Contents

topos theory

# Contents

## 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 $\mathcal{E}$ a topos, an internal site in $\mathcal{E}$ is an internal category $\mathbb{C} = C_1 \rightrightarrows C_0$ 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 $Sv(\mathbb{C}) \hookrightarrow PC_1$ of sieves, where a subobject $S \hookrightarrow C_1$ is a sieve if the composite

$S\times_{C_0} C_1 \to C_1\times_{C_0} C_1 \to C_1$

factors through $S$. Also recall the usual membership relation $\in_{C_1} \stackrel{(n,e)}{\to} PC_1 \times C_1$.

###### Definition

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

• The square

$\array{ T \times_{PC_1} \in_{C_1} & \stackrel{e pr_2}{\to} & C_1 \\ {}^{pr_1}\downarrow & {} & \downarrow^{s} \\ T & \stackrel{b}{\to} & C_0 }$

commutes, where the pullback in the top left corner is of the map $\in_{C_1} \to PC_1$ along $T \to Sv(\mathbb{C}) \hookrightarrow PC_1$.

• If we define the subobject $Q\hookrightarrow T\times_{C_0} C_1 \times_{C_0} T$ as

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

(in the internal language), the composite $Q \hookrightarrow T\times_{C_0} C_1 \times_{C_0} T \stackrel{pr_{23}}{\to} C_1 \times_{C_0} T$ 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 $\mathcal{C}$ be a small category and let $\mathcal{E} := [\mathcal{C}^{op}, Set]$ be its presheaf topos. Let $\mathbb{D} \in \mathcal{E}$ be an internal site. Regarded, by the Yoneda lemma, as a functor $\mathbb{D} : \mathcal{C}^{op} \to Cat$, this induces via the Grothendieck construction a fibered category which we denote

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

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 $\mathcal{C} := \mathbf{B}G$ be its delooping groupoid. Then

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

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

In this case we have that

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

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

Generally we have

###### Remark

The category $\mathcal{C} \rtimes \mathbb{D}$ from def. is described as follows:

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

• morphisms $(U',V') \to (U,V)$ are pairs $(a,b)$ where $a : U' \to U$ is in $\mathcal{C}$ and $b : V' \to \mathbb{D}(a)(V)$ in $\mathbb{D}(U')$.

###### Proposition

We have an equivalence of categories

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

between the category of internal presheaves in $\mathcal{E}$ over the internal category $\mathbb{D}$, and external presheaves over the semidirect product site $\mathcal{C} \rtimes \mathbb{D}$.

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

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

###### Definition

If $\mathcal{C}$ is equipped with a coverage $J$ and $\mathbb{D}$ is equipped with an internal coverage $K$ , define a coverage $J \rtimes K$ on $\mathcal{C} \rtimes \mathbb{D}$ 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 $\mathcal{E} = Sh_J(\mathcal{C})$ be a sheaf topos and $(\mathbb{D}, K)$ an internal site in $\mathcal{E}$. Then with def. we have an equivalence of categories

$Sh_{K}(\mathbb{D}) \simeq Sh_{J \rtimes K}(\mathcal{C} \rtimes \mathbb{D})$

between internal sheaves in $\mathcal{E}$ on $\mathbb{D}$ and external sheaves on the semidirect product site $J \rtimes K$.

Moreover, the projection functor $P : \mathcal{C} \rtimes \mathbb{D}$ is cover-reflecting and induces a geometric morphism

$\Gamma \colon Sh_K(\mathbb{D}) \stackrel{}{\to} \mathcal{E} \,.$

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

Section C2.4 and C2.5 of

The semidirect product externalization of internal sites is due to

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