Table Of ContentGrowing Sheaves on fertile Categories
JeskoHu¨ttenhain
Spring2010
Abstract
A sheaf (on a topological space) encompasses data which is attached to
the open sets of some space. Well-known examples include the sheaves
ofcontinuousmapsfromonespacetoanotherorthestructuresheafofan
affinescheme.Wegiveaself-containedintroductiontotherequiredresults
ofcategorytheorybeforeweintroducethenotionofasheaftakingvalues
inanycategory. Themainresultstatesthatundercertaincircumstances,
everypresheafhasasheafassociatedtoit.Theseresultsaredirectlybased
ontheoriginalworksofGray(see[Gr65]).
You should note that even of this very abstract setting, there have been
significantgeneralizations–ifyouareinterested,[KaSch06]isaverythor-
ough textbook. Most of the proofs of the category-theoretical results are
takenfrom[Brc94],whichisalsoveryrecommendableforfurtherreading.
Contents
1 Categories 2
2 Limits 4
3 IntersectionsandGenerators 9
4 Sheaves 15
5 Sheafification 20
1
1 Categories
Definition1.1. AcategoryC consistofthefollowingdata:
• AclassOb(C),theobjectsofC.
• AsetC(X,Y)forallX,Y ∈Ob(C),theC-morphismsfromXtoY.
• ForallX,Y,Z ∈Ob(C)amap
C(X,Y)×C(Y,Z) −→ C(X,Z) , (f,g) (cid:55)−→ g◦ f
whichwecallthecompositioninC.
suchthatthefollowingholds:
• ThecompositioninC isassociative: (f ◦g)◦h = f ◦(g◦h).
• Forall X ∈ Ob(C) thereexistsid ∈ C(X,X) withthepropertythatfor
X
all f ∈ C(X,Y)andg ∈ C(Z,X): f ◦id = f andid ◦g = g.
X X
AcategoryissmallwhenOb(C)isaset. Forourpurposes,acategoryiscalled
concreteifitsobjectsaresets,morphismsaremapsbetweenthemandthecom-
positionofmorphismsispreciselythecompositionofthesemaps.
Notation. Werequiresomecommonterminologyfordealingwithcategories:
• WereservetherighttowriteX ∈ C insteadofX ∈Ob(C).
• Thenotation f : X →Ymeans f ∈ C(X,Y).
Definition1.2. LetC beacategoryand f : X →Y. Thenwecall f
1. an epimorphism if for all h ,h : Y → Z, the equality h ◦ f = h ◦ f
1 2 1 2
alreadyimpliesh = h .
1 2
2. a monomorphism if for all g ,g : Z → X, the equality f ◦g = f ◦g
1 2 1 2
alreadyimpliesg = g .
1 2
3. an isomorphism if there exists a morphism f−1 : Y → X such that f ◦
f−1 =id and f−1◦ f =id . Wecall f−1theinverseof f.
Y X
Remark. Itiseasytoseethatifaninverseexists,itisunique: Assumingthat g
andharebothinverseto f yieldsh = h◦ f ◦g = g.
Definition1.3. WesaythattwoobjectsX,Y ∈ C areisomorphicifthereexists
anisomorphismX →Y. WedenotethisbyX ∼=Y.
2
Notation. Wewrite
1. f : X (cid:44)→Yif f isamonomorphism
(cid:16)
2. f : X Yif f isanepimorphism
3. f : X −→∼ Yif f isanisomorphism.
Wealsosaythatamorphismisepicormonicwhenwewanttosaythatitisan
epimorphismormonomorphism,respectively.Wesometiemssaythatepimor-
phisms(resp. monomorphisms)canberight-cancelled(resp. left-cancelled),
whichisjustanotherwaytodescribetheirdefiningproperties.
Definition1.4. LetC andDbecategories. AfunctorFfromC toDconsistsof
thefollowingdata:
• ForeveryX ∈Ob(C),anobjectF(X) ∈Ob(D).
• ForeverypairofobjectsX,Y ∈Ob(C),amapofsets
F = F : C(X,Y) −→ D(F(X),F(Y)).
