I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism I shall silently assume from now on enough universes to work around size issues). Neither are regular categories, so we don't have the usual pretopology consisting of regular epimorphisms, but we do have the canonical pretopology
- What is the canonical pretopology on $Cat$?
And by this I don't just mean 'the regular epimorphisms that are stable under pullback', but a characterisation of these.
There are a number of model structures on $Cat$, namely the Thomason model structure and the canonical model structure (often called the 'folk' model structure) and presumably others. The fibrations in any of these model structures constitute a pretopology. Given a (blah) model structure, call the associated pretopology of fibrations the (blah) pretopology.
- What do the sheaves look like for the Thomason and 'folk' pretopologies?
If we consider $CAT$, then there are number of interesting classes of functors one could conjecture to form a pretopology, like those that are projections from a topological concrete category (these may be called 'topological' in the literature, but this is ambiguous terminology), functors with left (or right) adjoints, monadic functors, localisation functors (or families of localisation functors, as sometimes occurs in non-commutative algebraic geometry a la Rosenberg/Gabriel/et al, but strictly speaking this is a co-pretopology), bifibrations, and so on. If we look at subcategories of $CAT$ then there are more options: geometric morphisms (with certain properties) between topoi come to mind.
- Do any of these (or any others you can think of) form pretopologies? Are their categories of sheaves interesting at all?