nLab
injective 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

(,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

An injective topos is a topos-theoretic generalization of the notion of an injective object familiar from algebra, or in other words, a topos with good extension properties along subtopos inclusions.

Definition

Definition

A Grothendieck topos \mathcal{E} is called injective (with respect to subtopos inclusions) in the 2-category of Grothendieck toposes GrTopGrTop if for every inclusion i:𝒢i:\mathcal{F}\hookrightarrow\mathcal{G} and 1-cell f:f:\mathcal{F}\to\mathcal{E} there exists a 1-cell g:𝒢g:\mathcal{E}\to\mathcal{G} and a 2-isomorphism gifg\circ i\cong f .

Properties

The main characterization of injective toposes is achieved by the following result due to Johnstone-Joyal (1982).

Theorem

There is an equivalence of 2-categories between the full sub-2-category of GrTopGrTop with 0-cells the injective toposes and the 2-category of cocomplete ind-small continuous categories with 1-cells the filtered colimit preserving functors. This equivalence sends an injective topos to its category of points.

Note, that this gives also a characterization of the continuous categories involved as category of points of injective toposes.

Proposition

The lex totally distributive categories with a small dense generator are precisely the injective Grothendieck toposes.

This occurs as theorem 4.5 in Lucyshyn-Wright (2011).

Example

References

Last revised on March 19, 2018 at 17:47:09. See the history of this page for a list of all contributions to it.