γcocart

(Cotimes timesLop

Lopint)


Lopint) Lop












118









O Cotimes

Oid p






119






HomαCotimes( m An)


HomαCotimes( m An)


HomαCn( n An)


HomC(1C A1)n


timesLop is















120









Fun(CnO1)



Day(CotimesO)ni






Fun(CO1) Fun(CO1)

Fun(CO1) Fun(CnO1)

prlowastiprlowasti

121



Cn C

C C

pri

pri











)

122









(d1d0)

(FG)




Cotimesm Om

Cotimesα Oα

d1





123











HomactDay(CotimesO)

((F1 Fn) G

) Homfn


















124



HomactO

((colimI

F x2 xn

) y) limIop


)Homact

O

((x1 xn) lim

IF) limI













limIop

HomαO

((x1 xn) lim

IF) limI










125








([1]minO




infin we let





minus SpOpinfin(O)










126















127



Day(( op)and C













Lop

timesLop

AtimesB timesSet

AotimesB



128









X times lowast

X or Y










129







E colimnisinN


)[minusn]


HomSp

(colimnisinN


)[minusn] Eprime

) limnisinNop


) limnisinNop







nisinN


)[minusn]






130







)op An minus Spop














131


















132



AlgComm(








LZ)minus CAlg







133





CAlg(






)= TorSlowast

(S[minus] E


)= ExtlowastS

(S[minus] E

)








134






) H

(RHomZ(Cbull(E) A)




)







135




HomactAssoc


lowast)





HomactLMod

((b1 bn) a



HomactLMod

((b1 bn)m

)=



if bi = m





alowast

A


136













Bi otimes


A





∆∆op AssocLop


Cut

minustimes1 a

LCut

p

137












F([n] 0

) (A AM) isin Cn+1


Bi otimes


A and N

( otimesα(n)ltj6m

A

)otimesM





(minustimes1)lowast

Cutlowast(A)

138







F (i)([n] 1

) (A A) and F

([n] 0

)(A AM(i)

)




(i [n] 1) 7minus






colimiisinI

M(i) colimiisinI

M(i)


139









)[1] Fun

(∆∆op times [1] r ([0] 0)Lop)

constF ([0]0)

qlowast


ηF





LModA(Cotimes)

C Cmlowast

Aotimesminus

Aotimesminus


140



















) πjH

(minus [minusi]


141




HomModT(T [i]minus

) HomSp

(S[i]minus

) Ωinfin+i




colimXisinCM

X simminusM


colimXisinCM

π0 HomModT(T [i] X





142











([n]timesO Cotimes








([k]Fun(O Cotimes)




) HomCatinfin


n times

))





143








) FunOpinfin


) Fun

(Comm1CAlg(Cotimes)

) CAlg(Cotimes)




alowast

〈1〉 A






Bar(MAN)




144







P([n])

N([n] 0

)MA

([n] 0

)A([n])







in Fun(Lopact






145



N colim∆∆op

Bar(AAN)


∆∆op ∆∆op

C

σop

B Bprime




colim∆∆op

B colim∆∆op









(MhomR(NL)

)




146



FSR ModS minus ModR













(D(R)otimes





H(R[0]


147











)otimes π0

(S[minusj]otimesM

)minus π0


) microminus π0


)




) πi+j

(S[i+ j]

) Z





148




) Hlowast(MR)




















) Hminuslowast(XR)



149







150

Chapter III









(M

middotsminusM

middotsminusM

middotsminus

)




151




)[sminus1]




M [Tminus1]












is an equivalence

152












m xotimesm m


fotimesidm

fotimesidm

micro

sim






153









)sube HomCAlg(R T )







)LModR

(ModotimesRR


sim


colim∆∆op


Bar(MR[Sminus1] N

)

154









HomCAlg(ModotimesRR

)

(R[Sminus1] T


R)(R T )





] simminus HQ



H D(Q) minus Sp



155












(S[M ] R

)


π0(S[M ]

) H0(MZ) Z

[π0(M)






156





)sube HomCAlg

(S[M ] R






) colim






πi(BSn lowast)

0 if i 6= 1Sn if i = 1







)



) colim



157





middot[1]minusminus

)minus colim

nisinN

(ΩinfinS middot[1]


) ΩinfinS





) colim


whereas


) colim

nisinNH1(SnZ) colim

nisinNSabn Z2Z




Hlowast(k(R)0Z


Hgrplowast(

GLn(R)Z)




Hlowast(k(R)Z

) colim

nisinN

(Hlowast(


(Proj(R)Z


)We have Proj(R)



158




Hlowast(k(R)0Z

) colim


) colim

nisinNHgrplowast(

GLn(R)Z)





Aut(P ) π1(

Proj(R) P)minus π1

(k(R) P

) K1(R)



infin we obtain




0 r


= n + reij







159

















sim

1otimesminus

∆lowast

minusotimes1

pr2lowast

pr1lowast

εotimesid

idotimesε

sim

sim



160






f X minusrprodi=1

K(Q ni)



Hlowast(K(Z n)Q

)= Hlowast

(K(Q n)Q




flowast rotimesi=1

Hlowast(K(Q ni)Q

) Hlowast

(rprodi=1




161







AotimesA AotimesA

A Q A

AotimesA AotimesA

(idAi)

micro

∆

∆

ε u

(iidA)

micro






) Hlowast(MQ)


162




microminus Agt0

)






GLinfin(R)Q)







Ki(OF )otimesQ =




163




T (M s) colimN

(M

sminusM

sminusM

sminus

)




A 7minus

(A 00 1

)



T (M s)

T (M s)

r colimN

M M M

M M M

r

s

τ r

s

τ

s

r

s s s




A 0 00 m 00 0 n


A 0 00 n 00 0 m

164




T (M s)

T (M s)

t colimN

M M M

M M M

s

id

s s

id

s

s

s

s

s

s


T (M s) colimN

(M

snminusM

snminusM

snminus




)HomAn(MnMn) HomAn

((m m)Mn

)Mn


)HomAn(MnM) HomAn

((m m)M

)M


fn



165







Minfin-grp T (M s)





(T (M s) s


minus )


Now we use that





166














S[X] simminus S[X+]





lowast X

167






iisinI S2 X

lowast X+



to check that

HomAn(KHomAn(X+ Z)

)minus π0 HomAn

(KHomAn(XZ)



)= 0








168




)= Flowast

((Xx)Sing(Y y)

)



Y

|X| times |∆n| Yf

exist f



X X+

X X+







169





) HomAn

(YHomAn(XZ)

) HomAn(X times YZ)





HomAn(T (M+ s) Z

) lim

Nop

(


minus HomAn(M+ Z))

limNop

(


minus HomAn(MZ))

HomAn(T (M s)+ Z


T (M s)+ Minfin-grp


sim sim

sim





170




(Proj(R) middot[R]


)+









VectR ∐ngt0


BU(n)


T(VectR [R]

) ZtimesBO and T

(VectC [C]

) ZtimesBU




) T


)

171







)grp





)= (Binfinko)i(X)





R(S8)



172











and πn(ko)





173








[V ] 7minussumigt0

[ΛiV ]ti












174



Elowast(BU(n)



prodnisinZE




ku0(ku) =prodnisinZ

Zqβici

∣∣ i gt 1y



prod06k6m

ku0(Grk(Cm))



