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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 .