Re: Teaching Category Theory

["Michael Shulman" <shulman@math.uchicago.edu>]

[Note from moderator: a response to this item from Jeff Egger was posted
earlier, but the original was not... ]

On 8/31/07, Jeff Egger <jeffegger@yahoo.ca> wrote:
> Actually, what I find intriguing is that it is the definition of
> enriched category which seems to have priority over the definition
> of internal category.  There are, I suppose, historical reasons for
> this (pre-1960 the focus tended to be on AbGp-enriched categories)
> ---but I think it fair to say that (for as long as I can remember,
> which obviously isn't that long from a "historical" perspective)
> the majority of category theorists tend to adopt the internal
> category style of definition (of category) as more primitive.

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).  Even when working over Set, I think it's fair
to say that the vast majority of categories arising in mathematical
practice are locally small.

Since in general, neither enriched nor internal category theory is a
special case of the other, it doesn't seem justified to me to consider
either one as "more primitive".  However, it's worth pointing out that
both are a special case of categories enriched in a bicategory, or in
a double category.

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.  I
wrote a bit about this at the end of "Framed Bicategories and Monoidal
Fibrations" (arXiv:0706.1286), but I intend to say more in a
forthcoming paper.

Mike