λn(x+ y) =sumi+j=n





175



λ(x) =sumigt0



`(t) =sumigt1

(minus1)iminus1ti

iisin QJtK



minus`(λ(x)minus 1

)prime =sumigt0





`(λ(x)minus 1


λ(x)= minus

sumigt0

(1minus λ(x)






)= ker


)



176







minuspminus1sumi=0

(1minus λ(x)


(1minus λ(x)

)p1minus

(1minus λ(x)

) middot λ(x)prime

equiv minus λ(x)p





λ(x)p equivsumigt0






)prime =sumigt0

[L]i+1ti



177



S2n βn

minus ku ψi

minus ku




iminus1sumj=0

[γC1 ]j = iβ





)

Consequently

Ki(Fq) =










178






Hlowast(

fib(ψq minus id)Q)

= 0 and Hlowast(


= 0




)= colim

nisinNHgrplowast(

GLn(Fq)Q)

= 0







179







χ(ψn(V )

)(g) = χ(V )(gn)


χ(λ(V )

)(g) =

dsumm=0


(1minus λit)




sumngt0

χ(ψn+1(V )

)(g)tn =

dsumi=1

λi1minus λit

=sumngt0

dsumi=1

λn+1i tn





Brρ RFp

(GLn(Fq)

)minus RC

(GLn(Fq)

)





180





RFp

(GLn(Fq)

)RC(

GLn(Fq))

RFq(

GLn(Fq))

ku0(BGLn(Fq))

Brρ

repFpotimesFqminus


RFq(

GLn(Fq))minus RFp

(GLn(Fq)

) Brρminus RC

(GLn(Fq)





χρFpotimesFqV

(g) =sum[micro]


ρ(λ)





RFq(



)ψq


bijections



)ψq




)

181










πi+1(Xn) minus πi

(limnisinN

Xn

)minus lim



R1 limnisinN


)= R1 lim

nisinNku1(BGLn(Fq)

)


R1 limnisinN

RC(

GLn(Fq))I

= 0




mn RnIn 0 show RnI



RlimnisinN

(limmisinN

RnImn

) Rlim

nisinN

(R limmisinN

RnImn

) Rlim

nisinNRnI

nn

RlimnisinN

ZoplusRlimnisinN

InInn

182











)cup(

idpartn timesγ)

(n times 0





n times∆1 B

γmiddotα

p

idn timesp(γ)



prodmicroinfin`





Brρ

sim

flowast

Brρ

simsumnminus1k=0

qki

183




on K-theory



k(Fq)0 ku ku

k(Fq)0 ku ku

Brρ

flowast


ψqk

Brρ ψqnminusid






k(Fq)0 ku ku

k(Fq)0 ku ku

Brρ

Froblowast ψp

ψqminusid

ψp

Brρ ψqminusid



Ki(Fp)

Z if i = 0oplus






i times

)



(FreeAlg

D(Z)otimesLZ

(Ftimesp [1]

))


184






Z((pn)i minus 1

)




p-sZs





Ms = limnisinN

MsnM


M M

0 MsnM

sn


185



Ms homR

((Rsinfin)[minus1]M

)

















M Ms


186






sn homR


)minus homR




(RsnhomR(MN)

) homR

(RsnhomR(Ms N)





homR

(R[sminus1]M

) homR(RM) M



R R R

0 Rsn Rsn+1

0 Rs

sn

s

s



Ms limnisinN

(M otimesR Rsn) 0

187




RHomR

(R[sminus1]K

) 0






R R[sminus1]

0 C[1]









Ks RlimnisinN

(K otimesLR Rsn

)

188















A[pn]








RlimnisinN

A[pn] 0


189






(πiminus1(E)

)minus 0





πi(EpnE) minus 0




0 minus Λp(πi(E)

)minus lim


(πiminus1(E)

)minus 0


0

0 R1 limnisinNop


)Λp(πi(E)

)0

0 R1 limnisinNop


πi(EpnE) 0

L1Λp(πiminus1(E)

)0

sim


190









Eprodp

Ep

E otimesHQ

(prodp

Ep

)otimesHQ







)

191











πi((Binfinku)) =




L0Λ`

(oplusq 6=p

microqinfin

)= 0 and L1Λ`

(oplusq 6=p

microqinfin

)= Z`


πi(K(Fp)) =


192









193

Chapter IV








K1(Z) K1(Q)oplus


plusmn1 Qtimesoplus

p primeZ Z Zsim



p primeZ minus 0

194





Then we can put

[M ] =nsumi=0







simminus K0(R)




K noplus


noplusi=minusm

Pi[i]


i=minusm+1Pi[i]


195








c minus d minus c



K0(D) minus K0(C)



CH =x isin C









Cmin = C0 =x isin C

∣∣ [x] = 0 in K0(C)


196



HD = im(K0(D) K0(C)











197








Sn(C) sube Fun(

Ar([n]) C)



(0 6 0)

(0 6 1) (0 6 2) (0 6 3) (0 6 4) (0 6 nminus 1)(0 6 n)

(1 6 n)

(2 6 n)

(3 6 n)

(nminus 2 6 n)

(nminus 1 6 n)

(n 6 n)

(1 6 1)

(2 6 2)

(3 6 3)



198







infin


0 a b

0 ba

0







K0(C) π0(k(C)

) π1

(|core(C)|

)








199







(minus [n]











(IV71)


Sn timesXn+1 |sknX|










200




core(C) t lowast)








Ztimes π1(




Ztimes π1(

coreS2(C) s)


π1(

coreS2(C) s)

pr2




201



π1(|core(C)|

)= π1

(colimnisinN

|skn coreS2(C)|)

= colimnisinN

π1(|sk2 coreS2(C)|

)= π1

(|sk2 coreS2(C)|

)


| | colim∆∆op

inj







Nr(C)esd Nr(

TwAr(Cop))


|X| |Xesd|








202





(0` 6 0`)

(0` 6 1`) (0` 6 2`) (0` 6 2r) (0` 6 1r)(0` 6 0r)

(1` 6 0r)

(2` 6 0r)

(2r 6 0r)

(1r 6 0r)

(0r 6 0r)

(1` 6 1`)

(2` 6 2`)

(2r 6 2r)

(1r 6 1r)





Fun(In C) Fun(

Ar([n] [n]op) C)



Lan

Ran Ran

Lan



203

























minus k(Dperf(R)

)



204
















205





0 x 0

z y z
















k coregrp



206






A B

C D




infin E) aredenoted






infin E)





0 C



Atimes C C

A 0

207




infinCMon(E)) CMon

(Funadd(Catst

infin E))




infinCGrp(E)) CGrp

(Funadd(Catst

infin E))

























HomCatstinfin


(B C)


HomCatstinfin

(cofib(f) C

) HomCatst



infin(B C)











A B Cf p

g q

or A B Cf p

gprime qprime


C B Aq g

p f


p f

209



A B Cf p

g

gprime

q

qprime


b fg(b)



qp(b) b







g(g(a)

) cofib


) g(a)


qF (a) a g(a)

qF(g(a)

)g(a) g

(g(a)

)qF (ηa) ηa

sim

g(ηa)

210






b

b c b

b0

s







) Homim(p)

(p(b) p(fib(ϕ))

)



A B Cf p

g q

211



C B Aq g

p f





Finally suppose

A B Cf p

g

gprime

q

qprime


p(









212




qp(b) b qprimep(b)

0 fg(b)



A B

C D




A B Cf p

g

gprime

q

qprime


B Ar(A)

C A

grArrcp

t

c

213



A Ar(A) AidArArr0 t

s

fib(minus)

