nLab 1-topos

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

A 11-topos, or (1,1)(1,1)-topos, is simply a topos in the usual sense of the word. The prefix 11- may be added when also discussing higher categorical types of topoi in higher topos theory such as 2-topos, (,1)(\infty,1)-topos, or even \infty-topos. Compare that a (0,1)-topos is a Heyting algebra.

Similarly, a Grothendieck 11-topos, or Grothendieck (1,1)(1,1)-topos, is simply a Grothendieck topos. Compare that a Grothendieck (0,1)-topos is a frame (or locale).

Note that a 1-topos is not exactly a particular sort of 2-topos or \infty-topos, just as a Heyting algebra is not a particular sort of 1-topos. The (1,2)-category of locales (i.e. (0,1)-topoi) embeds fully in the 2-category of Grothendieck 1-topoi by taking sheaves, but a locale is not identical to its topos of sheaves (and in fact no nontrivial 1-topos can be a poset), in that the following diagram of functors can not be filled by a natural isomorphism:

Likewise, one expects every Grothendieck 1-topos to give rise to a 2-topos or \infty-topos of stacks, hopefully producing a full embedding of some sort.

flavors of higher toposes

Last revised on September 18, 2022 at 04:44:04. See the history of this page for a list of all contributions to it.