nLab injective topos

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 $GrTop$ 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).

Example

• $Set$ is the only injective Boolean topos (cf Johnstone 2002, p.740).

Related entries

• injective object

• continuous category

• totally distributive category

• exponentiable topos

• De Morgan topos

References

• Peter Johnstone, Injective Toposes , pp.284-297 in LNM 871 Springer Heidelberg 1981.

• Peter Johnstone, Sketches of an Elephant vol. 2 , Cambridge UP 2002. (section C4.3, pp.738-745)

• Peter Johnstone, André Joyal, Continuous categories and exponentiable toposes , JPAA 25 (1982) pp.255-296.

• Rory Lucyshyn-Wright, Totally distributive toposes , arXiv:1108.4032 (2011). (abstract)