From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3874 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: Teaching Category Theory Date: Fri, 31 Aug 2007 09:37:06 -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 1241019574 10667 80.91.229.2 (29 Apr 2009 15:39:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:34 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Aug 31 17:02:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Aug 2007 17:02:13 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IRCYb-00009V-Oy for categories-list@mta.ca; Fri, 31 Aug 2007 16:53:29 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 55 Xref: news.gmane.org gmane.science.mathematics.categories:3874 Archived-At: --- Peter LeFanu Lumsdaine wrote: > >> [In my experience, non-category-theorists, when asked to > >> provide a definition of category, almost uniformly supply (what amou= nts > >> to) the definition of an enriched category, in the case V=3DSet---wh= ich I > >> find quite intriguing.]=20 >=20 > Surely the intriguing thing here is not (as I understand you to be > suggesting) the set-centricity that they're imposing, but rather that t= hey're > not imposing it as far as usual?=20 Actually, what I find intriguing is that it is the definition of=20 enriched category which seems to have priority over the definition of internal category. There are, I suppose, historical reasons for=20 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,=20 which obviously isn't that long from a "historical" perspective) the majority of category theorists tend to adopt the internal=20 category style of definition (of category) as more primitive. =20 The issue at stake may seem minor: do we think of a class of arrows=20 (which can later be partitioned into homsets), or do we think of the=20 homsets first (and take their disjoint union later)? But perhaps=20 the fact that one group of people prefers one approach and everyone else the other is symptomatic of a psychological divide?=20 It's also worth noting, perhaps, how flukey it is that in the case=20 V=3DSet, V-internal and small V-enriched categories happen to coincide. Consider V=3DCat, for example. Or, note how different the requirements=20 on V are, for V-internal and V-enriched categories to be defined.=20 >When asked to define pretty much any > algebraic gadget, most mathematicians will define a model of that algeb= raic > gadget in Set (see e.g. en.wikipedia.org/wiki/Group_%28mathematics%29 )= . It is true that one would expect set-theoretic conservatives to deal with small categories (~internal categories in the case V=3DSet), and more fle= xible mathematicians to use arbitrary large categories (~internal categories, w= here=20 V is a category of "large sets", or classes). This only re-inforces the=20 points made above. Cheers, Jeff.