nLab
2-Giraud theorem

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 KK, the following are equivalent.

In fact, it is not hard to prove the same theorem for n-categories, for any 1n21\le n\le 2.

Theorem

For an n-category KK, the following are equivalent.

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

For n=2n=2 this is Street’s theorem; for n=1n=1 it is the classical theorem. The other values included are of course n=(1,2)n=(1,2) and n=(2,1)n=(2,1).

References

Created on March 9, 2012 18:53:53 by Urs Schreiber (82.113.106.131)