nLab
2-Giraud theorem

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.

• $K$ 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 $[C^{\mathrm{op}},\mathrm{Cat}]$ 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 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 $[C^{\mathrm{op}},n\mathrm{Cat}]$ 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

• Mike Shulman, 2-Giraud theorem