nLab
site

Idea

A site is a presentation of a sheaf topos as a structure freely generated under colimits from a category, subject to the relation that certain covering colimits are preserved.

As such, sites generalise topological spaces and locales, which present localic sheaf toposes. More precisely, sites generalise and categorify posites, which present localic toposes but also present locales themselves in a decategorified manner.

In technical terms, a site is a small category equipped with a coverage or Grothendieck topology. The category of sheaves over a site is a sheaf topos and the site is a site of definition for this topos.

Definition

Definition

A site $(C,J)$ is a category $C$ equipped with a coverage $J$.

For $ℰ$ a topos equipped with an equivalence of categories

$$ℰ\simeq \mathrm{Sh}(C,J)$$\mathcal{E} \simeq Sh(C,J)

to the sheaf topos over a site, one says that $(C,J)$ is a site of definition for $ℰ$.

Some classes of sites have their special names

Definition

A site is called

• a small site, large site, essentially small site if the underlying category is a small category, large category, essentially small category, respectively;

• a cartesian site if the underlying category is finitely complete (which the Elephant calls a cartesian category);

• a standard site if it is a cartesian site equipped with a subcanonical coverage.

The term standard site appears in (Johnstone, example A2.1.11).

Remark

Often a site is required to be a small category. But also large sites play a role.

Remark

Every coverage induces a Grothendieck topology. Often sites are defined to be categories equipped with a Grothendieck topology. Some constructions and properties are more elegantly handled with covergaes, some with Grothendieck topologies.

Notice that there are many equivalent ways to define a Grothendieck topology, for instance in terms of a system of local isomorphisms, or in terms of a system of dense monomorphisms in the category of presheaves $\mathrm{PSh}(S)$.

Definition

For $(C,J)$ a site, we write ${\mathrm{Sh}}_J(C)$ for the category of sheaves on $C$ with respect to the coverage $J$.

Definition

A morphism of sites $f:(C,J)\to (D,K)$ is

• a functor $f:C\to D$;

• such that

  1, $f$ is a representably flat functor;

  1. $f$ preserves covers

     in that for every covering $\{p_i:U_i\to U\}$ of an object $U\in C$ with respect to the coverage $J$, we have that $\{f(p_i):f(U_i)\to f(U)\}$ is a covering of $f(U)\in D$ with respect to the covering $K$.

Remark

If $C$ has finite limits then the condition that $f:C\to D$ is a flat functor is equivalent to $f$ preserving these finite limits: to $f$ being a left exact functor.

Properties

Subcanonical sites

Proposition

For $ℰ$ a sheaf topos, the essentially small sites of definition $(𝒞,J)$ of $ℰ$ such that $J$ is a subcanonical coverage are precisely the full subcategories on generating families of objects equipped with the coverages induced from the canonical coverage of $ℰ$.

This appears as (Johnstone, prop. C2.2.16).

Morphisms of sites and geometric morphisms

We discuss how morphisms of sites, def. 4, induce geometric morphisms of the corresponding sheaf toposes, and how the converse holds, too, under certain conditions. The reader might want to first have a look at the discussion of Geometric morphisms between presheaf toposes.

Proposition

Let $f:(𝒞,J)\to (𝒟,K)$ be a morphism of sites.

Notice that precomposition with $f$ defines a functor between categories of presheaves $(-)\circ f:\mathrm{PSh}(𝒟)\to \mathrm{PSh}(𝒞)$.

There is a geometric morphism between the categories of sheaves

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

where $f_{*}$ is the restriction of $f$ to sheaves.

This appears for instance as (Johnstone, lemma C2.2.3, cor. C2.2.4).

Proof

By the assumption that $f$ preserves covers we have that the restriction of $(-)\circ f$ to ${\mathrm{Sh}}_K(𝒟)$ indeed factors through $\mathrm{Sh}(𝒞)\hookrightarrow \mathrm{PSh}(𝒞)$.

