SKETCH OF A PROOF OF A DEVISSAGE THEOREM

C. BARWICK

This is an outline of some of the work to appear in more complete form later this fall in joint work withJohn Rognes.

1. Exact ∞-categories

Definition 1.1. A triple C = (C ,C †,C†) consists of an ∞-category C and two subcategories C † ⊂ C andC† ⊂ C , each of which contains all the equivalences of C . A map of C † will be said to be egressive or afibration, and a map of C† will be said to be ingressive or a cofibration. A map of C † ∩ C† will be calledbigressive or a bifibration. A fibration will be denoted X Y , and a cofibation will be denoted X Y .

If C and D are triples, a functor of triples C D is simply a functor that preserves both egressive andingressive maps. A functor f : C D of triples is said to be strict if the squares

C † C

D† D

and

C† C

D† D

are pullbacks.A pullback (respectively, pushout) square

X0 X1

X2 X∞

in C is said to be ambigressive if X2 X∞ is ingressive and X1 X∞ is egressive (resp., if X0 X1

is ingressive and X0 X2 is egressive).An exact ∞-category is a triple A = (A ,A †,A†) that satisfies the following four hypotheses.

(1) The ∞-category A contains a zero object.(2) The ∞-category A † admits a terminal object, and the inclusion A † A preserves it; dually, the∞-category A† admits an initial object, and the inclusion A† A preserves it.

(3) Pullbacks of fibrations exist and are fibrations; dually, pushouts of cofibrations exist and are cofibrations.(4) A square in A is an ambigressive pullback if and only if it is an ambigressive pushout.

Suppose that A and B are exact ∞-categories. A functor f : A B of triples is said to be exact iff sends a zero object to a zero object and if f carries ambigressive pushout squares in A to ambigressivepushout squares in B.

It is not hard to see that in an exact ∞-category, a pushout square of the form

X0 X1

X2 X∞

1

2 C. BARWICK

is also a pullback square. Dually, a pullback square of the form

X0 X1

X2 X∞

is also a pushout square.

Example. (1) The nerve NE of an ordinary category E can be endowed with a triple structure yielding anexact ∞-category if and only if E is an (ordinary) exact category in the sense of Quillen, wherein theadmissible monomorphisms are exactly the cofibrations, and the admissible epimorphisms are exactlythe fibrations.

(2) At the other extreme, any stable ∞-category is an exact ∞-category in which all morphisms are bothegressive and ingressive, and, conversely, any∞-category that can be regarded as an exact category withthe maximal triple structure (in which all maps are bigressive) is a stable.

2. K-theory via the Q construction

Consider the ∞-category Exact∞ of exact ∞-categories. One of our main results is that the algebraicK-theory functor K : Exact∞ Sp can be related to a Q construction in the following manner.

Notation. For any simplicial set T , write O(T ) for the edgewise subdivision of T , whose value at [n] ∈ ∆op

is the value of X at [n]op ∗ [n] ∼= [2n+ 1] ∈ ∆op; this description makes it clear what the behavior on maps in

∆op is. When T is an ∞-category, we call O(T ) the twisted arrow ∞-category ; this terminology is justifiedby the following easy lemma.

Lemma 2.1. When T in an∞-category, O(T ) is an∞-category as well. When T is the nerve of an ordinary

category C, the simplicial set O(T ) is naturally isomorphic to the nerve of the twisted arrow category of C.

When T in an ∞-category, the ∞-category O(T ) can thus be described in the following manner. Theobjects are the maps of T , and a morphism from an object X Y to an object X ′ Y ′ is a diagram

X X ′

Y Y ′

We can give O(T ) the structure of a triple by declaring maps of the form above to be egressive if Y Y ′ isan equivalence and ingressive if X ′ X is an equivalence. More relevant to our purposes will be the dual

triple W (T ) := O(T )op with W (T )† :=(O(T )†

)op

and W (T )† :=(O(T )†

)op

.

Definition 2.2. For any exact∞-category C , define QC as the simplicial set whose n-simplices are functorsof triples

X : W (∆n) C

SKETCH OF A PROOF OF A DEVISSAGE THEOREM 3

that carry ambigressive pullback squares to ambigressive pullback squares. Thus an n-simplex of QC is adiagram

X00

X01 3 X10

. ..

3. . .. .

