nLab internal site

Redirected from "internal sites".
Contents

Context

Categorical algebra

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

Definition

For \mathcal{E} a topos, an internal site in \mathcal{E} is an internal category =C 1C 0\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()PC 1Sv(\mathbb{C}) \hookrightarrow PC_1 of sieves, where a subobject SC 1S \hookrightarrow C_1 is a sieve if the composite

S× C 0C 1C 1× C 0C 1C 1 S\times_{C_0} C_1 \to C_1\times_{C_0} C_1 \to C_1

factors through SS. Also recall the usual membership relation C 1(n,e)PC 1×C 1\in_{C_1} \stackrel{(n,e)}{\to} PC_1 \times C_1.

Definition

An internal sifted coverage is given by a span C 0bTcSv()C_0 \stackrel{b}{\leftarrow} T \stackrel{c}{\to} Sv(\mathbb{C}) subject to:

  • The square

    T× PC 1 C 1 epr 2 C 1 pr 1 s T b C 0 \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 C 1PC 1\in_{C_1} \to PC_1 along TSv()PC 1T \to Sv(\mathbb{C}) \hookrightarrow PC_1.

  • If we define the subobject QT× C 0C 1× C 0TQ\hookrightarrow T\times_{C_0} C_1 \times_{C_0} T as

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

    (in the internal language), the composite QT× C 0C 1× C 0Tpr 23C 1× C 0TQ \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 𝒞\mathcal{C} be a small category and let :=[𝒞 op,Set]\mathcal{E} := [\mathcal{C}^{op}, Set] be its presheaf topos. Let 𝔻\mathbb{D} \in \mathcal{E} be an internal site. Regarded, by the Yoneda lemma, as a functor 𝔻:𝒞 opCat\mathbb{D} : \mathcal{C}^{op} \to Cat, this induces via the Grothendieck construction a fibered category which we denote

𝒞𝔻𝒞. \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

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

is the topos of permutation representations of GG. Let HH \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:=H\mathbb{D} := \mathcal{B}H in \mathcal{E}.

In this case we have that

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

is the delooping groupoid of the semidirect product group of the GG-action on HH.

Generally we have

Remark

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

  • objects are pairs (U,V)(U,V) with UOb𝒞U \in Ob \mathcal{C} and VOb𝔻(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:UUa : U' \to U is in 𝒞\mathcal{C} and b:V𝔻(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]][(𝒞𝔻) op,Set] [\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 𝔻\mathbb{D}, and external presheaves over the semidirect product site 𝒞𝔻\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 JKJ \rtimes K on 𝒞𝔻\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 SK(U)S \in K(U) with b(S)=Vb(S) = V, …

Proposition

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

Sh K(𝔻)Sh JK(𝒞𝔻) Sh_{K}(\mathbb{D}) \simeq Sh_{J \rtimes K}(\mathcal{C} \rtimes \mathbb{D})

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

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

Γ:Sh K(𝔻). \Gamma \colon 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) (NUMDAM)

Last revised on March 9, 2023 at 02:31:40. See the history of this page for a list of all contributions to it.