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

$$\begin{array}{rl}{\mathrm{PSh}}_{𝒞}(\underrightarrow{\lim }(\coprod _{i,j}{U}_i\prod _U{U}_j)\overrightarrow{\to }\coprod _i{U}_i),F(f(-)))& \simeq \underleftarrow{\lim }(\prod _{i,j}{\mathrm{PSh}}_{𝒞}(U_i\prod _U{U}_j,F(f(-)))\overleftarrow{\leftarrow }\prod _i{\mathrm{PSh}}_C(U_i,F(f(-))))\\ & \simeq \underleftarrow{\lim }(\prod _{i,j}F(f(U_i\prod _U{U}_j)))\overleftarrow{\leftarrow }\prod _{i}F(f(U_i)))\\ & \simeq \underleftarrow{\lim }(\prod _{i,j}F(f(U_i)\prod _{f(U)}f(U_j))))\overleftarrow{\leftarrow }\prod _{i}F(f(U_i)))\\ & \simeq {\mathrm{PSh}}_{𝒟}(\underrightarrow{\lim }(\coprod _{i,j}f(U_i)\prod _{f(U)}f(U_j))\overrightarrow{\to }\coprod _{i}f(U_i)),F)\end{array},$$\begin{aligned} PSh_{\mathcal{C}}( \lim_\to ( \coprod_{i, j} U_i \prod_{U} U_j) \stackrel{\to}{\to} \coprod_{i} U_i ) \;,\; F(f(-)) ) & \simeq \lim_{\leftarrow} \left( \prod_{i,j} PSh_{\mathcal{C}}( U_i \prod_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 \prod_U U_j))) \stackrel{\leftarrow}{\leftarrow} \prod_i F(f(U_i)) \right) \\ & \simeq \lim_{\leftarrow} \left( \prod_{i,j} F(f(U_i) \prod_{f(U)} f(U_j)))) \stackrel{\leftarrow}{\leftarrow} \prod_i F(f(U_i)) \right) \\ & \simeq PSh_{\mathcal{D}}( \lim_\to ( \coprod_{i, j} f(U_i) \prod_{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 $\mathrm{PSh}(-,-)$ sends colimits in the first argument to limits, and the assumption that $f$ preserves the pullbacks involved.

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

$$f_{*}:{\mathrm{Sh}}_K(𝒟)\hookrightarrow \mathrm{PSh}(𝒟)\stackrel{(-)\circ f}{\to }{\mathrm{Sh}}_J(𝒞)$$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 $(-)\circ f$ and to sheaf inclusion. The first is left Kan extension ${\mathrm{Lan}}_f$ along $f$ and the second is sheafification $L_J$ on $(𝒞,J)$, so the left adjoint is the composite

$$f^{*}:{\mathrm{Sh}}_J(𝒞)\hookrightarrow \mathrm{PSh}(𝒞)\stackrel{{\mathrm{Lan}}_f}{\to }S\mathrm{PSh}(𝒟)\stackrel{L_J}{\to }{\mathrm{Sh}}_J(𝒟).$$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 ${\mathrm{Lan}}_f$ does. This is the case if $f$ is a flat functor.

(Because the left Kan extension is given by the colimit ${\mathrm{Lan}}_{f}X:d\mapsto {\lim }_{\to }((f^{\mathrm{op}}/d)\to {𝒞}^{\mathrm{op}}\stackrel{X}{\to }\mathrm{Set})$ over the comma category $f^{\mathrm{op}}/d$ which is a filtered category if $f$ 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 $(𝒞,J)$ and $(𝒟,K)$ be cartesian sites such that

• $𝒞$ is a small category,

• $𝒟$ is a possibly large site but with a small dense subsite (an “essentially small site” in (Johnstone, p. 548))

• and the coverage $K$ is subcanonical.

Then a geometric morphism of the corresponding sheaf toposes

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

is induced by a morphism of sites, def. 4,

$$(𝒟,K)\leftarrow (𝒞,J):F$$(\mathcal{D}, K) \leftarrow (\mathcal{C}, J) : F

precisely if the inverse image of $f$ respects the Yoneda embeddings $j$ as

$$\begin{array}{ccc}𝒟& \stackrel{F}{\leftarrow }& 𝒞\\ {}^{j_{𝒟}}\downarrow & & {\downarrow }^{j_{𝒞}}\\ \mathrm{Sh}(𝒟,K)& \stackrel{f^{*}}{\leftarrow }& \mathrm{Sh}(𝒞,J)\end{array}.$$\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) } \,.

This appears as (Johnstone, lemma C2.3.8).

Proof

It suffices to show that given $f$, the factorization $F$ is, if it exists, necessarily a morphism of sites: because since $f^{*}$ is left adjoint and thus preserves all colimits and every object in $\mathrm{Sh}(C)$ is a colimit of representables, $f^{*}$ 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. 2.