X,Y
suchthatF(id ) =id andF(g◦ f) = F(g)◦F(f). Thelatter(compatibility
X F(X)
withcomposition)isalsobeingreferredtoasfunctoriality. Wewritefunctors
asF : C → D.
Thecompositionoftwofunctors F : C → D and G : D → E isdefinedto
bethefunctorG◦Fwhichisdefinedby(G◦F)(X) = G(F(X))onobjectsand
thecomposition G◦F (asmapsofsets) onmorphisms. Itis obviousthatthis
definesafunctorC → E.
Remark. Beware: Categories and functors together do not form a category, as
hinted at before. However, one can consider the category Cat of small cate-
gorieswithfuntorsasmorphisms.
Definition1.5. LetFandGbetwofunctorsC → D. Anaturaltransformation
ϕ : F → G is a class {ϕ : F(X) → G(X) | X ∈Ob(C)} of morphisms such
X
that
F(X) ϕX (cid:47)(cid:47) G(X)
F(f) (cid:9) G(f)
(cid:15)(cid:15) (cid:15)(cid:15)
F(Y) ϕY (cid:47)(cid:47) G(Y)
we have G(f)◦ ϕ = ϕ ◦F(f) for all morphisms f : X → Y in C. This
X Y
propertyisalsoreferredtoasnaturality.
Ifthe ϕ areD–isomorphismsforallX ∈Ob(C),wesaythat ϕisanatural
X
∼
isomorphism(ornaturalequivalence). WethensometimeswriteF = G. We
ϕ
defineNat(F,G)tobetheclassofallnaturaltransformationsF → G.
3
Definition 1.6. Let C be a small category and D a category. We denote by
Fun(C,D)thefunctorcategorywhere
• ObjectsarefunctorsC → D.
• MorphismsfromFtoGarethenaturaltransformationsNat(F,G).
• Composition of natural transformations ϕ : F → G and ψ : G → H is
definedby(ψ◦ϕ) := ψ ◦ϕ foreveryX ∈Ob(C).
X X X
Remark. Note that C has to be small for this definition to make sense; other-
wise, the class of natural transformations from one functor to another is not
necessarilyaset.
Fact1.7. ThenaturalisomorphismsaretheisomorphismsinFun(C,D).
Proof. Simplynotethat ϕ : F → G isanaturaltransformationwithinverse ψ
inFun(C,D)ifandonlyifψ ◦ϕ =id forallX.
X X F(X)
2 Limits
Definition2.1. LetI beasmallcategoryandF : I → C afunctor. Wesaythat
(X,ϕ)isaconeoverFwhen{ϕ : X → F(I) | I ∈ I}isafamilyofmorphisms
I
suchthatforallI–morphismsι : I → J,theequality F(ι)◦ϕ = ϕ holds. We
I J
saythat(L,λ)isalimitofFifanyothercone(X,ϕ)factorsuniquelythrough
Linthesensethat
X
∃!µ
ϕI (cid:15)(cid:15) ϕJ
L
λI λJ
(cid:15)(cid:15) (cid:125)(cid:125) (cid:33)(cid:33) (cid:15)(cid:15)
(cid:47)(cid:47)
F(I) F(J)
F(ι)
Weoftenwritelim(F) := L and λF insteadof λ todenotethedependenceon
F. Thedualnotionisthatofacoconeandacolimit:
(cid:56)(cid:56)Y(cid:79)(cid:79) (cid:102)(cid:102)
∃!ν
ψI ψJ
(cid:61)(cid:61)C (cid:97)(cid:97)
γI γJ
(cid:47)(cid:47)
F(I) F(J)
F(ι)
4
Again,wewritecolim(F) :=CandγF insteadofγsometimes.
Example2.2. Pullbacksarelimitsoffunctorsfromthepathcategoryof
•
(cid:15)(cid:15)
(cid:47)(cid:47)
• •
whereasequalizersarelimitsoffunctorsfromthepathcategoryof
(cid:47)(cid:47)
• (cid:47)(cid:47) •
Also, products indexed by some set I are limits of functors from the discrete
categoryover I.
