Is.I

Lasttohe Limits of functors . A limit of a functor F : I - B is an

object L off and a collection of morphisms htt : Le Fli ) lies . so that

it i -8J c- It Ig÷gyYpig , commutes,ie ( L ,Misia, ) in a cone overF,

and the pair ( L , { Mi :L→ Flik ice ) has the appropriate universal property :tf ee fo and any set of morphisms { fi :c- Fli ) lie. so so that t it j e- I,

C

retry,

commutes,i.e

.

, (c , Hi :c → Feistier.) is a cone on F

,

c te LF ! f : c - L such that f ,} In, commutes tieIo

.

Fli)

More abstractly a limit of a functor F : I - to a'

a terminal object in thecategory Cone ( F1 of cones on F ( and so the ldncit An f a uniqueup to a unique isomorphism in cone (f ) )

.

Spew.

I in discrete lie . I only has the identity morphisms) . Thenldn (f : I - 8 ) is the product III.Flit of the collection h Flik , ←Eo .

Anotherspecialcase.me have a category G, a, bebo,f , g : a- b two morphisms .

Then I -- h andgsb2idb4 is a subcategory of 8 which comes with the

inclusion functor e : I - E .The limit of F (when it exists ) is

, bydefinition the equalizer of a Ig b : it's an object eel ,a morphism e Isa so that fop - gop which has the followinguniversal property :

z! ! Is a be tf a

Exercise An equaliser eEea of a Ig' b -

is monic .

152

Pullbacks ( fiber products .-

Definition Let to be a category , a f- tee three objects and two morphisms .These data define a subcategory I

-

- §¥fq←g qq.de of 8 and the

inclusion functor F : In 8.

The limit of f cwhen it exists ) is called the feberprodncte of a.Iob Ec.

In more detail,It's an object axbc a axqb.ge of b and three morphisms

Ma : axbc-a, Mb ii. axbc - b , Tlc : axbc- c so that axbcE

c commutes

Mad ga - b

f-

and the appropriate universal property hold :S .

Note that since Mb -- fora -- gonewe may omit this morphism .

With this caveat the universal property reads :He I.e.

, given debo , kid - c , Ua :D- a so that

d -#X fo la = go Cee F! 4 : de c so that Meal = Cec2. y

--

s axbc - Cand Mao 4 - Ya .£4 Is

af- b .

Example Fiber products exist in the category Set of sets and functions ,

given× qg in Sed let Xxzy - exist Xx Y / text -- gosh .

Define tix : XxzY -X , Tey : Xxzy - Y by restrictingthe projections : tix (x is) - x , My Cay) -- y for all x. 9) C- X*zY .We check the universal property : given two functions ex : W- X

, Y : W- Y

with fo of -- goofy F ! 4 : W- Xx Y with 10×04=4 , pyo 4--4( by the universal property of the product (Xx Y , 9 Px : xx Y - X, py : Xx Y - YE ) )

.

Since folk - go Yy , t we W 4 Cw) - @× Cw), Uy (w )) satisfies

fog (w) -- go 441W) , ie . Ucw) t XXZY .

Remade c) axf.b.ge depends on f and g even if we omit it in notation.

21 Given X f- Y Y in Set the fiber product Xyy,iY"

is" f-

'

(Y) :

f-'ft) Ey 153i I did

X g- Y commutes ( where i : f-' '

Hl-X is the inclusion ) YV

and given W 4- X,W -94 so that fo 4× - Yy , then f (Ux cut ) = hey H)

for all we w . Hence 4× Cw) e f- '

(Y) , which gives us 9±4× : wetly).

Note that f-'(Y ) # { *→ c- Xx Y l text -- y 's ( as sets) . But limits are unique

only up to an isomorphism and h : f- '

(Y)- 34,931 fat --S4,had = (x , fGI) in a

bijection .

3) If f :X - Z is the inclusion of a subsets and g : Y - Z in a function then

Xxqz ,gY =L y c- Y l GH c- XG = g-'

(x) , whith tix : g-'

(Xl -X and Ty : g-'

IX)- Y

being g and the inclusion , respectively : g-'

(x) y

ar f bX Cf Z .

Example Fiber products exist in top, the category of topological spaces andcontinuous maps : gun XI Z g- Y ni top we define

Xxf,z,gY - s dis) C-XxY l fcxtgcyjl E Xxi and give it the subspace

topology ( the topology on Xx Y is the product topology , of course ) .The space Xxz Y come with evident continuous maps tix ! Xxx Z-X,Ty ! Xxz Y - Y ( as in the case of sets ) . Check that

( X * z Y , fix ,Mys ) has the desired universal property.--

Definition A category to is complete if for any functor F : I -T, where I

is a small category , the limit of F exists .

Theoretic Suppose a category G has all equalizers and all smallproducts . Then 8 in complete .

< proof next time>

Exampled The category Set of sets has products and quakers .

15.4

Hence by 15,1 Set has all small ban its,ie

. Set in complete .

The category Tpe has equalizers and small products . Hence top is

complete .

The category Vest has equalizers and small products . Hence Ved is

complete .

Lett's check that the category Groupe of groups is complete . If I Gi lies in

a family of groups indexed by a set I, thenIII Gi = G n : I - U Gi

,Nile Gi tis

is a group under' " coordinate - wise " multiplication .

if G , K are twogroups and f,ohio TG - K are two homomorphisms

consider L -- 2 ge G l f Cg) -- hcg) 4 .Then L-40 since the identity ee E L . Also

, given a, BE Lf-Cale heal

, f Cbl-- hcbl

.

⇒ flab" ) - feat#b)5 ' -- health (BD" -- hCab

" ).

⇒ abt C-L.

.

.. L in a subgroup of LG

.

It's easy to check that the inclusion map M : L- G,that = a txe L

makes L into the equalizer of G ⇒ K :

For any group N and

for any homomorphism 4 : N→G with foie -- holewe have f 1641 ) - h ( 6h11 the N .

→ the image of y ins in L.Hence

⇒ &yanH.

N

we conclude that Groupe has all small limits .