Michael Shulman
2-Giraud theorem

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

Theorem

For a 2-category KK, the following are equivalent.

  • KK is equivalent to the 2-category of 2-sheaves on a small 2-site.
  • KK is an infinitary 2-pretopos with a small eso-generator.
  • KK is a reflective sub-2-category of a category [C op,Cat][C^{op},Cat] of 2-presheaves with left-exact reflector.

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

Theorem

For a nn-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).

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