0rArridA

idArArridA





SotimesRminus

homR(Sminus)

incl

homR(Iminus)











214



n and in general




divs(M) = homR

(R[sminus1]M

) lim

Nop

(

sminusM

sminusM

sminusM




simsim Mods-comp



M Ms






R








215




[0] [nminus i]

[i] [n]

0

i +i


Qnminusi(C) Q0(C)




)










216






gf(x) gf(y)

x y

gf(ϕ)

c c

ϕ


fibration



gf(x) g(yprime)

x y

c ϕ


xprime yprime

x y

ϕprime

f(ϕ)



217








x z v

y bull

w

(1) (2)

(3) (4)

p7minus

p(x) p(z) p(v)

p(y) u

p(w)




E C

Q1(D) D

p

d1

p

d1


218







p

(idE d1)

p

(d2d1)





F (E) minus F(Q1(C)

)minus HomCatinfin

([1]SpanF (C)




A B

C D

p

219





|D|

F | |

|F |




|p|




| |minus An

) colim

(|D| |F |minus An

) Unleft (|F |)




A B

C D



220



F(Q(A)

)F(Q(B)

)F(Q(C)

)F(Q(D)

)







xy1minus s1

y2 minus s2y3minus

y2n minusminus s2nw

z




F(


(Qn(C)

) F (C)2n+1



221



F (C)5 F(Q1(C)

)timesd1F (C)d0 F

(Q1(C)

)F(Q2(C)

) d1minus F(Q1(C)

) F (C)3(d0d2)

sim













F(Q1(C)

)times F (Q1(C))




∣∣ ∣∣F (Q(C))∣∣ ∣∣F (S(C)

)esd∣∣ ∣∣F (S(C))∣∣


222



C Q1(C)

0 Q0(C)timesQ0(C)

(d1d0)



F (C) F(Q0(C)

) simminus F(Q1(C)

)times F (C)3





lowast SpanF (C)

0


F (C) lowast


∣∣ 0

0




∣∣ BF (C)



223






infinAn)






A B

C D

σ τ


Bi Bj

Di Dj


colimiisinI

A colimiisinI

Bi

colimiisinI

Ci colimiisinI

Di

σ

τ

is a pullback in An

224




Bi K

Di colimiisinI

Di



F colimiisinI

Di minus An


Fun(

colimiisinI

DiAn) limiisinIop


AnDi



Iop minus An




F colimiisinI

colimDi

Di

colimiisinI

Unleft(Fi)

colimiisinI

Bi


225





Bi


Di

τ



Aj colimiisinI


Ci colimiisinI

Bi

Cj colimiisinI

Ci colimiisinI

Di

τ



asscat(F (Null(C))

) 0 SpanF (C)




asscat(


x asscat(X)

asscat(X)

d0

sim

226





0



F(

Null(C)) fib0


)


constF (C) F(

Null(C))

const lowast F(Q(C)

)F (d0)

0



0 0 0 0 0

x 0 0 0

x 0 0

x 0

x


227






∣∣F (Q(C))∣∣

|F (d0)|

0


asscat(F (Null(C))


(F (Q(C))

) SpanF (C)



F(

Nullm(C))

F(

Nulln(C))

F(Qm(C)

)F(Qn(C)

)






228




(Dperf(R)





Ω∣∣F (Q(minus)






infin minus PSp



infinPSp)











229






Q(f) Null(B)

Q(A) Q(B)

d0

f









)∣∣ minus ∣∣F (Q(B))∣∣




equivalent

230
















231








HomD(x cofib(c y)

)






)plowast



LF (x) colimcisinCx

F(

cofib(c x))


F (x) colimcisinCx


F(

cofib(c x))


232




HomD(x cofib(c y)

)



ϕlowastc x 0

c y cofib(x y)

0 cofib(c y)

ϕ



F(


cprimeisinCxF(





233











Buy Hardcover Book

eBook $8999











b1

b2nminus1

b2n



234




coreQnQ(i) (

Nr FunA(JnB))esd





(Q(i)












(Q(i)


A2n+1



b1 b3 b2nminus1


235



dj(a0 a2n) =









(Dperf(R)

)minus k

(Dperf(S)

)



p primeK(Fp) K

(Dperf(Z)Q-tors)




Catstinfin = Catst


236


















infin Catstinfin




infin Catstinfin as








(minus) Catstinfin

simminus Catstinfin





237

























238













239














HomCatstinfin



(DopSp)


HomCatstinfin


(EopSp)


HomCatstinfin


(EopSp)

240





disimminus e
















241









(Rsn)[minus1]

















M colimjisinJ


Ci otimesR Dj


R )

242





















(Dperf(Z)

)minus K

(Dperf(Q)


243















244













ibprime J ib r

s



245







G(R) = K(Daperf(R)

)


G(R) K(R)




m






R

246



Rm








K

(oplusm


(Dperf(R)tors)


mK(Dperf(Rm)) oplus





infinAn)






247
















248

Bibliography
















249

Bibliography




250

List of Lectures

Introduction


A Fairytale









Kan Extensions











Day Convolution






Miscellanea





Rational K-Theory










Verdier Sequences




Bibliography











iii

Contents





Bibliography 248

iv











v

List of Lectures













vi

List of Lectures








vii

Chapter 0









1

A Fairytale






D(f)s =

















2

A Fairytale








χ(XE) =dsumi=0


)







3

A Fairytale




K0(X) K0(Y )


f


flowast

















(R[Sminus1]

)


4





Ki(R) = πi N(






sube

sube sube

(minus)grp

sube

(minus)infin-grp








5







6

Chapter I






Y lowast FunL(





| |F sSet C SingF


∆∆ C

sSet

F

Y LanY F=| |F



)= HomC

(F ([n]) X

)

7







Λni C

∆n




0

1

2

Λ20

0

1

2

Λ21

0

1

2

Λ22



8






)


F(∆n X) = F(

N([n])N(C))

= N Fun([n] C)













9




∆0 C times C

(st)(ab)








core C C

N(coreπC) N(πC)







10













An


Cat(2)1

subesube

sube

sube sube





simpbull (∆n) D

)




11








(Hom(CD)

)for all i gt 0









| | Z[minus]

forget

| |Cbull

SingCbull




τgtiC =(




)

12





)[minus1]

)











| | An Nc(CW) Sing




13




Un(F ) lowastAn

C An

F











14




x z

y

g

f

exist()

p7minus

p(x) p(z)

p(y)

p(g)

p(f)σ



HomE(y z) HomE(x z)

HomC(p(y) p(z)

)HomC

(p(x) p(z)

)flowast

plowast plowast

p(f)lowast





p-Cocart

P Ar(C)

E C

simq

s

p

s

p



15












HomP

(q(f) q(g)


(p(f) p(g)

)


16







x y

xprime yprime

f σ g



x x

xprime yprime

f




17





z

x y

zprime

xprime yprime

h

f

g









18





C[Cop times C] Kan


Exinfin






Ar(C) C times C

C

(st)

t pr2




19




Ar(C) C times C

C

(st)

s pr1




TwAr(C) lowastAn

Cop times C An

HomC



x xprime

y yprime

f

ms

g

mt


∆0

mlowastsmtlowast

f g


HomTwAr(C)(f g) g


h


20






evidx Nat(




21








minus HomC(yRminus)







L CR minus D












π0 | |

core

22




HomC(

colimI

F x) Nat(F constx)




colimI

F (

hocolimC[I]

