nLab
classifying (infinity,1)-topos

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

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

For objects and pointed objects

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

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

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

is the classifying \infty-topos for objects, and

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

is the classifying \infty-topos for pointed objects.

For KGrpd fin */K \in \infty Grpd_{fin}^{\ast/}, write R(K)(Grpd fin */) opH[X *]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 H[X *]\mathbf{H}[X_\ast] is that represented by the 0-sphere:

X *=R(S 0). 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 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.

Revised on January 12, 2016 08:47:00 by Urs Schreiber (195.37.209.180)