morphism of sites in nLab

Context

Topos Theory

• Idea
• Definition
• Examples
• Properties
• Relation to geometric morphisms
• Between κ\kappa-ary sites
• References

Definition

A functor f:C→Df:C\to D is a morphism of sites if

1. ff is covering-flat, and

2. ff preserves covering families, i.e. for every covering {p i:U i→U}\{p_i : U_i \to U\} of an object U∈CU \in C, the family {f(p i):f(U i)→f(U)}\{f(p_i) : f(U_i) \to f(U)\} is a covering of f(U)∈Df(U) \in D.

Example

If AA and BB are frames regarded as sites via their canonical coverages, then a morphism of sites A→BA \to B is equivalently a frame homomorphism, a function preserving finite meets and arbitrary joins.

Example

(slice sites)

For CC a site and U∈CU \in C, the comma category (C/U)(C / U) inherits a topology from CC, such that the forgetful functor (C/U)→C(C/U) \to C constitutes a morphism of sites. This is also called the big site of UU. There are natural operations for restriction and extension of sheaves from a sub-site UU to XX and back.

For instance, if XX is a topological space and U∈Op(X)U \in Op(X) is an open subset, then UU regarded as a topological space in its own right has corresponding to it the site Op(U)=Op(X)↓UOp(U) = Op(X) \downarrow U.

Example

For CC and DD regular categories equipped with their regular coverages, a morphism of sites is the same as a regular functor, i.e. a functor preserving finite limits and covers.

More generally, if CC and DD are ∞-ary regular categories with their κ\kappa-canonical topologies, then a morphism of sites is the same as a κ\kappa-ary regular functor (preserving finite limits and κ\kappa-ary effective-epic families).

Example

For CC any site with finite limits, there is canonically a morphism of sites to its tangent category. See there for details.

Proposition

There is a geometric morphism between the categories of sheaves

(f *⊣f *):Sh(𝒟,K)→f *←f *Sh(𝒞,J) (f^* \dashv f_*) : Sh(\mathcal{D},K) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} Sh(\mathcal{C},J)

where f *f_* is the restriction of (−)∘f(-)\circ f to sheaves.

Proof

By the assumption that ff preserves covers we have that the restriction of (−)∘f(-)\circ f to Sh K(𝒟)Sh_K(\mathcal{D}) indeed factors through Sh(𝒞)↪PSh(𝒞)Sh(\mathcal{C}) \hookrightarrow PSh(\mathcal{C}).

Because for {U i→U}\{U_i \to U\} a cover in 𝒞\mathcal{C} and FF a sheaf on 𝒟\mathcal{D}, we have that (assuming here for simplicity that 𝒞\mathcal{C} has finite limits)

PSh 𝒞(lim→(∐ i,jU i× UU j)→→∐ iU i),F(f(−))) ≃lim ←(∏ i,jPSh 𝒞(U i× UU j,F(f(−)))←←∏ iPSh C(U i,F(f(−)))) ≃lim ←(∏ i,jF(f(U i× UU j)))←←∏ iF(f(U i))) ≃lim ←(∏ i,jF(f(U i)× f(U)f(U j))))←←∏ iF(f(U i))) ≃PSh 𝒟(lim→(∐ i,jf(U i)× f(U)f(U j))→→∐ if(U i)),F), \begin{aligned} PSh_{\mathcal{C}}( \underset{\to}{\lim} ( \coprod_{i, j} U_i \times_{U} U_j) \stackrel{\to}{\to} \coprod_{i} U_i ) \;,\; F(f(-)) ) & \simeq \lim_{\leftarrow} \left( \prod_{i,j} PSh_{\mathcal{C}}( U_i \times_U U_j, F(f(-))) \stackrel{\leftarrow}{\leftarrow} \prod_i PSh_C(U_i , F(f(-))) \right) \\ & \simeq \lim_{\leftarrow} \left( \prod_{i,j} F(f(U_i \times_U U_j))) \stackrel{\leftarrow}{\leftarrow} \prod_i F(f(U_i)) \right) \\ & \simeq \lim_{\leftarrow} \left( \prod_{i,j} F(f(U_i) \times_{f(U)} f(U_j)))) \stackrel{\leftarrow}{\leftarrow} \prod_i F(f(U_i)) \right) \\ & \simeq PSh_{\mathcal{D}}( \underset{\to}{\lim} ( \coprod_{i, j} f(U_i) \times_{f(U)} f(U_j)) \stackrel{\to}{\to} \coprod_{i} f(U_i) ) \;,\; F ) \end{aligned} \,,

where we used the Yoneda lemma, the fact that the hom functor PSh(−,−)PSh(-,-) sends colimits in the first argument to limits, and the assumption that ff preserves the pullbacks involved.

