nLab
classifying (infinity,1)-topos

Idea

The notion of a classifying (∞,1)-topos is the vertical categorification of the notion of classifying topos to the context of (∞,1)-category theory.

Any (∞,1)-topos KK by definition classifies the ∞-geometric morphisms into it in that it is the representing object of geom(,K)geom(-,K).

Examples

A special case of this is the is the notion of a classifying (∞,1)-topos for a geometry in the sense of structured spaces:

The geometry 𝒢\mathcal{G} is the (∞,1)-category that plays role of the syntactic theory. For 𝒳\mathcal{X} an (∞,1)-topos, a model of this theory is a limits and covering-preserving (∞,1)-functor

𝒢𝒳. \mathcal{G} \to \mathcal{X} \,.

The Yoneda embedding followed by ∞-stackification

𝒢YPSh (,1)(𝒢)(¯)Sh (,1)(𝒢) \mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G}) \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G})

constitutes a model of 𝒢\mathcal{G} in the (Cech) ∞-stack (∞,1)-topos Sh (,1)(𝒢)Sh_{(\infty,1)}(\mathcal{G}) and exhibits it as the classifying topos for such models (geometries):

This is Structured Spaces prop 1.4.2.

Created on March 10, 2012 19:18:45 by Stephan Alexander Spahn (79.227.163.200)