#
nLab

2-Giraud theorem

### Context

#### 2-Category theory

**2-category theory**

## Definitions

## Transfors between 2-categories

## Morphisms in 2-categories

## Structures in 2-categories

## Limits in 2-categories

## Structures on 2-categories

#### $(\infty,2)$-Topos theory

# Contents

## Idea

The *2-Giraud theorem* is the generalization of Giraud's theorem from topos theory to 2-topos theory.

## Statement

The following theorem, which generalizes the classical Giraud theorem, is due to StreetCBS.

###### Theorem

For a 2-category $K$, the following are equivalent.

In fact, it is not hard to prove the same theorem for n-categories, for any $1\le n\le 2$.

###### Theorem

For an n-category $K$, the following are equivalent.

- $K$ is equivalent to the $n$-category of n-sheaves on a small n-site.
- $K$ is an infinitary n-pretopos with a small eso-generator?.
- $K$ is a reflective sub-$n$-category of a category $[C^{op},n Cat]$ of $n$-presheaves with left-exact reflector.

For $n=2$ this is Street’s theorem; for $n=1$ it is the classical theorem. The other values included are of course $n=(1,2)$ and $n=(2,1)$.

## References

Created on March 9, 2012 18:53:53
by

Urs Schreiber
(82.113.106.131)