nLab
internal site

Idea

The notion of site may be internalized in any topos to yield a notion of internal site.

Definition

The definition of internal site is obvious and straightforward.

Definition

For $ℰ$ a topos, an internal site in $ℰ$ is an internal category $ℂ=C_1\rightrightarrows C_0$ equipped with an internal coverage.

Spelled out in components, this means the following (as in (Johnstone), we shall only define sifted coverages). First, we define the subobject $\mathrm{Sv}(ℂ)\hookrightarrow {\mathrm{PC}}_1$ of sieves, where a subobject $S\hookrightarrow C_1$ is a sieve if the composite

$$S{\times }_{C_0}{C}_1\to C_1{\times }_{C_0}{C}_1\to C_1$$S\times_{C_0} C_1 \to C_1\times_{C_0} C_1 \to C_1

factors through $S$. Also recall the usual membership relation ${\in }_{C_1}\stackrel{(n,e)}{\to }{\mathrm{PC}}_1\times C_1$.

Definition

An internal sifted coverage is given by a span $C_0\stackrel{b}{\leftarrow }T\stackrel{c}{\to }\mathrm{Sv}(ℂ)$ subject to:

• The square

  $$\begin{array}{ccc}T{\times }_{{\mathrm{PC}}_1}{\in }_{C_1}& \stackrel{e{\mathrm{pr}}_2}{\to }& C_1\\ {}^{{\mathrm{pr}}_1}\downarrow & & {\downarrow }^s\\ T& \stackrel{b}{\to }& C_0\end{array}$$\array{ T \times_{PC_1} \in_{C_1} & \stackrel{e pr_2}{\to} & C_1 \\ {}^{pr_1}\downarrow & {} & \downarrow^{s} \\ T & \stackrel{b}{\to} & C_0 }

  commutes, where the pullback in the top left corner is of the map ${\in }_{C_1}\to {\mathrm{PC}}_1$ along $T\to \mathrm{Sv}(ℂ)\hookrightarrow {\mathrm{PC}}_1$.

• If we define the subobject $Q\hookrightarrow T{\times }_{C_0}{C}_1{\times }_{C_0}T$ as

  $$Q:=\{(t\prime ,a,t)\mid \mathrm{aa}\prime \in t\forall a\prime \in t\prime \}$$Q := \{(t',a,t) | aa' \in t \forall a'\in t'\}

  (in the internal language), the composite $Q\hookrightarrow T{\times }_{C_0}{C}_1{\times }_{C_0}T\stackrel{{\mathrm{pr}}_{23}}{\to }{C}_1{\times }_{C_0}T$ is required to be an epimorphism.

We can additionally ask that more saturation conditions (as discussed at coverage) hold.

(…)

Properties

Externalization

We discuss how to every internal site there is a corresponding external site such that the internal sheaf topos on the former agrees with the external sheaf topos on the latter.

Definition

Let $𝒞$ be a small category and let $ℰ:=[{𝒞}^{\mathrm{op}},\mathrm{Set}]$ be its presheaf topos. Let $𝔻\in ℰ$ be an internal site. Regarded, by the Yoneda lemma, as a functor $𝔻:{𝒞}^{\mathrm{op}}\to \mathrm{Cat}$, this induces via the Grothendieck construction a fibered category which we denote

$$𝒞⋊𝔻\to 𝒞.$$\mathcal{C} \rtimes \mathbb{D} \to \mathcal{C} \,.

This is reviewed for instance in (Johnstone, p. 596).

The notation is motivated from the following example.

Example

Let $G$ be a group (in Set, hence a discrete group) and let $𝒞:=BG$ be its delooping groupoid. Then

$$ℰ\simeq [BG,\mathrm{Set}]$$\mathcal{E} \simeq [\mathbf{B}G , Set]

is the topos of permutation representations of $G$. Let $H\in ℰ$ be a group object. This is equivalently a group in $\mathrm{Set}$ equipped with a $G$-action. Its internal delooping gives the internal groupoid $𝔻:=ℬH$ in $ℰ$.

In this case we have that

$$𝒞⋊𝔻\simeq B(G⋊H)$$\mathcal{C} \rtimes \mathbb{D} \simeq \mathbf{B}(G \rtimes H)

is the delooping gorupoid of the semidirect product group of the $G$-action on $H$.

Generally we have

Note

The category $𝒞⋊𝔻$ from def. 3 is described as follows:

• objects are pairs $(U,V)$ with $U\in \mathrm{Ob}𝒞$ and $V\in \mathrm{Ob}𝔻(U)$;

• morphisms $(U\prime ,V\prime )\to (U,V)$ are pairs $(a,b)$ where $a:U\prime \to U$ is in $𝒞$ and $b:V\prime \to 𝔻(a)(V)$ in $𝔻(U\prime )$.

Proposition

We have an equivalence of categories

$$[{𝔻}^{\mathrm{op}},[{𝒞}^{\mathrm{op}},\mathrm{Set}]]\simeq [(𝒞⋊𝔻{)}^{\mathrm{op}},\mathrm{Set}]$$[\mathbb{D}^{op}, [\mathcal{C}^{op}, Set]] \simeq [(\mathcal{C} \rtimes \mathbb{D})^{op}, Set]

between the category of internal presheaves in $ℰ$ over the internal category $𝔻$, and external presheaves over the semidirect product site $𝒞⋊𝔻$.

This appears as (Johnstone, lemma C2.5.3).

This result generalizes straightforwardly to an analogous statement for internal sheaves.

Definition

If $𝒞$ is equipped with a coverage $J$ and $𝔻$ is equipped with an internal coverage $K$ , define a coverage $J⋊K$ on $𝒞⋊𝔻$ by declaring that a sieve on an object $(U,V)$ is $(J\times K)$-covering if there exists an element $S\in K(U)$ with $b(S)=V$, …

Proposition

Let $ℰ={\mathrm{Sh}}_J(𝒞)$ be a sheaf topos and $(𝔻,K)$ an internal site in $ℰ$. Then with def. 4 we have an equivalence of categories

$${\mathrm{Sh}}_K(𝔻)\simeq {\mathrm{Sh}}_{J⋊K}(𝒞⋊𝔻)$$Sh_{K}(\mathbb{D}) \simeq Sh_{J \rtimes K}(\mathcal{C} \rtimes \mathbb{D})

between internal sheaves in $ℰ$ on $𝔻$ and external sheaves on the semidirect product site $J⋊K$.

Moreover, the projection functor $P:𝒞⋊𝔻$ is cover-reflecting and induces a geometric morphism

$$\Gamma :{\mathrm{Sh}}_K(𝔻)\stackrel{}{\to }ℰ.$$\Gamma : Sh_K(\mathbb{D}) \stackrel{}{\to} \mathcal{E} \,.

This appears as (Johnstone, prop. C2.5.4).

Related concepts

• site

• internal locale, internal sheaf

• internalization, internal logic

References

Section C2.4 and C2.5 of

• Peter Johnstone, Sketches of an Elephant

The semidirect product externalization of internal sites is due to

• Ieke Moerdijk, Continuous fibrations and inverse limits of toposes, Composition Math. 68 (1986)