For any small category, C, the presheaf category of functors from C^op to Set is a topos (and given a coverage on C restricting to those functors that are sheaves gives rise to a topos). Are there any known results of this type where C is not necessarily small?
A category is equivalent to a small category if and only if both it and its presheaf category are locally small. For a simple proof, see [P. Freyd and R. Street, On the size of categories, Theory and Applications of Categories 1 (9) (1995) 174-178]. So the only way a presheaf category on a locally small category C can be a Grothendieck topos is for C to be equivalent to a small category. I have a very old related conjecture: if the category E of sheaves on a locally small site is locally small then E is a Grothendieck topos. Ross 20-Jun-2002 09:56:36 -0300,3138;000000000000-00000000