F)f

and limIF

(holimC[I]

F)c







23





Γ(p) Fun(SE)

idS Fun(S S)

plowast



colimI


limIF Γcocart

(Uncocart(F )

)


colimI

F ∣∣Un(F )

∣∣ and limIF Γ

(Un(F )

)


Fun(ICatinfin)

Catinfin

Cocart(I)

sim

St

const



) HomCatinfin(C X)


24







I I

pr2 p












25



W C

|W | C[Wminus1]

F

p



)minus Fun(CD)


(C[Wminus1]





∆0 ∆1

∆1

1

0






Λ20

D Dprime


26



















Fun(IFun(D C)

)Fun(D C)

Fun(DFun(I C)

)const

const lowast

colimlowast

limlowast

27




colimI


F



colimE

F colimC

G





Fun(CP(D)

)Fun

(E P(D)

)Fun

(cP(D)

)Fun

(E|cP(D)

)plowast

clowast ilowast

plowast|c


28




EF




Cprime Dprime

C D

q

g f

p


Right(C) Right(D)

qlowast

qlowast

glowast

plowast

flowast

plowast



lowast






29








in this casecolimI

F α colimJ

F


jα Ar(J )

j times I J times J

(st)(inclα)





|I| colimI

const lowast colimJ

const lowast |J |


30



C


st const

αlowast

colimJ colimI







colimJ

F colimI

H

if either exists


colimJ

F colimUncocart(G)

F







31



E E prime

C D

p pprime

f










)=rArr HomCatinfinC

(E flowast(minus)

)


32





C D

P(C) P(D)

f

Y C Y D

f




Cx Df(x)

C Df


∆0

Cx Df(x)

x f(x)

f




HomD(F (x) G(y)

)



33












HomC(F (i) x





HomC(F (i) x)




34





) ηlowast









) simminus Fun(CD)







F (c)



35





fE Ar(E)

C E

s

f





C fE C

Ef

r

t

s

flArr=η


f s f s

f s t

η η

η


36




colim(ccE)isinY CE


is an equivalence





U xC

Y CE C

s



) E(x)





J Y CP(C) C D

I P(C)

t

s F

Θ LanY C F

37



LanY C F(

colimiisinI

Ei


(Y CE minus C F

minus D)


colimiisinI


minus D)



J C

P(C)Θ P(C)

Y C

s

andY CE C

P(C)E P(C)

Y C

s


Stcart(J C) =rArr E


I An

P(C)

Θ

evxΘ

HomP(C)

(Y C(x)minus




)

38




colimnisinN


empty = empty





)Y Clowast







39




on objects d isin D


HomD(


(minus R(d)












40




)= π0 HomDgt0(Z)

(Cbull(Si lowast) C

)= π0 HomDgt0(Z)

(Z[i] C

)= Hi(C)


πi HomAn(XK(An)

)= πi HomDgt0(Z)

(Cbull(X) A[n]

)= HiminusnC





HomlowastAn(XK(An)

) Hn(XA) HomAb

(Hn(X) A

)



πi(τ6nX) =

0 if i gt n

πi(X) if i 6 n


41




) HomAb

(Hn+1(F ) πn+1X

)


The diagram





C infinoplusi=0


Hi(C)[i]


X infinprodi=0

K(πiX i)


τgt0(C[minus1]

)0

0 C




42




0 Cbull(Xx)



) HomDgt0(Z)

(Cbull(X) D


HomAn(XK(D)



)(all pullbacks are taken in An as announced on 5) The complex Cbull(Xx) computesreduced homology For example if Si denotes the i-sphere then Cbull(Si lowast) cofib(Z[0] Z[0]oplus Z[i]) Z[i] as one would expect



| |F sAn D SingF




incl

| |1F

Sing1F






43



Nrn(C) HomCatinfin

([n] C

) core Fun

([n] C

)










HomC(yRd) HomC(xRd)


flowast

sim sim

Lflowast




HomC(RdRx)

clowastd

Rsim

44



evidd Nat(





cLxminusminus Lx











C[Wminus1]op

pop

HomC(minusc)


45



C D

C[Wminus1]

p

Fp

F





) G(pc) Nat










colimI

F L(

colimI

RF)

and limIF L

(limIRF)


46



HomD(L(

colimI

RF)minus) HomC

(colimI

RFRminus) Nat

(RF const(Rminus)



) Nat(F constminus)




HomC(minus xj)



(limIRF)


HomD(minus L

(limIRF)) HomC

(Rminus RL

(limIRF)) HomC

(Rminus lim

IRF)

Nat(

const(Rminus) RF)

Nat(constminus F )









)

47










π0

sube

| |

core




]









48













D(R)ϕ

ϕlowastD(S)

ϕlowast

ϕlowastD(R)



R S

0 SLR

ϕ


49










S S otimesLR S

0 S otimesLR (SLR)

SotimesLRϕ






Nrm

([n]) HomCatinfin

([m] [n]

) core Fun

([m] [n]

)


50







X3 X1 timesX1

s

∆

(ss)(d02d13)




Pwz X1

lowast X0 timesX0

(d1d0)(wz)



(d0d2)sim







51





0

1

2

3

id

id








52




lowast X0 timesX0

(d1d0)

(xy)

(I652)



core asscat(


0(C)






Nr1(C) core Ar(C)

lowast lowast

Nr0(C)timesNr


sim

(xy)(xy)

sim



colim∆∆op

T (

hocolimC[∆∆op]

T)f T f Sing |T |


| asscat(T )| |T |

53





Nrminus CSAn







s




πminus Cat(2)

1






54




Nrn

(TwAr(C)

) HomCatinfin

([n]op [n] C

) HomsAn

((∆n)op ∆nNr(C)

)



TwAr(C) lowastAn

Cop times C An

HomC


Nrn

(TwAr(C)

) HomCatinfin

([n]TwAr(C)

)(1) HomCatinfin

([n] Cop times C


([n] lowastAn

)(2) HomCatinfin

([n] Cop times C


(3) HomCatinfin

([n] Cop times C



)(4)(

HomCatinfin([n]op C

)timesHomCatinfin

([n] C


(5)(

HomCatinfin([n]op C

)timesHomCatinfin

([n] C


(6) core Fun

([n]op t0 [1] t0 [n] C

)(7) HomCatinfin

([n]op [n] C

)(8) HomsAn

((∆n)op ∆nNr(C)

)




)core(lowastAn)

HomCatinfin([n]An

)core(An)

ev0

ev0


55



core Fun([n+ 1]An

) core Fun

([1]An


([1 n+ 1]An

)



) core Funlowast

([1]An


([1 n+ 1]An

)





([n]op C

)








Qn(C) sube Fun(

TwAr([n])op C)



56











0 6 0 1 6 1 2 6 2 3 6 3

0 6 1 1 6 2 2 6 3

0 6 2 1 6 3

0 6 3

J3




57







s

∆

(ss)(d02d13)




Q0(C) P

Q3(C)s



F (0 6 0) F (1 6 1) F (2 6 2) F (3 6 3)

F (0 6 1) F (1 6 2) F (2 6 3)

F (0 6 2) F (1 6 3)

F (0 6 3)

(1) (2)(4) (4)

(2) (1)

(1) (3) (3) (1)

(1) (1)

58





F (0 6 3) F (1 6 3)

F (0 6 1) F (1 6 1)

sim

sim


59

Chapter II








Xnsimminus

nprodi=1

X1





60










Yn Xn

lowast Xn+10

(xx)




| |X sAn minus Anop


∆([n])

