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 $1\le n\le 2$.

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 $n=(2,1)$.