# 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 $K$ by definition classifies the ∞-geometric morphisms into it in that it is the representing object of $geom(-,K)$.

## Examples

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

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

$\mathcal{G} \to \mathcal{X} \,.$

The Yoneda embedding followed by ∞-stackification

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

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

This is Structured Spaces prop 1.4.2.

Revised on January 15, 2015 13:20:11 by David Corfield (129.12.18.108)