category of sheaves in nLab

Context

Topos Theory

Locality and descent

• Idea
• Definition
• Equivalent characterizations
• As localizations
• As toposes
• As accessible reflective subcategories
• Properties
• Dependence on the site
• Limits and colimits
• Epi- mono- and isomorphisms
• Exactness properties
• Related concepts
• References

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)[local isomorphisms] −1. Sh(S) \simeq Ho_{PSh(S)} = PSh(C)[\text{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.

Proposition

A presheaf A∈[C op,Set]A \in [C^{op}, Set] is in ℰ\mathcal{E} (meaning: in the essential image of ii) precisely if for all f:X→Yf : 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 A≃iA^A \simeq i \hat A, then by the (L⊣i)(L \dashv i)-adjunction isomorphism we have

Hom(f,iA^)≃ℰ(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:A→iL(A) \epsilon_A : A \to i L(A) be the (L⊣i)(L \dashv i)-unit.

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

Li→Id L i \to Id

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

LA→Lϵ ALiLA→≃LA 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:iLA→Ak_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:X→Yf : 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 ℰ\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.

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. .

Proof

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

(…)

Definition

For (L⊣i):ℰ↪←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 ℰ≃Sh J(C)\mathcal{E} \simeq Sh_J(C) we say that CC is a site of definition for the topos ℰ\mathcal{E}.

Definition

For (C,J)(C,J) a site with coverage JJ and D→CD \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 D→CD \to C (not necessarily full) is called a dense sub-site with the induced coverage J DJ_D if

1. every object U∈CU \in C has a covering {U i→U}\{U_i \to U\} in JJ with all U iU_i in DD;

2. for every morphism f:U→df : U \to d in CC with d∈Dd \in D there is a covering family {f i:U i→U}\{f_i : U_i \to U\} such that the composites f∘f if \circ f_i are in DD.

Theorem

(comparison lemma)

Let (C,J)(C,J) be a (possibly large) site with CC a locally small category and let f:D→Cf : D \to C be a small dense sub-site. The 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} \,.

Examples

• Let XX be a locale with frame Op(X)Op(X) regarded as a site with the canonical coverage ({U i→U}\{U_i \to U\} covers if the join of the U iU_i is UU). Let bOp(X)bOp(X) be a basis for the topology of XX: a complete join-semilattice such that every object of Op(X)Op(X) is the join of objects of bOp(X)bOp(X). Then bOp(X)bOp(X) is a dense sub-site.

  • For XX a locally contractible space, Op(X)Op(X) its category of open subsets and cOp(X)cOp(X) the full subcategory of contractible open subsets, we have that cOp(X)cOp(X) is a dense sub-site.

• For C=TopManifoldC = TopManifold the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp top{}_{top} is a dense sub-site: every paracompact manifold has a good open cover by open balls homeomorphic to a Cartesian space.

• Similaryl for C=C = Diff the category of smooth manifolds equipped with the good open cover coverage, the full subcategory CartSp smooth{}_{smooth} is a dense sub-site.

Proposition

Let 𝒞\mathcal{C} be a small site and let Sh(𝒞)Sh(\mathcal{C}) be its category of sheaves. Let f:X→Yf \colon X \to Y be a homomorphism of sheaves, hence a morphism in Sh(𝒞)Sh(\mathcal{C}). Then:

1. ff is a monomorphism or isomorphism precisely if it is so globally in that for each object U∈𝒞U \in \mathcal{C} in the site, then the component f U:X(U)→Y(U)f_U \colon X(U) \to Y(U) is an injection or bijection of sets, respectively.

2. ff is an epimorphism precisely if it is so locally, in that: for all U∈CU \in C and every element y∈Y(U)y \in Y(U) there is a covering {p i:U i→U} i∈I\{p_i : U_i \to U\}_{i \in I} such that for all i∈Ii \in I the element Y(p i)(y)Y(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).

But if {x i} i∈I\{x_i\}_{i \in I} is a set of points of a topos for Sh(𝒞)Sh(\mathcal{C}) such that these are enough points (def.) then the morphism ff is epi/mono/iso precisely it is is so an all stalks, hence precisely if

x i *f:x i *X⟶x i *Y x_i^\ast f \;\colon\; x_i^\ast X \longrightarrow x_i^\ast Y

is a surjection/injection/bijection of sets, respectively, for all i∈Ii \in I.