= ∆n and ∆0([n])

=n∐i=0

∆0






61



Xn+10

∣∣∣∣∣n∐i=0

∆0

∣∣∣∣∣X












colim∆∆op

π0X = Coeq(π0X1

d1

d0π0X0

)



62



HomlowastCSAn(


((T 1T ) (Xx)




) HomsAn









comp(

decomp(Xx)) (Xx)






T decomp(compT )

63








(lowastCatinfin)gt1

sube sim








Mon(An) Grp(An)



sim

supe

sim

sim

supe

sim

supe


64













ΩkK lowast

lowast K

k

k

simminus

HomLK(k k) Kk

lowast K

k



65














(ΩBMΩxXx

) Hom(lowastAn)gt1

(BMXx

) HomlowastAn

(BMX

)



66




FreeMon(K)1 ∐ngt0

Kn













int[n]Xk colim


Xk1







∐kgt0X


(LaniX)n ∐


1


∐ngt0K

n as claimed

67



Set Mon(Set)

An Mon(An)

Free

FreeMon






X lowast

lowast ΣX


colimI

Fun(IAn) An const


colimI




68



Grp(An) An

(lowastAn)gt1

B

ev1

Ω









prodni=1 S1







69




N N

lowast N2

middot2


Z Z

lowast N2

middot2


S1 S1

D2 RP2

middot2






obj(C)n



)=

(α f)



70




i=1


)




)











1)sube Mon(An)

71






)


K0(R) = π0(


)grp =


grp




(m 7 (r otimes 1)m








Nm XZ2 minus XZ2

[x] 7minus x+ τ(x)



72





Unimodr(PM) =









minus C


b(p p) = Nm(q(p)







73



VectR =


EuclR =


VectC =


SphR =










R



VectR ∐ngt0


BU(n)


Grp(CW) minus Grp(

Nc(CW)) Grp(An)





74







simminus Grp(An)











75




and πn(ko)



πn(

fib(ko ktop)) Θn


πn(

fib(ktop ksph)) Lq

n(Z)








Cut(αop)(i) =



76





ρi(j) =

1 if i = j

undefined else



(lr)


X1 timesX1

X1

X1 timesX1

flip factors

micro

micro









Cotimesmon Cotimes

∆∆op Lop

potimesmon

potimes

Cut

77



CMon(Cat(2)

1)

Mon(Cat(2)

1)


1

Cutlowast

sim sim

forget




idx 00 idy














78




HomC

(noplusi=1

xi

noplusi=1

yi

)

nprodij=1

HomC(xi yj)





Fun(CAn) minus Fun(


(CGrp(C)Fun(LopAn)

)



)







79





Xi



61[n] ([n]∆∆61)op


0 010 11

01 1

1 1

0 0

0





80


12

n

empty empty









Proof We compute



(Mn timesNm)


(Mn times colim

[m]isin∆∆opNm

)(


Mn

)times(


Nm

) |M | times |N |



81








B CMon(Mon(An)



CMon(Mon(C)

)CMon(C)

Mon(CMon(C)

)

sim

forgetlowastsim

simforget

(II18b2)



CMon(An) CMon(An)

Mon(An) lowastAn

B

CutlowastΩ

ev1

B

Ω


82


















83



πi((BX)1

)



Dgt0(Z) CGrp(An)

Dgt0(Z) CGrp(An)

[1]=Σ

K

BΩ

K

Ω












CMon(CMon(C)

)CMon(C)

simforgetlowast

simforget



84





CMon(C) limnisinNop

Mon(n)(C)










microxtimesy





85










simtimes

idsim

(I) (III)


sim







times

times

sim

times

86










)HomC


)


(V)

microa (VI)microa





sim

sim

microctimesd





LopXn


87







)









XnhSn


FhG colimBG

F


88





∐ngt0X

nhSn

As an example



)= S







(

Ωminus CGrp(An) Ω



limNop

(

Ωminus CGrp(An)gt2

Ωminus CGrp(An)gt1

Ωminus CGrp(An)

)


Sp(C) = limNop

(

Ωminus lowastC Ω

minus lowastC)


89



Sp(CGrp(C)

) simminus Sp(C)

is an equivalence


limNop

(



(Lop limNop

(

Ωminus lowastC Ω

minus lowastC))




) ev1minus Sp(C)








)Grp

(Fun(Dop lowastAn)


ev1

sim

Ω

Ω


90

















91







)C 7minus




Sp(Dgt0(Z)

)Sp(An)

D(Z) Sp

Sp(K)

H


D(Z) Sp


H

Ωinfinminusi




Nop

(


minus (lowastAn)op)

limNop

(

Σminus lowastAn Σ



92



Cbull Sp D(Z) H



HomD(Z)(Cbull(X) D

) limiisinNop


) limiisinNop


) limiisinNop





)= colim



)






93





Ωa Ωc 0

Ωb x g 0

0 f a b

0 c



F =

a b

c

7minus Ωminus1

Ωa Ωb

Ωc x




Nat

a b

c

x x

x

Nat

a b

c d

x x

x x

HomC(d x)



94














infin




limI










95



C Cprime












Sp


HomC

homC

of the HomC functor





)and homK(R)(CD) H

(HomR(CD)

)


ΩinfinH(RHomR(CD)

) K

(RHomR(CD)

) HomAn

(lowastK(RHomR(CD))

) HomD(Z)

(Z[0] RHomR(CD)

) HomD(R)

(Z[0]otimesLZ CD

) HomD(R)(CD)

96














97





PSp(C) limnisinNop

Fun(N26(nn) lowastC

)

Sp(C) limnisinNop

lowastC

sube

sim

η=lim ηn





lowast lowast

lowast lowast


︸︷︷︸n+1

︸︷︷

︸n+1N2

6(nn)






HomlowastC(Xn Yn)




98




ΩnXn+i














Fun(N26(nn) lowastC

)

99




lowast lowast

lowast lowast


︸︷︷︸n+1

︸︷︷

︸

n+1N26(nn)





ΩnΣn(Xx)




minus Sp




πi+n(Sn lowast)





Hlowast(S[X] A


(S[X] A

)= Hminuslowast(XA)

100



CbullS colimiisinN


colimnisinNgti




) colim



colimnisinN

colimiisinN6n


colimnisinN

Hlowast+n(Sn A)












ΩXspi+1 Ω colim



nisinNΩnXn+i Xsp

i

101






limiisinNop

(colimnisinN

ΩnXn+i Yi

) limiisinNop

HomlowastC(

colimnisinNgti


limiisinNop

limnisinNop

gti


) limnisinNop

limjisinN6n


) limnisinNop

HomlowastC(Xn Yn)

HomPSp(C)(XY )






102




(F (S0) F (S1)

)


evS0 Sp(Sp(C)


) simminus Sp(C)



X otimes d = colimX


const d



colimx

const d colimX

d

)




F sp colimnisinN

ΩnF (Σnminus)



103



A C lowast

B D Q lowast

lowast P ΣA ΣB

lowast ΣC ΣD








(colimX

E)


(limXE)





)[minusi]



104












HomO(x y)nprodi=1

HomO(x yi)


) nprodi=1


)

(ρ1ρn)

p p

(ρ1ρn)






Lopint minus


Lop





105





HomO1

((αlowastx)i yi

)

HomαO(x y)

nprodi=1

HomρiαO (x y)

sim

sim sim











(F0F1)