Lemma2.3. LetF : I → C beafunctor.
• If (L,λ) isthelimitof F, thenanytwomorphisms f,g : X → L areequal ⇔
λ ◦ f = λ ◦gforall I ∈ I.
I I
• If(C,γ)isthecolimitofF,thenanytwomorphisms f,g :C →Yareequal⇔
f ◦γ = g◦γ forall I ∈ I.
I I
Proof. We prove only the first statement since the second follows dually (and
equallytrivial): Both f andgarefactorizationsofthecone(X,(λ ◦ f)).
I
Definition2.4. WesaythatacategoryC has(co)limitsoftypeI ifeveryfunc-
tor F : I → C has a (co)limit. A category is said to be (co)complete if it has
limitsoftypeI foranysmallcategoryI.
Proposition2.5. LetI andC besmallcategoriesandD anycategory. Let F : I →
Fun(C,D). WedenotebyF(−)(X) : I → Dthefunctordefinedby
(cid:31) (cid:47)(cid:47)
I F(I)(X)
(cid:31)
(cid:47)(cid:47)
f F(f)
X
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:31) (cid:47)(cid:47)
J F(J)(X)
If F(−)(X) has a limit for all X ∈ Ob(C), then F has a limit (L,(λ )) such that
I
(L(X),(λX))isalimitofF(−)(X).
I
Proof. Inthefollowing,letalways f : X →YbeaC–morphismandι : I → J a
I–morphism. Let(L(X),λX)denotethelimitofF(−)(X)where
λX : L(X) → F(I)(X) suchthat λX = F(ι) ◦λX. (1)
I J X I
5
ThenaturalityofF(ι)
F(I)(X) F(ι)X (cid:47)(cid:47) F(J)(X)
F(I)(f) (cid:9) F(J)(f)
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
F(I)(Y) F(J)(Y)
F(ι)Y
impliesthatthemapsF(I)(f)◦λX satisfy
I
F(ι) ◦F(I)(f)◦λX = F(J)(f)◦F(ι) ◦λX = F(J)(f)◦λX
Y I X I J
andthereforeconstituteaconeoverF(−)(Y). Consequently,
L(X)
λXI ∃!L(f) λXJ
(cid:126)(cid:126) (cid:15)(cid:15) (cid:32)(cid:32)
F(I)(X) L(Y) F(J)(X)
F(I)(f) F(J)(f)
λY λY
(cid:15)(cid:15) (cid:126)(cid:126) I J (cid:32)(cid:32) (cid:15)(cid:15)
(cid:47)(cid:47)
F(I)(Y) F(J)(Y)
F(ι)Y
thereexistsauniquemorphismL(f) : L(X) → L(Y)withthepropertythat
λY◦L(f) = F(I)(f)◦λX. (2)
I I
We now claim that L is functorial. Indeed, assume that g : Y → Z is another
C–morphism,thenwehave
λZ◦L(g◦ f) = F(I)(g◦ f)◦λX
I I
= F(I)(g)◦F(I)(f)◦λX
I
= F(I)(g)◦λY◦L(f)
I
= λZ◦L(g)◦L(f)
I
and by the uniqueness of (2), this asserts functoriality. We also observe that
theequality(2)meansthatλ : L → F(I)isanaturaltransformationforevery
I
I ∈ Ob(I) (or, look at the parallelograms in the diagram). By (1), the pair
(L,(λ ))constitutesaconeoverF.
I
ToverifythatitisalimitofF,assumethat(K,(κ ))isanotherconeoverF.
I
Sincetheκ alsosatisfiy
I
∀X ∈Ob(C) :κX = F(ι) ◦κX, (3)
J X I
6
wegetauniquemorphismµX : K(X) → L(X)
K(X)
∃!µX
(cid:15)(cid:15)
κXI L(X) κXJ
λXI λXJ
(cid:13)(cid:13) (cid:126)(cid:126) (cid:32)(cid:32) (cid:17)(cid:17)
(cid:47)(cid:47)
F(I)(X) F(J)(X)
F(ι)X
withthepropertythatκX = λX◦µX forall I ∈Ob(I). Also,
I I
λY◦L(f)◦µX (=2) F(I)(f)◦λX◦µX = F(I)(f)◦κ
I I X
(=3)κY◦K(f) = λY◦µY◦K(f)
I I
so L(f)◦µX = µY ◦K(f) by 2.3. This means that µ : K → L is the unique
naturaltransformationwithκ = λ ◦µ–andhence,wearedone.