.3

. . .

X02 3 X13 3 X31 3 X20

X01 3 X12 3. . .. .

.3 X21 3 X10

X00 X11 X22 X22 X11 X00

(Here we write n for p− n.)

Lemma 2.3. For any exact ∞-category C , the space QC is an ∞-category.

Theorem 2.4 (Comparison). The functor K := Ω∞K : Exact∞ S is naturally equivalent to the functorΩ Ex∞Q : Exact∞ S .

3. The general devissage theorem

Now we can work toward a devissage statement in the style of Quillen. The main point is that we seeka statement that will hold even in the absence of fully faithful hypotheses; this forces us to introduce someadditional technicalities.

The thrust of the argument is that if f : A B is a suitable exact functor of exact ∞-categories, andif every object of B admits a certain kind of finite length filtration whose associated graded lifts to A , thenall of the∞-categories (Qf/X) := (B/X)×B A are weakly equivalent. From this it follows that the functorQf : QA QB is cofinal if and only if (Qf/0) is weakly contractible; we will have to impose conditionsthat will ensure this.

We begin with some preliminary definitions that will be used to provide some of the technical hypothesesfor the devissage theorem. We begin by restricting the sorts of exact ∞-categories we will allow.

Definition 3.1. An exact ∞-category A is said to be tame if it satisfies the following two conditions.

(1) Pullbacks in A† exist and are preserved by the inclusion A† A .(2) A section of any cofibration is itself a cofibration.

Abelian categories are tame, since a monomorphism that admits a section is an isomorphism. Stable∞-categories are clearly also tame.

We now turn to the conditions on the sorts of exact functors with which we wish to deal.

Definition 3.2. Suppose A an exact ∞-category. A morphism Y X of A is said to be a counit if itfits into a commutative diagram

K Z

0 Y

X

4 C. BARWICK

in which

K Z

0 X

is a pullback square, and

K Z

0 Y

is an ambigressive pullback square.

Dually, a morphism X Y of A is said to be a unit if it fits into a commutative diagram

X

Y Z

0 C

in which

X Z

0 C

is a pushout square, and

Y Z

0 C

is an ambigressive pushout square.

In an exact ∞-category, the collection of counits (resp., of units) forms a subcategory. If a given counit(resp., a given unit) admits a kernel (resp., a cokernel), then that kernel (resp., that cokernel) is 0. Oneconcludes that an egressive counit (resp., an ingressive unit) is an equivalence.

Observe that, in an abelian category, any cofibration is a counit, and any fibration is a unit. On the otherhand, in a stable ∞-category, the only counits or units are equivalences.

Definition 3.3. Suppose now that A and B are tame exact ∞-categories, and suppose that f : A Bis an exact functor. Then f is said to be devissable if the following conditions are satisfied.

(1) The underlying functor f is conservative; that is, a map φ of A is an equivalence if and only if its imagef(φ) in B is so.

(2) The functor of triples f is strict.(3) For any object X of A , the canonical morphism fΩX ΩfX is an equivalence.(4) Every object X of B admits a filtration

0 = X0 X1 · · · Xn−1 Xn = X,

each of whose successive quotients Xi/Xi−1 can each be written as f(Yi) for some object Yi of A .(5) For any functor X : K B, denote by Fun(K,B)c/X the full subcategory of Fun(K,B)/X spanned by

those natural transformations Y X such that for any object k ∈ K, the morphism Yk Xk is acounit. Then for any functor X : K A , the functor

Fun(K,A )/X ×Fun(K,B)/fXFun(K,B)c

/fX Fun(K,B)c/fX

is an equivalence.

Happily, the first three conditions are considerably milder than full faithfulness. The price we have to payfor this added generality is the rather subtle final condition, which seems difficult to motivate. In effect, itasserts that for any diagram X of B and any lift X of that diagram to A , any objectwise counit Y Xcan be lifted to a map Y X in an essentially unique manner.

For abelian categories A and B, this just amounts to the condition that A be closed under formation ofsubobjects in B, in the sense that if an object X of B lifts to an object X of A , then the subobjects of Xlift in an essentially unique fashion to subobjects of X. If B is a stable ∞-category, this is no condition.

Lemma 3.4. Suppose A and B two tame exact ∞-categories. Suppose f : A B an exact, devissablefunctor. Then the ∞-categories (Qf/X) are all weakly equivalent.

