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 $\mathrm{geom}(-,K)$.

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

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

The Yoneda embedding followed by ∞-stackification

$$𝒢\stackrel{Y}{\to }{\mathrm{PSh}}_{(\mathrm{\infty },1)}(𝒢)\stackrel{\overline{(}-)}{\to }{\mathrm{Sh}}_{(\mathrm{\infty },1)}(𝒢)$$\mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G}) \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G})

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

This is Structured Spaces prop 1.4.2.

category: higher topos theory