106







C0 C1Cyl[(α)

0 1(∆1)

subeC0 C1

C0

Cyl[(α)

C0 times (∆1)]

0 1(∆1)

subeC0 C1

C0

Cyl[(α)


0 1(∆1)

supeC0 C1

C0


0 1(∆1)



HomactO((a1 an) b

)HomO

((a1 an) b

)lowast HomLop

(〈n〉〈1〉

)

fn

107








)is an equivalence







Opinfin dOpinfin

Catinfin Catinfin

sim

(minus)1 (minus)1

(minus)op






108

















109






) Homact

Cotimes((d1 dn) d



)


)




) Homα





)





) Homα




) HomC


)



) HomD


)


110














111








HomP(E)X(

HomE(eminus) F)

HomP(E)(

HomE(eminus) F)

F (e)

lowast HomP(E)(

HomE(eminus) X)

X(e)

sim

sim





112











(∆3(∆02 cup∆13) X

)






([n]timesLop

Lopt C






HomsAn(∆nNr(Lop)

)HomsAn

(InNr(Lop)

)sim

113







Lopt C

) simminusnprodi=1


Lopt C

)


Lopt in



t )





sAn

sim






[k] Lop

pr1

const 〈n〉

114



Yk FCt(


Lopt C

) HomCatinfin

([k]times 〈n〉 C

) HomCatinfin

([k] Cn







([1]timesLop

Lopt C

)lowast HomCatinfin

([0]timesLop

Lopt C

)timesHomCatinfin

([0]timesLop

Lopt C

) (d1d0)

(xy)



Lopt C



) HomsAnNr(Lop)



([1] Ct


Lop) α




Lopt C


[0]timesLopLopt C

[1]timesLopLopt

d1




Lopt C to be




)

115







) Homfn

Ct((a1 an)minus

)




)

nprodi=1

HomC(aiminus)








([n]timesLop

Loptimes C

)




)


prodiisinS F (i)


infin-operad)

116








lowast Fun(OLop)

pprimelowast

p


prodni=1 F (ai)















117





α





(b) If O is Lopint


FunLop(Lop

int Cotimes) FunLop

int


Lopint)



) Γcocart

(Cotimes timesLop

Lopint)


Lopint) Lop












118









O Cotimes

Oid p






119






HomαCotimes( m An)


HomαCotimes( m An)


HomαCn( n An)


HomC(1C A1)n


timesLop is















120









Fun(CnO1)



Day(CotimesO)ni






Fun(CO1) Fun(CO1)

Fun(CO1) Fun(CnO1)

prlowastiprlowasti

121



Cn C

C C

pri

pri











)

122









(d1d0)

(FG)




Cotimesm Om

Cotimesα Oα

d1





123











HomactDay(CotimesO)

((F1 Fn) G

) Homfn


















124



HomactO

((colimI

F x2 xn

) y) limIop


)Homact

O

((x1 xn) lim

IF) limI













limIop

HomαO

((x1 xn) lim

IF) limI










125








([1]minO




infin we let





minus SpOpinfin(O)










126















127



Day(( op)and C













Lop

timesLop

AtimesB timesSet

AotimesB



128









X times lowast

X or Y










129







E colimnisinN


)[minusn]


HomSp

(colimnisinN


)[minusn] Eprime

) limnisinNop


) limnisinNop







nisinN


)[minusn]






130







)op An minus Spop














131


















132



AlgComm(








LZ)minus CAlg







133





CAlg(






)= TorSlowast

(S[minus] E


)= ExtlowastS

(S[minus] E

)








134






) H

(RHomZ(Cbull(E) A)




)







135




HomactAssoc


lowast)





HomactLMod

((b1 bn) a



HomactLMod

((b1 bn)m

)=



if bi = m





alowast

A


136













Bi otimes


A





∆∆op AssocLop


Cut

minustimes1 a

LCut

p

137












F([n] 0

) (A AM) isin Cn+1


Bi otimes


A and N

( otimesα(n)ltj6m

A

)otimesM





(minustimes1)lowast

Cutlowast(A)

138







F (i)([n] 1

) (A A) and F

([n] 0

)(A AM(i)

)




(i [n] 1) 7minus






colimiisinI

M(i) colimiisinI

M(i)


139









)[1] Fun

(∆∆op times [1] r ([0] 0)Lop)

constF ([0]0)

qlowast


ηF





LModA(Cotimes)

C Cmlowast

Aotimesminus

Aotimesminus


140



















) πjH

(minus [minusi]


141




HomModT(T [i]minus

) HomSp

(S[i]minus

) Ωinfin+i




colimXisinCM

X simminusM


colimXisinCM

π0 HomModT(T [i] X





142











([n]timesO Cotimes








([k]Fun(O Cotimes)




) HomCatinfin


n times

))





143








) FunOpinfin


) Fun

(Comm1CAlg(Cotimes)

) CAlg(Cotimes)




alowast

〈1〉 A






Bar(MAN)




144







P([n])

N([n] 0

)MA

([n] 0

)A([n])







in Fun(Lopact






145



N colim∆∆op

Bar(AAN)


∆∆op ∆∆op

C

σop

B Bprime




colim∆∆op

B colim∆∆op









(MhomR(NL)

)




146



FSR ModS minus ModR













(D(R)otimes





H(R[0]


147











)otimes π0

(S[minusj]otimesM

)minus π0


) microminus π0


)




) πi+j

(S[i+ j]

) Z





148




) Hlowast(MR)




















) Hminuslowast(XR)



149







150

Chapter III









(M

middotsminusM

middotsminusM

middotsminus

)




151




)[sminus1]




M [Tminus1]












is an equivalence

152












m xotimesm m


fotimesidm

fotimesidm

micro

sim






153









)sube HomCAlg(R T )







)LModR

(ModotimesRR


sim


colim∆∆op


Bar(MR[Sminus1] N

)

154









HomCAlg(ModotimesRR

)

(R[Sminus1] T


R)(R T )





] simminus HQ



H D(Q) minus Sp



155












(S[M ] R

)


π0(S[M ]

) H0(MZ) Z

[π0(M)






156





)sube HomCAlg

(S[M ] R






) colim






πi(BSn lowast)

0 if i 6= 1Sn if i = 1







)



) colim



157





middot[1]minusminus

)minus colim

nisinN

(ΩinfinS middot[1]


) ΩinfinS





) colim


whereas


) colim

nisinNH1(SnZ) colim

nisinNSabn Z2Z




Hlowast(k(R)0Z


Hgrplowast(

GLn(R)Z)




Hlowast(k(R)Z

) colim

nisinN

(Hlowast(


(Proj(R)Z


)We have Proj(R)



158




Hlowast(k(R)0Z

) colim


) colim

nisinNHgrplowast(

GLn(R)Z)





Aut(P ) π1(

Proj(R) P)minus π1

(k(R) P

) K1(R)



infin we obtain




0 r


= n + reij







159

















sim

1otimesminus

∆lowast

minusotimes1

pr2lowast

pr1lowast

εotimesid

idotimesε

sim

sim



160






f X minusrprodi=1

K(Q ni)



Hlowast(K(Z n)Q

)= Hlowast

(K(Q n)Q




flowast rotimesi=1

Hlowast(K(Q ni)Q

) Hlowast

(rprodi=1




161







AotimesA AotimesA

A Q A

AotimesA AotimesA

(idAi)

micro

∆

∆

ε u

(iidA)

micro






) Hlowast(MQ)


