Re: Teaching Category Theory

[Jeff Egger <jeffegger@yahoo.ca>]

--- 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'm not sure I agree with that: internal groupoids, at the very least,
show up in a variety of situations which non-category theorists can be,
and are, interested in.  Perhaps one of the reasons why some people try
to deal with groupoids as if they weren't a special case of categories
is because they never thought of categories in any other way than as a 
mass of hom-sets.

> 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.

Now I do think there is a good reason for this, which is the fact 
that in functorial semantics (by which I don't just mean the original, 
universal-algebraic, case), the domain category is typically small.
Raising to a small power does not destroy local smallness.

> 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". 

I agree with this entirely, of course.  It follows that, in a first 
course on category theory, one should present both styles of definition 
as soon as possible.  This, in turn, suggests (but does not prove) that 
one should not sweep size distinctions under the carpet.  

> 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.

I look forward to it!

Cheers,
Jeff.