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Lattin we proved that if a category G has all equalizers and all small productsthen it has all small limits : for any functor f : I e f from a small

category I the limit ldnf exists.

We now take up colimits . They are defined by dualiz.mg limits :

let F ' I- to be a functor from a category I to a category L. Write FOP for the

corresponding functor for : IP→ b?PA cocaine on F is a one on FOP in GOP

,

a colimit of F is a limit of Food.In more details :

Definition A cocaine on a functor F : I -8 is a pair (c , hi : Rli) - c sie⇒ )where C C- too

,and qi 's are morphisms in f so that t i-j c- I, the diagram

C

p%¥g¥Yj, commutes- A morphism of cocones from (c,# s ) to K's i's )

c f- Clis a morphism fi. c- c' ni b so that

q? Iq,' commutes for all ie IoFli )

Consequently we have the category CoconeEl of cocones onF .

A coheir of F : I - to is an initial co cone (d ai : Fli)- d lieIo) :for any co cone (c , IFis ) on F F ! morphism at id-c so that

! -

¥

Cq,commutes ti c- Io .

Fli)#

Renard Since colimites of F : I-8 are initial in Coone(F ), they are unique

up to a unique isomorphism ( in Coone (f ) ).

Notation ( colin F, hi : Fci)- colinHielo) , or just Colin F for"the" colimit

of f : I- E.

17.2"

Example " If I is discrete,colin F = ¥⇒ Fli ) , the co product of

h F Lillie Io .

"

Example" let I -01

,the empty category (so Io -- 0, I, -0 ) .

The colimit of F : Q- 8 in an object eef so that tee to F ! g : e - c

Therefore colin E :O-e ) is an initial object of 8 .

Example let X be a set,RE Xxx an equivalence relation , Xlv the set

of equivalence classes of R and g :X→ Xlv , at =

,the quotient map .

We have two functions pi, Pz : R-X given by p , Cxyxzkx , , Pz Cx. , Kal

-

- Xz.

As in the case of equalizers we have a subcategory I-- R ÷g X of fed ,and the inclusion functor Fi . I - Sed

.

Cloud ( HR , Lg :X-Xlr , gopi : R-XIN ) is the cohimit of F . So quotients areco limits

.

Pi

Boot suppose REX is a co cone on F : 1 RIX IS Set.

Then

flu legC

g op, = f -- gopz . Hence if (x , ,xr) ER ( ie .X , nxz ) , g Gi) - g ka) .

Therefore F a well-defined map of :X In → c which is given by of Gtx) ) e g (x ) ,for any equivalence class Ex] -- f Cx) .Moreover such map g- : XX - c is unique : if h : Xf - c is another

function so that hog - g then h ( CD) - g Cx) for all at X. ⇒ h =J .

Definition het to be a category , a,b tho two objects , f,g. a- b two morphisms.

The colimit of the inclusion functor p : lajaby c. f is called the coequalizer ofthe diagram, a Tgtsb .

That a,the coefualizer of a JIB is an object d off and a morphism b ked

with the following universal property: given an object e off and a morphism

173

b hee so that hof -- hog , I ! morphism d-Ma so that the diagram

at b Ee d commutes.

g- ×, Lin

Definition A category f in cocompkte if it has all small colour its i. for any small categoryIand any functor F : I- to the Colinit 1 Colin F

, Lij : Fcj ) - ColinF)jet ) exists .

theorem A category L is cocomplete it and only if it has all small coproducts and allcoequalizers .Proof ⇐) since coproducts and coequalizers are coleworts

, any cocomplete category has all(small) co products and coequalizers .

⇐) A category b is acomplete ⇒ GOP is complete ⇒ b"has all products and

equalizers ⇐ b has all coproducts and coequalizers . D

It is useful to know that the category Set of (small ) sets is cocomplete .

By Theorem 17.I it is enough to prove that Set has coequalizers .

Coegfalours in Set are equivalence classes of appropriate equivalence relations . To constructthese equivalence relations we need

Lemmens let X be a set and I RakeA a family of equivalence relations on X.

Then the intersection S : = In.

R2 E Xxx is also an equivalence relation .

Proof Suppose Ca, b), lb, c) E S then V-2 La, b) ER, and lb, c)tRd. Since Ra is an

equivalence relation , la, c) C- Ra .

Since la, e) C- Re for all x, Ca, c) C-¥, Ra = S

.

⇒ S.

is transitive . Similarly S is reflexive and symmetric . D

Corollary 't:3 Suppose X is a set and S e Xxx a relation .Let A be the set of all

equivalence relations R containing S .Then '

- = LIAR is an equivalence relation .Ls> is the smallest equivalence relation containing S.

174

Proof Note first that A ¥0 since Xxx c-A . By lemma , 457 is an equivalence relation .Since t Re A

,SER

,we have Sen R - SSS

.

So S E SSS.

REA

tf U c-Xx X is an equivalence relation with SEU , then U ⇐ A.⇒ U 2¥

.

13=557.

. :(S ) is the smallest.

equivalence relation containing S . a

Lemme 17.4 Set has all coequalizers , hence in cocomplete .

We'll prove it next time.

-

D