Re: Teaching Category Theory

[Michal Przybylek <mrp@neostrada.pl>]

"Jeff Egger" <jeffegger@yahoo.ca> wrote:

> --- Michael Shulman <shulman@math.uchicago.edu> wrote:

> My guess would be that it's because for non-category theorists, many
> (perhaps most) categories which arise in practice are enriched (over
> something more exotic than Set), while few are internal (to something
> more exotic than Set).

I think it's not that.

It's just because the concept of internal category is in some sense subsumed
by the concept of fibration. And in practical situations we prefer to work
with fibrations rather than internal categories.

> 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.

What do you mean by "you get internal/enriched categories" ? Do you have a
2-equivalence between the 2-category of all S-internal (resp. enriched)
categories (S-internal functors, S-internal natural transformations) and the
2-category of your categories ? I'm asking because I have encountered some
difficulties here (i.e. in my framework some diagrams are not willing to
commute "on the nose").


Best regards,
Michal R. Przybylek