From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3882 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: Teaching Category Theory Date: Tue, 4 Sep 2007 12:30:00 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019580 10711 80.91.229.2 (29 Apr 2009 15:39:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:40 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 4 21:45:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Sep 2007 21:45:26 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISiuA-00053K-HH for categories-list@mta.ca; Tue, 04 Sep 2007 21:38:02 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 49 Xref: news.gmane.org gmane.science.mathematics.categories:3882 Archived-At: --- Michael Shulman 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).=20 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=20 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=20 that in functorial semantics (by which I don't just mean the original,=20 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".=20 I agree with this entirely, of course. It follows that, in a first=20 course on category theory, one should present both styles of definition=20 as soon as possible. This, in turn, suggests (but does not prove) that=20 one should not sweep size distinctions under the carpet. =20 > 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=3DSet, you get > classical enriched categories. This could be regarded as "explaining" > the coincidence of internal and enriched categories for V=3DSet. 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.