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 $GrTop$ if for every inclusion $i:\mathcal{F}\hookrightarrow\mathcal{G}$ and 1-cell $f:\mathcal{F}\to\mathcal{E}$ there exists a 1-cell $g:\mathcal{E}\to\mathcal{G}$ and a 2-isomorphism $g\circ i\cong f$ .

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 $GrTop$ 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).