162




microminus Agt0

)






GLinfin(R)Q)







Ki(OF )otimesQ =




163




T (M s) colimN

(M

sminusM

sminusM

sminus

)




A 7minus

(A 00 1

)



T (M s)

T (M s)

r colimN

M M M

M M M

r

s

τ r

s

τ

s

r

s s s




A 0 00 m 00 0 n


A 0 00 n 00 0 m

164




T (M s)

T (M s)

t colimN

M M M

M M M

s

id

s s

id

s

s

s

s

s

s


T (M s) colimN

(M

snminusM

snminusM

snminus




)HomAn(MnMn) HomAn

((m m)Mn

)Mn


)HomAn(MnM) HomAn

((m m)M

)M


fn



165







Minfin-grp T (M s)





(T (M s) s


minus )


Now we use that





166














S[X] simminus S[X+]





lowast X

167






iisinI S2 X

lowast X+



to check that

HomAn(KHomAn(X+ Z)

)minus π0 HomAn

(KHomAn(XZ)



)= 0








168




)= Flowast

((Xx)Sing(Y y)

)



Y

|X| times |∆n| Yf

exist f



X X+

X X+







169





) HomAn

(YHomAn(XZ)

) HomAn(X times YZ)





HomAn(T (M+ s) Z

) lim

Nop

(


minus HomAn(M+ Z))

limNop

(


minus HomAn(MZ))

HomAn(T (M s)+ Z


T (M s)+ Minfin-grp


sim sim

sim





170




(Proj(R) middot[R]


)+









VectR ∐ngt0


BU(n)


T(VectR [R]

) ZtimesBO and T

(VectC [C]

) ZtimesBU




) T


)

171







)grp





)= (Binfinko)i(X)





R(S8)



172











and πn(ko)





173








[V ] 7minussumigt0

[ΛiV ]ti












174



Elowast(BU(n)



prodnisinZE




ku0(ku) =prodnisinZ

Zqβici

∣∣ i gt 1y



prod06k6m

ku0(Grk(Cm))



λn(x+ y) =sumi+j=n





175



λ(x) =sumigt0



`(t) =sumigt1

(minus1)iminus1ti

iisin QJtK



minus`(λ(x)minus 1

)prime =sumigt0





`(λ(x)minus 1


λ(x)= minus

sumigt0

(1minus λ(x)






)= ker


)



176







minuspminus1sumi=0

(1minus λ(x)


(1minus λ(x)

)p1minus

(1minus λ(x)

) middot λ(x)prime

equiv minus λ(x)p





λ(x)p equivsumigt0






)prime =sumigt0

[L]i+1ti



177



S2n βn

minus ku ψi

minus ku




iminus1sumj=0

[γC1 ]j = iβ





)

Consequently

Ki(Fq) =










178






Hlowast(

fib(ψq minus id)Q)

= 0 and Hlowast(


= 0




)= colim

nisinNHgrplowast(

GLn(Fq)Q)

= 0







179







χ(ψn(V )

)(g) = χ(V )(gn)


χ(λ(V )

)(g) =

dsumm=0


(1minus λit)




sumngt0

χ(ψn+1(V )

)(g)tn =

dsumi=1

λi1minus λit

=sumngt0

dsumi=1

λn+1i tn





Brρ RFp

(GLn(Fq)

)minus RC

(GLn(Fq)

)





180





RFp

(GLn(Fq)

)RC(

GLn(Fq))

RFq(

GLn(Fq))

ku0(BGLn(Fq))

Brρ

repFpotimesFqminus


RFq(

GLn(Fq))minus RFp

(GLn(Fq)

) Brρminus RC

(GLn(Fq)





χρFpotimesFqV

(g) =sum[micro]


ρ(λ)





RFq(



)ψq


bijections



)ψq




)

181










πi+1(Xn) minus πi

(limnisinN

Xn

)minus lim



R1 limnisinN


)= R1 lim

nisinNku1(BGLn(Fq)

)


R1 limnisinN

RC(

GLn(Fq))I

= 0




mn RnIn 0 show RnI



RlimnisinN

(limmisinN

RnImn

) Rlim

nisinN

(R limmisinN

RnImn

) Rlim

nisinNRnI

nn

RlimnisinN

ZoplusRlimnisinN

InInn

182











)cup(

idpartn timesγ)

(n times 0





n times∆1 B

γmiddotα

p

idn timesp(γ)



prodmicroinfin`





Brρ

sim

flowast

Brρ

simsumnminus1k=0

qki

183




on K-theory



k(Fq)0 ku ku

k(Fq)0 ku ku

Brρ

flowast


ψqk

Brρ ψqnminusid






k(Fq)0 ku ku

k(Fq)0 ku ku

Brρ

Froblowast ψp

ψqminusid

ψp

Brρ ψqminusid



Ki(Fp)

Z if i = 0oplus






i times

)



(FreeAlg

D(Z)otimesLZ

(Ftimesp [1]

))


184






Z((pn)i minus 1

)




p-sZs





Ms = limnisinN

MsnM


M M

0 MsnM

sn


185



Ms homR

((Rsinfin)[minus1]M

)

















M Ms


186






sn homR


)minus homR




(RsnhomR(MN)

) homR

(RsnhomR(Ms N)





homR

(R[sminus1]M

) homR(RM) M



R R R

0 Rsn Rsn+1

0 Rs

sn

s

s



Ms limnisinN

(M otimesR Rsn) 0

187




RHomR

(R[sminus1]K

) 0






R R[sminus1]

0 C[1]









Ks RlimnisinN

(K otimesLR Rsn

)

188















A[pn]








RlimnisinN

A[pn] 0


189






(πiminus1(E)

)minus 0





πi(EpnE) minus 0




0 minus Λp(πi(E)

)minus lim


(πiminus1(E)

)minus 0


0

0 R1 limnisinNop


)Λp(πi(E)

)0

0 R1 limnisinNop


πi(EpnE) 0

L1Λp(πiminus1(E)

)0

sim


190









Eprodp

Ep

E otimesHQ

(prodp

Ep

)otimesHQ







)

191











πi((Binfinku)) =




L0Λ`

(oplusq 6=p

microqinfin

)= 0 and L1Λ`

(oplusq 6=p

microqinfin

)= Z`


πi(K(Fp)) =


192









193

Chapter IV








K1(Z) K1(Q)oplus


plusmn1 Qtimesoplus

p primeZ Z Zsim



p primeZ minus 0

194





Then we can put

[M ] =nsumi=0







simminus K0(R)




K noplus


noplusi=minusm

Pi[i]


i=minusm+1Pi[i]


195








c minus d minus c



K0(D) minus K0(C)



CH =x isin C









Cmin = C0 =x isin C

∣∣ [x] = 0 in K0(C)


196



HD = im(K0(D) K0(C)











197








Sn(C) sube Fun(

Ar([n]) C)



(0 6 0)

(0 6 1) (0 6 2) (0 6 3) (0 6 4) (0 6 nminus 1)(0 6 n)

(1 6 n)

(2 6 n)

(3 6 n)

(nminus 2 6 n)

(nminus 1 6 n)

(n 6 n)

(1 6 1)

(2 6 2)

(3 6 3)



198







infin


0 a b

0 ba

0







K0(C) π0(k(C)

) π1

(|core(C)|

)








199







(minus [n]











(IV71)


Sn timesXn+1 |sknX|










200




core(C) t lowast)








Ztimes π1(




Ztimes π1(

coreS2(C) s)


