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Context

Topos Theory

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 $\mathbb{C}$ equipped with an internal coverage.

Spelled out in components, this means the following:

(…)

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

Generally we have

Note

The category $\mathcal{C} \rtimes \mathbb{D}$ from def. 2 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. 3 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 : Sh_K(\mathbb{D}) \stackrel{}{\to} \mathcal{E} \,.

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

Related concepts

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)

Revision on January 28, 2012 at 19:24:56 by Stephan Alexander Spahn See the history of this page for a list of all contributions to it.

nLab internal site (Rev #5, changes)

Context

Topos Theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

Definition

Definition

Properties

Externalization

Definition

Example

Note

Proposition

Definition

Proposition

Related concepts

References