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Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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 ℰ\mathcal{E} a topos, an internal site in ℰ\mathcal{E} is an internal category ℂ\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.

Definition

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

𝒞⋊𝔻→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 GG be a group (in Set, hence a discrete group) and let 𝒞:=BG\mathcal{C} := \mathbf{B}G be its delooping groupoid. Then

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

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

In this case we have that

𝒞⋊𝔻≃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 GG-action on HH.

Generally we have

Note

The category 𝒞⋊𝔻\mathcal{C} \rtimes \mathbb{D} from def. 2 is described as follows:

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

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

Proposition

We have an equivalence of categories

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

between the category of internal presheaves inℰ\mathcal{E}category of internal presheaves over in the internal category 𝔻 ℰ \mathbb{D} \mathcal{E} , and external presheaves over the semidirect internal product category site𝒞⋊𝔻 \mathcal{C} \rtimes \mathbb{D}, and external presheaves over the semidirect product site 𝒞⋊𝔻\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 𝒞\mathcal{C} is equipped with a coverage JJ and 𝔻\mathbb{D} is equipped with an internal coverage KK , define a coverage J⋊KJ \rtimes K on 𝒞⋊𝔻\mathcal{C} \rtimes \mathbb{D} by declaring that a sieve on an object (U,V)(U,V) is (J×K)(J \times K)-covering if there exists an element S∈K(U)S \in K(U) with b(S)=Vb(S) = V, …

Proposition

Let ℰ=Sh J(𝒞)\mathcal{E} = Sh_J(\mathcal{C}) be a sheaf topos and (𝔻,K)(\mathbb{D}, K) an internal site in ℰ\mathcal{E}. Then with def. 3 we have an equivalence of categories

Sh K(𝔻)≃Sh J⋊K(𝒞⋊𝔻) Sh_{K}(\mathbb{D}) \simeq Sh_{J \rtimes K}(\mathcal{C} \rtimes \mathbb{D})

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

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

Γ:Sh K(𝔻)→ℰ. \Gamma : Sh_K(\mathbb{D}) \stackrel{}{\to} \mathcal{E} \,.

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

References

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)

Revision on March 16, 2012 at 14:32:55 by Urs Schreiber See the history of this page for a list of all contributions to it.

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