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

For objects and pointed objects

Write $\infty Grpd_{fin}$ for the (∞,1)-category of finite homotopy types and write $\infty Grpd_{fin}^{\ast/}$ for its pointed objects.

For $\mathbf{H}$ any base (∞,1)-topos, the (∞,1)-presheaf (∞,1)-topos

$$\mathbf{H}[X] \coloneqq PSh(\infty Grpd_{fin}^{op}, \mathbf{H})$$

is the classifying $\infty$-topos for objects, and

$$\mathbf{H}[X_\ast] \coloneqq PSh((\infty Grpd_{fin}^{\ast/})^{op}, \mathbf{H})$$

is the classifying $\infty$-topos for pointed objects.

For $K \in \infty Grpd_{fin}^{\ast/}$, write $R(K) \in (\infty Grpd_{fin}^{\ast/})^{op} \hookrightarrow \mathbf{H}[X_\ast]$ for its formal dual under (∞,1)-Yoneda embedding. The generic pointed object in $\mathbf{H}[X_\ast]$ is that represented by the 0-sphere:

$$X_\ast = R(S^0) \,.$$

For local structures

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.

category: higher topos theory