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Topos Theory

topos theory

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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 =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 gorupoid 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. 3 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. 4 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 : 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)

Revised on October 23, 2013 08:17:56 by Urs Schreiber (188.200.54.65)