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category of sheaves

Contents

Idea

This entry is about the properties and the characterization of the category Sh(S)Sh(S) of (set-valued) sheaves on a (small) site SS, which is a Grothendieck topos. Among other things it gives a definition and a characterization of the notion of sheaf itself, but for more details on sheaves themselves see there.

Definition

Let (C,J)(C,J) be a site: a (small) category equipped with a coverage.

Definition

The category of sheaves on (C,J)(C,J) is the full subcategory of the category of presheaves

i:Sh J(C)PSh(C) i : Sh_J(C) \hookrightarrow PSh(C)

on those presheaves which are sheaves with respect to JJ.

Proposition

Every category of sheaves is a reflective subcategory

(Li):Sh J(C)LPsh(C), (L \dashv i) : Sh_J(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} Psh(C) \,,

hence a subtopos of the presheaf topos. Moreover, every such subtopos arises in this way: there is a bijection between Grothendieck topologies on CC and equivalence classes of geometric embeddings into PSh(C)PSh(C).

This appears for instance as (Johnstone, corollary 2.1.11). See also Lawvere-Tierney topology.

Proof

Details on the first statement are at sheafification. A full proof for the second statement is at (∞,1)-category of (∞,1)-sheaves (there proven in (∞,1)-category theory, but the proof is verbatim the same in category theory).

Equivalent characterizations

As localizations

Proposition

The category of sheaves is equivalent to the homotopy category of the category with weak equivalences PSh(C)PSh(C) with the weak equivalences given by W=W = local isomorphisms

Sh(S)Ho PSh(S)=PSh(C)[localisomorphisms] 1. Sh(S) \simeq Ho_{PSh(S)} = PSh(C)[local isomorphisms]^{-1} \,.

The converse is also true: for every [left exact functor]] L:PSh(S)PSh(S)L : PSh(S) \to PSh(S) (preserving finite limits) which is left adjoint to the inclusion of its image, there is a Grothendieck topology on SS such that the image of LL is the category of sheaves on SS with respect to that topology.

We spell out proofs of some of the above claims.

Let CC be a small category and

i:[C op,Set] i : \mathcal{E} \hookrightarrow [C^{op}, Set]

a reflective subcategory, hence a full subcategory with a left adjoint L:[C op,Set]L : [C^{op}, Set] \to \mathcal{E}, such that moreover LL preserves finite limits.

Write W:=L 1(isos)Mor([C op,Set])W := L^{-1}(isos) \subset Mor([C^{op}, Set]) for the class of morphisms in [C op,Set][C^{op}, Set] that are sent to isomorphisms by LL.

Proposition

A presheaf A[C op,Set]A \in [C^{op}, Set] is in \mathcal{E} (meaning: in the essential image of ii) precisely if for all f:XYf : X \to Y in WW the induced function

Hom(f,A):Hom(Y,A)Hom(X,A) Hom(f,A) : Hom(Y,A) \to Hom(X,A)

is a bijection.

Proof

If AiA^A \simeq i \hat A, then by the (Li)(L \dashv i)-adjunction isomorphism we have

Hom(f,iA^)(L(f),A). Hom(f, i \hat A) \simeq \mathcal{E}(L(f), A) \,.

But by assumption L(f)L(f) is an isomorphism, so the claim is immediate.

Conversely, if for all ff the function Hom(f,A)Hom(f,A) is a bijection, define A^:=L(A)\hat A := L(A) and let ϵ A:AiL(A) \epsilon_A : A \to i L(A) be the (Li)(L \dashv i)-unit.

By the assumption that ii is a full and faithful functor and basic properties of adjoint functors we habe that the counit

LiId L i \to Id

is a natural isomorphism. By the zig-zag law the composite

LALϵ ALiLALA L A \stackrel{L \epsilon_A}{\to} L i L A \stackrel{\simeq}{\to} L A

is the identity and therefore LϵL \epsilon is an isomorphism and so ϵ A\epsilon_A is in WW, under our assumption on AA.

