"Jeff Egger" <jeffegger@yahoo.ca> wrote:
--- Michael Shulman <shulman@math.uchicago.edu> wrote:
[I don't see the original post, so I'm responding here]
Actually, currently my favorite level of generality is something I call a "monoidal fibration". Roughly, the idea is that you have two different "base" categories, S and V, such that the object-of-objects comes from S while the object-of-morphisms comes from V. When S and V are the same, you get internal categories, and when S=Set, you get classical enriched categories. This could be regarded as "explaining" the coincidence of internal and enriched categories for V=Set.
Heh... I'm studying the same problem as a part of my Master Thesis (under supervision of prof. Andrzej Tarlecki), but fortunately :-) in a bit different framework. The chief concept of my work is a definition of a category ("elementary category") in a fibred monoidal category (i.e. each fibre is monoidal and reindexing functors preserve the monoidal structure) over a base category with binary products. Than, roughly speaking, for a category C with finite limits, C-enriched categories are just "Fam : Fam(C) -> Set"-elementary categories, and C-internal categories are just "Cod : C^{->} -> C"-elementary categories. It turns out (if I didn't make mistakes :-)), that when C has Set-indexed coproducts, than there is an adjunction between the global section functor C(1, -) : Cod -> Fam and the "coproduct functor" \coprod_{-}(1) : Fam -> Cod. Furthermore, if the coproducts are universal, than these functors are fibred and preserves the monoidal structures, and if additionally all global sections in C are disjoint (i.e. the pullback of two different global section is an initial object) than this adjunction is an equivalence of categories (these results give us approximations C-internal categories ---> C-enriched categories and in the other direction). Best regards, Michal R. Przybylek