Also (−)∘f(-)\circ f preserves all limits, because for presheaves these are computed objectwise. And since the inclusion Sh K(𝒟)→PSh(𝒟)Sh_K(\mathcal{D}) \to PSh(\mathcal{D}) is right adjoint (to sheafification) we have that

f *:Sh K(𝒟)↪PSh(𝒟)→(−)∘fSh J(𝒞) f_* : Sh_K(\mathcal{D}) \hookrightarrow PSh(\mathcal{D}) \stackrel{(-)\circ f}{\to} Sh_J(\mathcal{C})

preserves all limits. Therefore by the adjoint functor theorem it has a left adjoint. Explicitly, this is the composite of the left adjoint to (−)∘f(-)\circ f and to sheaf inclusion. The first is left Kan extension Lan fLan_f along ff and the second is sheafification L JL_J on (𝒞,J)(\mathcal{C},J), so the left adjoint is the composite

f *:Sh J(𝒞)↪PSh(𝒞)→Lan fSPSh(𝒟)→L JSh J(𝒟). f^* :Sh_J(\mathcal{C}) \hookrightarrow PSh(\mathcal{C}) \stackrel{Lan_f}{\to}S PSh(\mathcal{D}) \stackrel{L_J}{\to} Sh_J(\mathcal{D}) \,.

Here the first morphism preserves all limits, the last one all finite limits. Hence the composite preserves all finite limits if the left Kan extension Lan fLan_f does. This is the case if ff is a flat functor.

(Because the left Kan extension is given by the colimit Lan fX:d↦lim→((f op/d)→𝒞 op→XSet)Lan_f X : d \mapsto {\underset{\to}{\lim}}((f^{op}/d) \to {\mathcal{C}}^{op} \stackrel{X}{\to} Set) over the comma category f op/df^{op}/d which is a filtered category if ff is flat, and filtered colimits are precisely those that commute with finite limits. For more details on this argument see the discussion at Geometric morphisms between presheaf toposes.)

Proposition

Let (𝒞,J)(\mathcal{C}, J) be a small site and let (𝒟,K)(\mathcal{D}, K) be a small-generated site. Then a geometric morphism

f:Sh(𝒟,K)→Sh(𝒞,J) f : Sh(\mathcal{D}, K) \to Sh(\mathcal{C}, J)

is induced by a morphism of sites (𝒟,K)←(𝒞,J):F(\mathcal{D}, K) \leftarrow (\mathcal{C}, J) : F precisely if the inverse image functor f *f^* respects the Yoneda embeddings, i.e. there is a functor FF making the following diagram commute:

𝒟 ←F 𝒞 j 𝒟↓ ↓ j 𝒞 Sh(𝒟,K) ←f * Sh(𝒞,J). \array{ \mathcal{D } &\stackrel{F}{\leftarrow}& \mathcal{C} \\ {}^{\mathllap{j_{\mathcal{D}}}}\downarrow && \downarrow^{\mathrlap{j_{\mathcal{C}}}} \\ Sh(\mathcal{D}, K) &\stackrel{f^*}{\leftarrow}& Sh(\mathcal{C}, J) } \,.

Proof

It suffices to show that given ff, the factorization FF is, if it exists, necessarily a morphism of sites: because since f *f^* is left adjoint and thus preserves all colimits and every object in Sh(C)Sh(C) is a colimit of representables, f *f^* is fixed by the factorization. By uniqueness of adjoint functors this means then that together with its right adjoint it is the geometric morphism induced from the morphism of sites, by prop. .

So we show that FF is necessarily a morphisms of sites:

1. since the Yoneda embedding and sheafification as well as inverse images preserve finite limits, so does f *j 𝒞f^* j_{\mathcal{C}} and hence FF preserves finite limits, hence is a flat functor;

2. f *h 𝒞f^* h_{\mathcal{C}} preserves coverings (maps them to epimorphisms in Sh(D,K)Sh(D, K)) and since KK is assumed to be subcanonical it follows from this prop. that j 𝒟j_{\mathcal{D}} also reflects covers. Therefore FF preserves covers.

Corollary

Let (𝒞,J)(\mathcal{C},J) be a small site and let ℰ\mathcal{E} be any sheaf topos. Then we have an equivalence of categories

Topos(ℰ,Sh(𝒞,J))≃Site((𝒞,J),ℰ) Topos(\mathcal{E}, Sh(\mathcal{C}, J)) \simeq Site((\mathcal{C}, J), \mathcal{E})

between the geometric morphisms from ℰ\mathcal{E} to Sh(𝒞,J)Sh(\mathcal{C}, J) and the morphisms of sites from (𝒞,J)(\mathcal{C}, J) to the big site (ℰ,C)(\mathcal{E}, C) for CC the canonical coverage on ℰ\mathcal{E}.

Proposition

If CC and DD are ∞-ary sites, then a functor f:C→Df:C\to D is a morphism of sites if and only if it preserves finite local κ\kappa-prelimits.