Using this it follows that

Hom(ϵ A,A):Hom(iLA,A)Hom(A,A) Hom(\epsilon_A, A) : Hom(i L A, A) \stackrel{\simeq}{\to} Hom(A,A)

is an isomorphism. Write k A:iLAAk_A : i L A \to A for the preimage of id Aid_A under this isomorphism, which is therefore a left inverse of ϵ A\epsilon_A. This immediately implies that also k Ak_A is in WW, and so we can enter the same argument with k Ak_A to find that it has a left inverse itself. But this means that k Ak_A is in fact an isomorphism and hence so is ϵA\epsilon A, which thus exhibits AA as being in the essential image of ii.

Proposition

A morphism f:XYf : X \to Y is in WW precisely if for every morphism z:j(c)Yz : j(c) \to Y with representable domain, the pullback z *fz^* f in

X× Yj(c) X z *f f j(c) z Y \array{ X \times_Y j(c) &\to& X \\ {}^{\mathllap{z^* f}}\downarrow && \downarrow^{\mathrlap{f}} \\ j(c) &\stackrel{z}{\to}& Y }

is in WW.

Proof

Assume first that ff is in WW. Since by assumption LL preserves finite limits, it follows that

L(X× Yj(c)) LX L(z *f) Lf L(j(c)) Lz LY \array{ L(X \times_Y j(c)) &\to& L X \\ {}^{\mathllap{L(z^* f)}}\downarrow && \downarrow^{\mathrlap{L f}} \\ L(j(c)) &\stackrel{L z}{\to}& L Y }

is still a pullback diagram in \mathcal{E} and hence that L(z *f)L(z^* f) is the pullback of the isomorphism LfL f and thus itself an isomorphism. Therefore z *fz^* f is in WW.

Conversely, suppose that all these pullbacks are in WW. Then use the “co-Yoneda lemma” to write the presheaf YY as a colimit over all representables mapping into it

lim j(c i)z iYj(c)Y. {\lim_\to}_{j(c_i) \stackrel{z_i}{\to} Y} j(c) \stackrel{\simeq}{\to} Y \,.

Forming the pullback along ff, using that in a topos (such as our presheaf topos) colimits are preserved by pullbacks, we get

lim if *j(c i) X lim iz i *f f lim ij(c i) Y. \array{ {\lim_\to}_i f^* j(c_i) &\stackrel{\simeq}{\to}& X \\ {}^{\mathllap{{\lim_\to}_i z_i^* f}}\downarrow && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_i j(c_i) &\stackrel{\simeq}{\to}& Y } \,.

Since LL preserves all colimits and finite limits, we also get

lim iL(f *j(c i)) L(X) lim iL(z i *f) L(f) lim iL(j(c i)) L(Y). \array{ {\lim_\to}_i L(f^* j(c_i)) &\stackrel{\simeq}{\to}& L(X) \\ {}^{\mathllap{{\lim_\to}_i L(z_i^* f)}}\downarrow && \downarrow^{\mathrlap{L(f)}} \\ {\lim_\to}_i L(j(c_i)) &\stackrel{\simeq}{\to}& L(Y) } \,.

Since by assumption now all L(z i *f)L(z_i^* f ) are isomorphisms, also lim iL(z i *f){\lim_\to}_i L(z_i^* f) is an isomorphism and hence three sides of the above square are isomorphisms. Therefore also L(f)L(f) is and hence ff is in WW.

Proposition

The collection of sieves in WW, hence the collection of monomorphisms in WW whose codomain is a representable, constitute a Grothendieck topology on CC.

Proof

We check the list of axioms, given at Grothendieck topology:

  1. Pullbacks of covering sieves are covering :

    First of all, the pullback of a sieve along a morphism of representables is still a sieve, because monomorphisms are (as discussed there) stable under pullback.

    Next, since LL preserves finite limits, LL applied to the pullback sieve is the pullback of an isomorphism, hence is an isomorphism, hence the pullback sieve is in WW.

  2. The maximal sieve is covering. Clear: LL applied to an isomorphism is an isomorphism.

  3. Two sieves cover precisely if their intersection covers. This is again due to the pullback-stability of elements of WW, due to the preservation of finite limits by LL.

  4. If all pullbacks of a sieve along morphisms of a covering sieve are covering, then the original sieve was covering .

    This is the same argument as in the second part of the proof of prop. 4.

Proposition

\mathcal{E} is a Grothendieck topos.

Proof

By prop. 5 and prop. 3 we are reduced to showing that an object AA is in \mathcal{E} already if for all monomorphisms ff in WW the function Hom(f,A)Hom(f,A) is a bijection.