I I
Corollary2.6. IfC is(co-)complete,thensoisFun(I,C).
Definition2.7. Adirectedset(I,≤)isasetIwithareflexive,transitiverelation
“≤” with the additional property that every pair of elements i,j ∈ I has an
upperbound,i.e. anelementk ∈ I suchthati ≤ kand j ≤ k. Asmallcategory
I withOb(I) = I and
(cid:40)
{i → j} ; i ≤ j
I(i,j) =
∅ ; otherwise
forsomedirectedset(I,≤)isthencalledadirectedcategory.
LetIbeadirectedcategory.AfunctorF : I → Ciscalledadirectedfunctor.It
canberepresentedasafamilyofobjectsX = F(i)andmorphisms f = F(i →
i ij
j). In this case, we say that (X, f ) is a directed diagram in C. A colimit of
i ij
typeI iscalledadirectlimit. Wethenwrite
limX :=colim(F)
−→ i
Dually, a functor G : Iop → C is given by a family of objects Y = G(i) and
i
morphisms g = F(i ← j). Wesaythat(Y,g )isaninversediagraminC. A
ij i ij
limitoftypeIopiscalledaninverselimitandwewrite
limY :=lim(F)
←− i
7
Fact2.8. InsideacategoryC,assumethat
f(cid:48) (cid:47)(cid:47)
P X
g(cid:48) × g
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
Y S
f
isapullbackdiagram. If gisamonomorphism(resp. isomorphism),thensois g(cid:48). In
otherwords,anypullbackofamonomorphism(resp. isomoprhims)ismonic(resp. an
isomorphism).
Proof. Assume that g is monic and let u,v : Q → P be two morphisms such
thatg(cid:48)◦u = g(cid:48)◦v. Wedefineg(cid:48)(cid:48) := g(cid:48)◦uand f(cid:48)(cid:48) := f(cid:48)◦u. Thus,
f ◦g(cid:48)(cid:48) = f ◦g(cid:48)◦u = g◦ f(cid:48)◦u = g◦ f(cid:48)(cid:48)
means that (Q, f(cid:48)(cid:48),g(cid:48)(cid:48)) is a cone. Therefore, u is the unique morphism that
makesthefollowingdiagramcommute:
Q f(cid:48)(cid:48)
u
(cid:31)(cid:31) (cid:30)(cid:30)
f(cid:48) (cid:47)(cid:47)
P X
g(cid:48)(cid:48)
g(cid:48) g
(cid:32)(cid:32) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
Y S
f
Ontheotherhand,wealsohaveg(cid:48)◦v = g(cid:48)◦u = g(cid:48)(cid:48). From
g◦ f(cid:48)◦v = f ◦g(cid:48)◦v = f ◦g(cid:48)◦u = f ◦g(cid:48)(cid:48) = g◦ f(cid:48)(cid:48)
wecancancelg(itismonic)toobtain f(cid:48)◦v = f(cid:48)(cid:48) = f(cid:48)◦u–andbyuniqueness
ofu,thisimpliesu = v.
Let us assume now that g is an isomorphism. It is then easy to see that
P :=Y, f(cid:48) := g−1◦ f andg(cid:48) := id isapullback,andanytwopullbacksdiffer
Y
byuniqueisomorphism(whichwillthenbeg(cid:48)).
Fact/Definition 2.9. The kernel pair of a C–morphism f : X → Y is a triple
(K,α,β)with
α (cid:47)(cid:47)
K X
β × f
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
X Y
f
iftheabovepullbackdiagramexists.
8
Fact2.10. For f : X →Y,thefollowingconditionsareequivalent:
1. f isamonomorphism.
