nLab
Giraud's theorem

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

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 ( , 1 ) (\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.

References

Textbook accounts:

following:

Lecture notes:

Last revised on November 13, 2021 at 07:15:49. See the history of this page for a list of all contributions to it.