From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3888 Path: news.gmane.org!not-for-mail From: "Michael Shulman" Newsgroups: gmane.science.mathematics.categories Subject: Re: Teaching Category Theory Date: Sep 2, 2007 1:36 AM Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019583 10739 80.91.229.2 (29 Apr 2009 15:39:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:43 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Sep 9 21:23:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Sep 2007 21:23:23 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IUWyx-0002Lc-Sn for categories-list@mta.ca; Sun, 09 Sep 2007 21:18:27 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 40 Xref: news.gmane.org gmane.science.mathematics.categories:3888 Archived-At: [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 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