On Wed, 19 Jun 2002, Christopher Townsend wrote:
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?
Thank you,
Christopher Townsend, OU
Two partial answers: (a) A result due to H. Engenes (I don't have the reference to hand, but it's early 1970s): if C is a category such that each slice C/A is (equivalent to) a small category, then [C^op,Set] is a topos. Such categories C need not be small (e.g. the ordered class of ordinals, or the category of sets and injective functions), and the resulting toposes need not be locally small (equivalently, defined over Set). I suspect that there may be a converse to this result, but I've never managed to prove it. (b) It is well known (in SGA4, and many other places) that if (the underlying category of) a site (C,T) has a small subcategory D which is dense for T, then the category of T-sheaves on C is a topos. Presumably one could weaken the smallness requirement on D to the condition in (a). Again, I don't know whether there is a converse. Peter Johnstone 20-Jun-2002 09:56:36 -0300,1592;000000000000-00000000