nLab
higher topos theory

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

(,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

Higher topos theory is the generalisation to higher category theory of topos theory. It is partly motivated by Grothendieck‘s program in Pursuing Stacks.

More generally, the concept (n,r)(n,r)-topos is to topos as (n,r)-category is to category.

Rather little is known about the very general notion of higher topos theory. A rich theory however exists in the context of (∞,1)-categories, see at (∞,1)-topos theory

Examples

Flavors of higher toposes

flavors of higher toposes

Archetypical higher toposes

Just as the archetypical example of an ordinary topos (i.e. a (1,1)-topos) is Set – the category of 0-categories – so the \infty-category of (n,r)-categories should form the archetypical example of an (n+1,r+1)(n+1,r+1)-topos:

Example

(examples of archetypical higher toposes)

References

Last revised on August 25, 2021 at 11:38:24. See the history of this page for a list of all contributions to it.