Re: Teaching Category Theory

[Michal Przybylek <mrp@neostrada.pl>]

"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