nLab
Giraud's theorem

Idea

A classical theorem of Giraud characterizes sheaf toposes abstractly as categories with certain properties known as Giraud’s axioms.

In higher topos theory there are corresponding higher analogs .

Statement

Proposition

(Giraud’s theorem)
A category is a Grothendieck topos if and only if it satisfies the following conditions:

1. it is locally presentable;

2. it has universal colimits;

3. it has disjoint coproducts;

4. it has effective quotients.

(e.g. Lurie 2009, Prop. 6.1.0.1);

Similarly:

Proposition

(Giraud-Rezk-Lurie theorem)
An $(\infty,1)$-category is an Grothendieck-Rezk-Lurie-(infinity,1)-topos if and only if it satisfies the following conditions:

1. it is locally presentable;

2. it has universal $\infty$-colimits;

3. it has disjoint coproducts;

4. it has effective groupoid objects.

(e.g. Lurie 2009, Prop. 6.1.0.6)

Similarly, in 2-category theory there is the analogous 2-Giraud theorem for Grothendieck 2-toposes.

Related concepts

• fundamental theorem of topos theory

References

Textbook accounts:

• Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

following:

• Jacob Lurie, Section 2.2 of: On $\infty$-Topoi (arXiv:math/0306109)

Lecture notes:

• Jacob Lurie, Giraud’s theorem, Lecture 10 in: Categorical Logic, 2018 (pdf)