(…)

As toposes

Categories of sheaves are examples of categories that are toposes: they are the Grothendieck toposes characterized among all toposes as those satisfying Giraud's axioms.

Proposition

Sheaf toposes are equivalently the subtoposes of presheaf toposes.

This appears for instance as (Johnstone, corollary 2.1.11).

As accessible reflective subcategories

See at reflective sub-(∞,1)-category.

Properties

Dependence on the site

Definition

For (Li):L(L \dashv i) : \mathcal{E} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} a category of sheaves and (C,J)(C,J) a site such that we have an equivalence of categories Sh J(C)\mathcal{E} \simeq Sh_J(C) we say that CC is a site of definition for the topos \mathcal{E}.

Remark

There are always different sites (C,J)(C,J) whose categories of sheaves are equivalent. First of all for fixed CC and given a coverage JJ, the category of sheaves depends only on the Grothendieck topology generated by JJ. But there may be site structures also on inequivalent categories CC that have equivalent categories of sheaves.

Definition

For (C,J)(C,J) a site with coverage JJ and DCD \to C any subcategory, the induced coverage J DJ_D on DD has as covering sieves the intersections of the covering sieves of CC with the morphisms in DD.

Definition

Let (C,J)(C,J) be a site (possibly large). A subcategory DCD \to C (not necessarily full) is called a dense sub-site with the induced coverage J DJ_D if

  1. every object UCU \in C has a covering {U iU}\{U_i \to U\} in JJ with all U iU_i in DD;

  2. for every morphism f:Udf : U \to d in CC with dDd \in D there is a covering family {f i:U iU}\{f_i : U_i \to U\} such that the composites ff if \circ f_i are in DD.

Remark

If DD is a full subcategory then the second condition is automatic.

Theorem

(comparison lemma)

Let (C,J)(C,J) be a (possibly large) site with CC a locally small category and let f:DCf : D \to C be a small dense sub-site. Then pair of adjoint functors

(f *f *):PSh(D)f *f *PSh(C) (f^* \dashv f_*) : PSh(D) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} PSh(C)

with f *f^* given by precomposition with ff and f *f_* given by right Kan extension induces an equivalence of categories between the categories of sheaves

(f *f *):Sh J D(D)f *f *Sh JC. (f_* \dashv f^*) : Sh_{J_D}(D) \underoverset {\underset{f_*}{\to}}{\overset{f^*} {\leftarrow}} {\simeq} Sh_J{C} \,.

This appears as (Johnstone, theorm C2.2.3).

Examples

Epi- mono- and isomorphisms

Let Sh(C)Sh(C) be a category of sheaves.

Proposition

A morphism f:XYf : X \to Y is a monomorphism if it is a monomorphism of presheaves, that is if for each UCU \in C the function f(U):X(U)Y(U)f(U) : X(U) \to Y(U) is injective.

Proposition

A morphism f:XYf : X \to Y in Sh(C)Sh(C) is an epimorphism precisely if it is locally surjective in the sense that:

for all UCU \in C there is a covering {p i:U iU} iI\{p_i : U_i \to U\}_{i \in I} such that for all iIi \in I and every element yY(U)y \in Y(U) the element f(p i)(y)f(p_i)(y) is in the image of f(U i):X(U i)Y(U i)f(U_i) : X(U_i) \to Y(U_i).

Exactness properties

Every sheaf topos satisfies the following exactness properties. it is an

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes\hookrightarrowalgebraic lattices\simeq Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes\hookrightarrowlocally presentable categories\simeq Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories\hookrightarrowaccessible categories
model category theorymodel toposes\hookrightarrowcombinatorial model categories\simeq Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes\hookrightarrowlocally presentable (∞,1)-categories\simeq
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories\hookrightarrowaccessible (∞,1)-categories

References

Secton A.4 and C.2 in

The characterization of sheaf toposes and Grothendieck topologies in terms of left exact reflective subcategories of a presheaf category is also in

where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.

The characterization of Sh(S)Sh(S) as the homotopy category of PSh(S)PSh(S) with respect to local isomorphisms is emphasized at the beginning of the text

Details are in

It’s a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed:

Revised on November 25, 2013 03:40:00 by Urs Schreiber (89.204.153.236)