2. Thekernelpairof f existsandisgivenby(X,id ,id ).
X X
3. Thekernelpair(K,α,β)of f existsandα = β.
Proof. Fortheimplication(1)⇒(2),itsufficestoverifytheuniversalproperty
ofthecone(X,id ,id ):Giventwomorphismsu,v : Z → Xwith f ◦u = f ◦v,
X X
since f is monic, the unique factorization is given by u = v. The implication
(2)⇒(3)isobvious. For(3)⇒(1),letu,v : Z → X besuchthat f ◦u = f ◦v.
Then,thereexistsauniqueh : Z → Ksuchthatu = α◦h = β◦h = v.
3 Intersections and Generators
Definition3.1. LetC beacategoryandX ∈Ob(C)anobject. Onthe(possibly
proper)class{(U,u) | u :U (cid:44)→ X},wedefineabinaryrelationby
(U,u) ≤ (V,v) :⇔ ufactorsthroughv ⇔ ∃ν :U →V : u = v◦ν
Itiseasytoseethatthisisapartialorder.Wecontinuetodefineanequivalence
relation
(U,u) ∼ (V,v) :⇔ u ≤ vandv ≤ u
The equivalence classes with respect to ∼ are called the subobjects of X, de-
notedbySub(X).
Fact 3.2. Let X be an object and (U,u) ∼ (V,v) two representatives of the same
subobjectofX.Then,UandVareisomorphicviathefactorizationofuthroughv(and
vthroughu).
Proof. Bydefinition, vfactorsthroughuviasomeν : V (cid:44)→ U whichhastobe
monicsincebothuandvaremonic. Equivalently,ufactorsthroughvviasome
µ :U (cid:44)→V,so
U(cid:95)(cid:95) (cid:111)(cid:95) u (cid:47)(cid:47)(cid:63)(cid:63)X
µ (cid:9)
ν (cid:31) (cid:31)(cid:31) (cid:127) v
V
Now u◦ν◦µ = v◦µ = u = u◦id and u canbeleft-cancelled, sowehave
U
ν◦µ =id andsymmetrically,wegetµ◦ν =id .
U V
9
Notation 3.3. For ease of notation, we mean byU ∈ Sub(X) the equivalence
class of a pair denoted by (U,ι ). This notation is slightly abusive because
U
there might be more than one monomorphism U (cid:44)→ X. However, we shall
neverabuseittoimplythecontrary.
Definition3.4. AcategoryC iscalledwell-poweredwhenSub(X)isasetfor
anyobjectX ∈Ob(C).
Example3.5. Clearly,thecategorySetsiswell-powered:Thesubobjectsofany
set are in bijection with its subsets. Similarly, the categories Grp and Ab are
well-powered.
Definition3.6. LetXbeanobjectofsomecategoryC. Theintersectionofany
familyofsubobjectsofU⊆Sub(X)isdefinedtobeitsinfimuminSub(X),ifit
exists. Wedenotethisby
(cid:92)
U:= inf (U).
Sub(X)
Furthermore,theunionofUisdefinedtobeitssupremum
(cid:91)
U:= sup (U).
Sub(X)
InthecaseU = {U,V},wewriteU∩V := (cid:84)UaswellasU∪V := (cid:83)Uand
similarlyforallfinitesetsU.
Fact 3.7. Let X be an object of a category C such that Sub(X) is a set. Then, the
intersection of any family of subobjects of X exists if and only if the union of any
familyofsubobjectsofXexists.
Proof. Thisfollowsbecause
inf(U) = sup{V | ∀U ∈U:V ≤U}
sup(U) = inf{V | ∀U ∈U:V ≥U}
holdsinanypartiallyorderedset.
Proposition3.8. LetU andV betwosubobjectsofX. Ifthepullbackofthediagram
(cid:111) (cid:47)(cid:47) (cid:111)(cid:111) (cid:47)
U ιU X ιV V exists,itistheintersectionU∩V.
Proof. Weconsiderthepullbackdiagram
W(cid:79) (cid:111) λU (cid:47)(cid:47)U(cid:79)
λV × ιU
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:111) (cid:47)(cid:47)
V X
ιV
10
