# nLab Giraud's theorem

Contents

topos theory

## Theorems

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## 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 $(\infty,1)$-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 disjoint coproducts;

3. 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.

Textbook accounts:

following:

Lecture notes:
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