Michael Shulman
2-Giraud theorem

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

Theorem

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 $1 \leq n \leq 2$ .

Theorem

For a $n$ -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 $n = (2, 1)$ .

Revised on June 12, 2012 11:10:00 by Andrew Stacey? (129.241.15.200)

Michael Shulman 2-Giraud theorem

Theorem

Theorem

Michael Shulman
2-Giraud theorem