π1(

coreS2(C) s)

pr2




201



π1(|core(C)|

)= π1

(colimnisinN

|skn coreS2(C)|)

= colimnisinN

π1(|sk2 coreS2(C)|

)= π1

(|sk2 coreS2(C)|

)


| | colim∆∆op

inj







Nr(C)esd Nr(

TwAr(Cop))


|X| |Xesd|








202





(0` 6 0`)

(0` 6 1`) (0` 6 2`) (0` 6 2r) (0` 6 1r)(0` 6 0r)

(1` 6 0r)

(2` 6 0r)

(2r 6 0r)

(1r 6 0r)

(0r 6 0r)

(1` 6 1`)

(2` 6 2`)

(2r 6 2r)

(1r 6 1r)





Fun(In C) Fun(

Ar([n] [n]op) C)



Lan

Ran Ran

Lan



203

























minus k(Dperf(R)

)



204
















205





0 x 0

z y z
















k coregrp



206






A B

C D




infin E) aredenoted






infin E)





0 C



Atimes C C

A 0

207




infinCMon(E)) CMon

(Funadd(Catst

infin E))




infinCGrp(E)) CGrp

(Funadd(Catst

infin E))

























HomCatstinfin


(B C)


HomCatstinfin

(cofib(f) C

) HomCatst



infin(B C)











A B Cf p

g q

or A B Cf p

gprime qprime


C B Aq g

p f


p f

209



A B Cf p

g

gprime

q

qprime


b fg(b)



qp(b) b







g(g(a)

) cofib


) g(a)


qF (a) a g(a)

qF(g(a)

)g(a) g

(g(a)

)qF (ηa) ηa

sim

g(ηa)

210






b

b c b

b0

s







) Homim(p)

(p(b) p(fib(ϕ))

)



A B Cf p

g q

211



C B Aq g

p f





Finally suppose

A B Cf p

g

gprime

q

qprime


p(









212




qp(b) b qprimep(b)

0 fg(b)



A B

C D




A B Cf p

g

gprime

q

qprime


B Ar(A)

C A

grArrcp

t

c

213



A Ar(A) AidArArr0 t

s

fib(minus)

0rArridA

idArArridA





SotimesRminus

homR(Sminus)

incl

homR(Iminus)











214



n and in general




divs(M) = homR

(R[sminus1]M

) lim

Nop

(

sminusM

sminusM

sminusM




simsim Mods-comp



M Ms






R








215




[0] [nminus i]

[i] [n]

0

i +i


Qnminusi(C) Q0(C)




)










216






gf(x) gf(y)

x y

gf(ϕ)

c c

ϕ


fibration



gf(x) g(yprime)

x y

c ϕ


xprime yprime

x y

ϕprime

f(ϕ)



217








x z v

y bull

w

(1) (2)

(3) (4)

p7minus

p(x) p(z) p(v)

p(y) u

p(w)




E C

Q1(D) D

p

d1

p

d1


218







p

(idE d1)

p

(d2d1)





F (E) minus F(Q1(C)

)minus HomCatinfin

([1]SpanF (C)




A B

C D

p

219





|D|

F | |

|F |




|p|




| |minus An

) colim

(|D| |F |minus An

) Unleft (|F |)




A B

C D



220



F(Q(A)

)F(Q(B)

)F(Q(C)

)F(Q(D)

)







xy1minus s1

y2 minus s2y3minus

y2n minusminus s2nw

z




F(


(Qn(C)

) F (C)2n+1



221



F (C)5 F(Q1(C)

)timesd1F (C)d0 F

(Q1(C)

)F(Q2(C)

) d1minus F(Q1(C)

) F (C)3(d0d2)

sim













F(Q1(C)

)times F (Q1(C))




∣∣ ∣∣F (Q(C))∣∣ ∣∣F (S(C)

)esd∣∣ ∣∣F (S(C))∣∣


222



C Q1(C)

0 Q0(C)timesQ0(C)

(d1d0)



F (C) F(Q0(C)

) simminus F(Q1(C)

)times F (C)3





lowast SpanF (C)

0


F (C) lowast


∣∣ 0

0




∣∣ BF (C)



223






infinAn)






A B

C D

σ τ


Bi Bj

Di Dj


colimiisinI

A colimiisinI

Bi

colimiisinI

Ci colimiisinI

Di

σ

τ

is a pullback in An

224




Bi K

Di colimiisinI

Di



F colimiisinI

Di minus An


Fun(

colimiisinI

DiAn) limiisinIop


AnDi



Iop minus An




F colimiisinI

colimDi

Di

colimiisinI

Unleft(Fi)

colimiisinI

Bi


225





Bi


Di

τ



Aj colimiisinI


Ci colimiisinI

Bi

Cj colimiisinI

Ci colimiisinI

Di

τ



asscat(F (Null(C))

) 0 SpanF (C)




asscat(


x asscat(X)

asscat(X)

d0

sim

226





0



F(

Null(C)) fib0


)


constF (C) F(

Null(C))

const lowast F(Q(C)

)F (d0)

0



0 0 0 0 0

x 0 0 0

x 0 0

x 0

x


227






∣∣F (Q(C))∣∣

|F (d0)|

0


asscat(F (Null(C))


(F (Q(C))

) SpanF (C)



F(

Nullm(C))

F(

Nulln(C))

F(Qm(C)

)F(Qn(C)

)






228




(Dperf(R)





Ω∣∣F (Q(minus)






infin minus PSp



infinPSp)











229






Q(f) Null(B)

Q(A) Q(B)

d0

f









)∣∣ minus ∣∣F (Q(B))∣∣




equivalent

230
















231








HomD(x cofib(c y)

)






)plowast



LF (x) colimcisinCx

F(

cofib(c x))


F (x) colimcisinCx


F(

cofib(c x))


232




HomD(x cofib(c y)

)



ϕlowastc x 0

c y cofib(x y)

0 cofib(c y)

ϕ



F(


cprimeisinCxF(





233











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b1

b2nminus1

b2n



234




coreQnQ(i) (

Nr FunA(JnB))esd





(Q(i)












(Q(i)


A2n+1



b1 b3 b2nminus1


235



dj(a0 a2n) =









(Dperf(R)

)minus k

(Dperf(S)

)



p primeK(Fp) K

(Dperf(Z)Q-tors)




Catstinfin = Catst


236


















infin Catstinfin




infin Catstinfin as








(minus) Catstinfin

simminus Catstinfin





237

























238













239














HomCatstinfin



(DopSp)


HomCatstinfin


(EopSp)


HomCatstinfin


(EopSp)

240





disimminus e
















241









(Rsn)[minus1]

















M colimjisinJ


Ci otimesR Dj


R )

242





















(Dperf(Z)

)minus K

(Dperf(Q)


243















244













ibprime J ib r

s



245







G(R) = K(Daperf(R)

)


G(R) K(R)




m






R

246



Rm








K

(oplusm


(Dperf(R)tors)


mK(Dperf(Rm)) oplus





infinAn)






247
















248

Bibliography
















249

Bibliography




250

List of Lectures

Introduction


A Fairytale









Kan Extensions











Day Convolution






Miscellanea





Rational K-Theory










Verdier Sequences




Bibliography

AlgebraicandHermitian K-TheoryList of Lectures Lecture21(21st January,2021)164 The Quillen plus...

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Transcript of AlgebraicandHermitian K-TheoryList of Lectures Lecture21(21st January,2021)164 The Quillen plus...