Spahn sheaf on a sheaf (Rev #4, changes)

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Motivation

Let XHX\in H be a space (an object of a category HH of spaces), let Sh(X)Sh(X) be the category of sheaves on the frame of opens on XX, let (H/X) et(H/X)^{et} denote the wide subcategory of H/XH/X with only étale morphisms. Then there is an adjoint equivalence

(LΓ):(H/X) etΓSh(X)(L\dashv \Gamma):(H/X)^{et}\stackrel{\Gamma}{\to}Sh(X)

where

  • Γ\Gamma sends an étale morphism f:UXf:U\to X to the sheaf of local sections of ff.

  • LL sends a sheaf on XX to its espace étale.

Très petit topos

We wish to clarify in which sense also the (,1)(\infty,1)- topos (H/X) fet(H/X)^{fet} can be regarded as an (,1)(\infty,1)-sheaftopos on XX.

We assume that fetfet is the admissible class defined by an infinitesimal modality \Box on HH.

Definition (universal GG-structure, classifying topos)

(1) A GG-structure OO on an (,1)(\infty,1)-topos is called universal if for every (,1)(\infty,1)-topos XX composition with OO induces an equivalence of (,1)(\infty,1)-categories if

Fun *(K,T)Str G(T)Fun^*(K,T)\to Str_G(T)

(2) In this case we say OO exhibits KK as classifying (,1)(\infty,1)-topos for GG-structures on XX.

Remark

HH, H/XH/X, and (H/X) fet(H/X)^fet are (H/X) fet(H/X)^{fet}-structured (,1)(\infty,1)-toposes.

Proof

The classifying topos for (G/X) fet(G/X)^{fet}-structures is HH and the (,1)(\infty,1)-toposes in question are linked with HH by geometric morphisms. We obtain the required structures as the image of

Fun *(H,H/X)Str (H/X) fet(H/X)Fun^*(H,H/X)\to Str_{(H/X)^{fet}}(H/X)

respectively for HH and (H/X) fet(H/X)^{fet} in place of “H/X”.

Local representability of the très petit topos

We assume that fetfet is the admissible class defined by an infinitesimal modality \Box on HH.

Definition

A GG-structured (∞,1)-topos (X,O G,X)(X,O_{G,X}) is called locally representable (aka a GG-scheme) if

  • there exists a collection {U iX}\{U_i \in X\}

such that

  • the {U i}\{U_i\} cover XX in that the canonical morphism iU i*\coprod_i U_i \to {*} (with *{*} the terminal object of XX) is an effective epimorphism;

  • for every U iU_i there exists an equivalence

    (X/U i,O G,X| U i)Spec GA i (X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G} A_i

    of structured (,1)(\infty,1)-toposes for some A iPro(G)A_i \in Pro(G) (in the (∞,1)-category of pro-objects of GG). In other words (X,O G,X)(X,O_{G,X}) is assumed to be locally equivalent to an absolute spectrum (aka affine scheme) of a pro object in GG.

Remark

HH, H/XH/X, and (H/X) fet(H/X)^fet are (H/X) fet(H/X)^fet-structured (,1)(\infty,1)-toposes.

Definition (universal GG-structure, classifying topos)

(1) A GG-structure OO on an (,1)(\infty,1)-topos is called universal if for every (,1)(\infty,1)-topos XX composition with OO induces an equivalence of (,1)(\infty,1)-categories if

Fun *(K,X)Str G(X)Fun^*(K,X)\to Str_G(X)

(2) In this case we say OO exhibits KK as classifying (,1)(\infty,1)-topos for GG-structures on XX.

Revision on December 16, 2012 at 01:29:19 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.