SKETCH OF A PROOF OF A DEVISSAGE THEOREM 5

Sketch of proof. The condition, combined with an induction argument, implies that it is enough to show thatif φ : X ′ X is a cofibration of B whose cofiber admits a lift to A , then (Qf/X ′) and (Qf/X) are weaklyequivalent.

We’ll use the identification of (Qf/X) with the following ∞-category J(X). The objects of J(X) arediagrams

X1 X2 X

2

0 X2/1

along with a lift of X2/1 to an object X2/1 of A. In situations in which the choice of object X2/1 and its lift

X2/1 is obvious, we may simply write

X1 X2 X

for this object. A map of J(X)

from an object

X1 X2 X

2

0 X2/1

to an object

X ′1 X ′2 X

2

0 X2′/1′

is a diagram

X ′1 X1 X2 X ′2 X

2 2 2

0 X1/X′1 X2/1′ X2′/1′

2

0 X2/1

along with a lift of

X2/1′ X2′/1′

X2/1

to a diagram

X2/1′ X2′/1′

X2/1

of A.

In situations in which the choice of objectsX2/1,X2/1′ ,X2′/1′ , and the liftsX2/1′ X2/1 andX2/1′ X2′/1′ ,is obvious, we may simply write

X ′1 X1 X2 X ′2 X

for this map.Now suppose φ : X ′ X a small map with a fixed lift X/X ′ of the cofiber X/X ′ to A . The cofiber

Y/Y ′ of the pullback Y ′ Y of any cofibration Y X along φ therefore admits a lift Y/Y ′ to A ; wewill use this observation repeatedly without comment.

We’ll now define four functors:

φ! : J(X ′) J(X) φ? : J(X) J(X ′)

λφ : J(X) J(X) µφ : J(X ′) J(X ′)

and four natural transformations:

α : φ!φ? λφ β : idJ(X) λφ

γ : φ?φ! µφ δ : idJ(X′) µφ

6 C. BARWICK

These will define maps

φ! : Ex∞ J(X ′) Ex∞ J(X) φ? : Ex∞ J(X) Ex∞ J(X ′)

λφ : Ex∞ J(X) Ex∞ J(X) µφ : Ex∞ J(X ′) Ex∞ J(X ′)

as well as four homotopies:

α : φ!φ? ' λφ β : idJ(X) ' λφ

γ : φ?φ! ' µφ δ : idJ(X′) ' µφ,

thereby completing the proof.First, the functor φ! carries

an object

X1 X2 X ′

2

0 X2/1

to the object

X1 X2 X

2

0 X2/1

The choice of lift of X2/1 does not change. The behavior under morphisms here is clear.The functor φ? carries

an object

X1 X2 X

2

0 X2/1

to the object

X1 ×X X ′ X2 ×X X ′ X ′

2

0 (X2 ×X X ′)/(X1 ×X X ′)

The choice of lift of (X2×XX ′)/(X1×XX ′) is guaranteed by the fact that (X2 ×X X ′)/(X1 ×X X ′) X2/1

is a counit. Once again the behavior under morphisms is clear.Now the functor λφ carries

an object

X1 X2 X

2

0 X2/1

to the object

X1 ×X X ′ X2 X

2

0 X2/(X1 ×X X ′),

where X2/(X1 ×X X ′) is lifted in a canonical manner, since X2/(X1 ×X X ′) X2/1 ∨X2/(X2 ×X X ′) isa counit, the target of which admits a canonical lift.

If

X ′1 X1 X2 X ′2 X

2 2 2

0 X1/X′1 X2/1′ X2′/1′

2

0 X2/1

SKETCH OF A PROOF OF A DEVISSAGE THEOREM 7

is a map in J(X), then we have the diagram

X ′1 ×X X ′ X1 ×X X ′ X2 X ′2 X

2 2 2

0 (X1 ×X X ′)/(X ′1 ×X X ′) X2/(X′1 ×X X ′) X ′2/(X

′1 ×X X ′)

2

0 X2/(X1 ×X X ′)

Since the vertical morphisms in the diagram

X2/(X1 ×X X ′) X2/(X′1 ×X X ′) X ′2/(X

′1 ×X X ′)

X2/1 ∨X2/(X2 ×X X ′) X2/1′ ∨X2/(X2 ×X X ′) X2′/1′ ∨X ′2/(X ′2 ×X X ′)

are all counits, and since there is a selected lift of the bottom row to A , there is a canonical lift of the toprow to A as well.

