From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3274 Path: news.gmane.org!not-for-mail From: Peter Freyd Newsgroups: gmane.science.mathematics.categories Subject: dualities Date: Sat, 29 Apr 2006 10:14:14 -0400 (EDT) Message-ID: <200604291414.k3TEEEPt012831@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241019198 8017 80.91.229.2 (29 Apr 2009 15:33:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:18 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Apr 30 13:28:23 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 30 Apr 2006 13:28:23 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1FaEYr-0002SH-6K for categories-list@mta.ca; Sun, 30 Apr 2006 13:14:17 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 72 Original-Lines: 25 Xref: news.gmane.org gmane.science.mathematics.categories:3274 Archived-At: On the subject of favorite dualities: Surely the most important are the self-dualities and the most important of these (so important we stop noticing it as we age) is the category of finite-dimensional vector spaces over a given field. Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules. (For 50 years I've been trying to turn this into an exercise in abelian categories. There's a nice reduction down to the proposition that R/Z is a cogenerator for the category of compact abelian groups, but that fact seems to require some non-trivial functional analysis.) Strange that two of the most important "dualities" are both Pontryagin's. The other is in algebraic topology theory. Then, of course there's my present favorite: the category of finitely presented group-valued functors from the category of finitely presented modules over a commutative ring.