definition rli and qi t the p%¥g¥yj
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171
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
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