The last of our functors is µφ; it carries

an object

X1 X2 X ′

2

0 X2/1

to the object

X1 X2 ×X X ′ X ′

2

0 X2/1 ∨ Ω(X2/X2 ×X X ′)

where X2/1 ∨Ω(X2/(X2×X X ′)) is lifted to X2/1 ∨Ω(X2/(X2 ×X X ′)) in A. The map X1 X2 ×X X ′ isingressive since it is the composite of X1 X2 along with the map X2 X2 ×X X ′, which is a sectionof the projection X2 ×X X ′ X2. This provides a splitting of the cofiber sequence X2/1 (X2 ×XX ′)/X1 Ω(X2/(X2 ×X X ′)). If

X ′1 X1 X2 X ′2 X ′

2 2 2

0 X1/X′1 X2/1′ X2′/1′

2

0 X2/1

is a map in J(X ′), then the squares of the diagram

X ′1 X1 X2 ×X X ′ X ′2 ×X X ′ X ′

2 2 2

0 X1/X′1 X2/1′ ∨ Ω(X2/(X2 ×X X ′)) X2′/1′ ∨ Ω(X ′2/(X

′2 ×X X ′))

2

0 X2/1 ∨ Ω(X2/(X2 ×X X ′))

8 C. BARWICK

are all cartesian, and we have a canonical lift of

X2/1 ∨ Ω(X2/(X2 ×X X ′)) X2/1′ ∨ Ω(X2/(X2 ×X X ′)) X2′/1′ ∨ Ω(X ′2/(X′2 ×X X ′))

Now it is time to describe the natural transformations α, β, γ, δ. For brevity, we’ll describe them compo-nentwise, omitting mention of the various quotients or their lifts to A; in each case these objects are specifiedin the obvious way.

On J(X), the natural transformation α carries

an object X1 X2 X to the map X1 ×X X ′ X1 ×X X ′ X2 ×X X ′ X2 X;

the natural transformation β carries

an object X1 X2 X to the map X1 ×X X ′ X1 X2 X2 X.

On J(X ′), the natural transformation γ carries an object

an object X1 X2 X ′ to the map X1 X1 ×X X ′ X2 ×X X ′ X2 ×X X ′ X ′;

the natural transformation δ carries an object

an object X1 X2 X ′ to the map X1 X1 X2 X2 ×X X ′ X ′.

Definition 3.5. An object X of a tame exact ∞-category is said to be subzero if the morphism X 0 isa cofibration.

One checks easily that the full subcategory A z of a tame exact ∞-category A spanned by the subzeroobjects is a semistable ∞-category: in A z, every morphism is a cofibration, and the natural map X ΩΣXis an equivalence for any object X.

In an abelian category, only 0 itself is subzero. By contrast, in a stable∞-category, every object is subzero.

Definition 3.6. Suppose A and B tame exact ∞-categories, and suppose f : A B an exact functor.Then denote by Πf the ∞-category whose objects are the subzero objects of B, in which a morphism froman object X ′ to an object X is an ambigressive pullback

X ′ X

2

Y ′ 0

in B along with a lift Y ′ of Y ′ to A .

The direct sum of objects in B endows Ex∞Πf with the structure of an E∞ space. It is easy to see thatif A and B are tame exact ∞-categories, and f : A B is a conservative, exact functor, then Ex∞Πf isthe cofiber of the map ιA z ιBz in the ∞-category of E∞ spaces, and thus ΩB Ex∞Πf is the cofiberof the map K⊕(A z) K⊕(Bz) in the ∞-category of connective spectra.

The additional hypothesis needed for our devissage theorem is the weak contractibility of Πf . Of courseΠf is weakly contractible if ιA z ιBz is an equivalence, but this is not a necessary condition. We actuallyneed slightly less.

Theorem 3.7 (General devissage). Suppose A and B a tame exact ∞-categories. Suppose f : A Ban exact, devissable functor. Then f induces a cofinal functor QA QB if and only if the following twoconditions are satisfied.

