Spahn sheaf on a sheaf (Rev #3)

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.

Local representability of the très petit topos

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

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 15, 2012 at 22:48:00 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.