So we show that $F$ is necessarily a morphisms of sites:

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

2. $f^{*}{h}_{𝒞}$ preserves coverings (maps them to epimorphisms in $\mathrm{Sh}(D,K)$) and since $K$ is assumed to be subcanonical it follows from prop. 1 that $j_{𝒟}$ also reflects covers. Therefore $F$ preserves covers.

Corollary

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

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

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

This appears as (Johnstone, cor. C2.3.9).

Proof

Since for the canonical coverage the Yoneda embedding is the identity, this follows directly from prop. 3.

Remark

Corollary 1 leads to the notion of classifying toposes. See there for more details.

Morita equivalent sites

Many inequivalent sites may have equivalent sheaf toposes.

Proposition

Every sheaf topos has a standard site of definition.

This appears as (Johnstone, theorem C2.2.8 (iii)).

Remark

By cor. 1 this means that every sheaf topos is the classifying topos for some theory of local algebras.

Examples

Classes of sites

• Every frame is canonically a site, where $U$ is covered by $\{U_i\}$ precisely if it is their join.

  A subclass of examples is the category of open subsets of a topological space.

  This are examples of posites/(0,1)-site.

• Various categories come with canonical structures of sites on them:

  • For every category $C$ there is its canonical coverage.

  • On every regular category there is its regular coverage.

  • On every coherent category there is its coherent coverage.

  If the category in question is the syntactic category of a theory, the corresponding site is the syntactic site.

• For every site there is the corresponding double negation topology that forces the sheaf topos to a Boolean topos.

Other classes of sites are listed in the following.

• big site

• dense sub-site

• large site, essentially small site

• concrete site

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site

Specific sites

• Sites for big toposes defining notions of geometry are:

  • The sites that define the geometry called differential geometry are CartSp${}_{\mathrm{smooth}}$, SmoothMfd, etc, equipped with the open cover coverage. Or more generally smooth loci etc.

  • The sites that induce [[topology]topological geometry]] are small versions of Top equipped with the open cover coverage.

  • The sites that induce the higher geometry modeled on Euclidean topology are the large site of paracompact manifolds and its dense sub-site CartSp${}_{\mathrm{top}}$.

• The sites that define the geometry called algebraic geometry are site structures on categories of formal duals of commutative rings or commutative associative algebras

  • fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site

    crystalline site

Morphisms of sites

Example

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

Example

(sub-sites)

For $X$ a presite and $U\in S_X$ an object in the corresponding category, the comma category $(Y_{S_X}/U)$ is naturally regarded as the presite defined by $U$, which by convenient abuse of notation one would just write $U$ itself, so that $S_U=S_X\downarrow U$.

For instance for $X$ a topological space and $U\in \mathrm{Op}(X)$ an open subset, $U$ regarded as a topological space in its own right has corresponding to it the site $U$ with $S_U=\mathrm{Op}(U)=\mathrm{Op}(X)\downarrow U$.

The forgetful functor

$$j_{U\to X}^t:(S_U=(Y_{S_X}/U))\to S_X$$j^t_{U \to X} : (S_U = (Y_{S_X} / U)) \to S_X

therefore constitutes a canonical morphism of pre-sites

$$j_{U\to X}:X\to U.$$j_{U \to X} : X \to U \,.

This induces a Grothendieck topology on the site $U$ whose local epimorphisms $(Y\to U)\in [S_U^{\mathrm{op}},\mathrm{Set}]$ are precisely those morphisms for which

$${\hat{j}}_{U\to X}^t(Y\to U)\in [S_X^{\mathrm{op}},\mathrm{Set}]$$\hat j^t_{U \to X}(Y \to U) \in [S_X^{op}, Set]

is a local epimorphism.

This is also called the big site of $U$

There are natural operations for restriction and extension of sheaves from a sub-site $U$ to $X$ and back.

Example

For $C$ and $D$ 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.

Example

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

Related concepts

• site

  • internal site

• 2-site, (2,1)-site

• (∞,1)-site

  • model site, simplicial site

References

Morphisms between sites are discussed for instance

in section 17.2 of

• Kashiwara-Schapira, Categories and Sheaves

(in terms of local isomorphisms)

as well as in section VII. 10 of

• Saunders MacLane Ieke Moerdijk, Sheaves in Geometry and Logic

(in terms of covering sieves), where also the relation to geometric morphisms is discussed.

In

• Peter Johnstone, Sketches of an Elephant

sites are discussed in section C2.1 and morphism of sites in C2.3.