(1) For every subzero object X of B, there exists a subzero object Y of A such that X ∨ fY lifts to A .(2) For any integer i ≥ 1, the induced homomorphism K⊕i (A z) K⊕i (Bz) is an isomorphism.

Sketch of proof. We’ll use the Joyal-Quillen Theorem A [HTT, 4.1.3.1]. The objective is thus to show thatthese conditions are satisfied if and only if, for any object X of B, the over ∞-category (Qf/X) is weaklycontractible.

SKETCH OF A PROOF OF A DEVISSAGE THEOREM 9

The previous lemma, along with the existence of “small” filtrations and an induction argument, impliesthat it’s now enough to check that (Qf/0) is weakly contractible if and only if our conditions are satisfied.To see this, consider the full subcategory R spanned by objects of the form

X 0.

0

Write i for the inclusion R (Qf/0). The ∞-category R can obviously be identified with the ∞-categoryΠf described above. We claim that i is a weak equivalence of simplicial sets. For this, let r : (Qf/0) Rbe the functor defined by the formula

r

X 0

Y

=

X 0

0

.

This functor has the property that ri ' id; moreover there is an obvious natural transformation i r id.It follows that i and r define a homotopy equivalence Ex∞(Qf/0) ' Ex∞R ' Ex∞Πf .

We now claim that the weak contractibility of Πf is equivalent to our conditions. Clearly the first conditionis equivalent to the vanishing of π0 Ex∞Πf . If the first condition holds, the second condition implies thatΩB Ex∞Πf is contractible, which suffices to show that Ex∞Πf vanishes by the group completion theorem.On the other hand, the weak contractibility of Πf implies the weak contractibility of ΩB Ex∞Πf , which inturn implies the second condition.

One way to describe this result is to say that it provides conditions under which statements about thedirect-sum K-theory of subzero objects can be lifted to statements about the K-theory of the entire ∞-category.

If the functor f has the additional property that it induces a homotopy monomorphism ιA z ιBz,then we can dispense with the second of the two conditions listed, as follows.

Corollary 3.8 (Special devissage). Suppose A and B tame exact ∞-categories. Suppose f : A B anexact, devissable functor that induces a homotopy monomorphism ιA z ιBz. Then f induces a cofinalfunctor QA QB if and only if, for every subzero object X of B, there exists a subzero object Y of Asuch that X ∨ fY lifts to A .

Let us look at the results in the two extremal cases. First, since no nonzero objects are subzero in anabelian category, we recover a slight generalization of Quillen’s devissage theorem.

Corollary 3.9. Suppose f : A B a conservative, exact functor between abelian categories. Suppose thatA is closed under subobjects in B, and suppose that every object X of B admits a filtration

0 = X0 X1 · · · Xn−1 Xn = X,

each of whose successive quotients Xi/Xi−1 can each be written as f(Yi) for some object Yi of A . Then finduces an equivalence of K-theory spectra K(A ) ∼ K(B).

Since every object of a stable ∞-category is a subzero object, our result is rather less satisfying in thiscase.

Corollary 3.10. Suppose f : A B a conservative, exact functor between stable ∞-categories. Supposethat every object X of B admits a filtration

0 = X0 X1 · · · Xn−1 Xn = X,

each of whose successive quotients Xi/Xi−1 can each be written as f(Yi) for some object Yi of A . Finally,suppose that the following two conditions are satisfied.

(1) For every object X of B, there exists an object Y of A such that X ∨ fY lifts to A .(2) For any integer i ≥ 1, the induced homomorphism K⊕i (A ) K⊕i (B) is an isomorphism.

10 C. BARWICK

Then f induces an equivalence of K-theory spectra K(A ) ∼ K(B).

4. A conjecture

I expect a rather more general devissage theorem.

Conjecture 4.1. Suppose A and B tame exact∞-categories, and suppose f : A B an exact, devissablefunctor that induces a homotopy monomorphism ιA z ιBz. Then f induces an equivalence of K-theoryspectra K(A ) ∼ K(B).

A strategy for proving this might proceed as follows. For any functor satisfying the hypotheses above,construct a new tame exact ∞-category E (f) along with a functor j : B E (f). Now one might hope toshow that Qj and Q(fj) are both cofinal. Significantly, cofinal functors do not satisfy the 2-out-of-3 axiom,so these assertions do not imply that Qf is cofinal.