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)