Is there a standard name for those categories, A, such that the slice category A/X is a topos for each object X in A (e.g. the category whose objects are topological spaces and whose maps are local homeomorphisms)? Are there any theorems?

You should try asking Jean Benabou about this (not, I guess, by email). I seem to remember he worked on this kind of thing a few years ago (but I can't seem to remember what the results were). Andy Pitts ==============================================================================