**Definitions**

**Transfors between 2-categories**

**Morphisms in 2-categories**

**Structures in 2-categories**

**Limits in 2-categories**

**Structures on 2-categories**

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

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

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

- $K$ is equivalent to the 2-category of 2-sheaves on a small 2-site.
- $K$ is an infinitary 2-pretopos with a small eso-generator?.
- $K$ is a reflective sub-2-category of a category $[C^{op},Cat]$ of 2-presheaves with left-exact reflector.

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

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 $n=(2,1)$.

Created on March 9, 2012 at 18:53:54. See the history of this page for a list of all contributions to it.