The following theorem, which generalizes the classical Giraud theorem, is due to StreetCBS.
For a 2-category , the following are equivalent.
- is equivalent to the 2-category of 2-sheaves on a small 2-site.
- is an infinitary 2-pretopos with a small eso-generator.
- is a reflective sub-2-category of a category of 2-presheaves with left-exact reflector.
In fact, it is not hard to prove the same theorem for -categories, for any .
For a -category , the following are equivalent.
- is equivalent to the -category of n-sheaves on a small n-site.
- is an infinitary n-pretopos with a small eso-generator.
- is a reflective sub--category of a category of -presheaves with left-exact reflector.
For this is Street’s theorem; for it is the classical theorem. The other values included are of course and .