From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7084 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Dualities arising via pairs of schizophrenic objects Date: Mon, 28 Nov 2011 12:12:09 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1322574465 3101 80.91.229.12 (29 Nov 2011 13:47:45 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 29 Nov 2011 13:47:45 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Tue Nov 29 14:47:41 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RVO27-0002UZ-SD for gsmc-categories@m.gmane.org; Tue, 29 Nov 2011 14:47:40 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:55991) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RVO10-00022c-Dn; Tue, 29 Nov 2011 09:46:30 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RVO0y-0004UU-RB for categories-list@mlist.mta.ca; Tue, 29 Nov 2011 09:46:28 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7084 Archived-At: There's something odd about when this term is used (under whatever name). The implication is that it's two manifestations of the "same" object, one in each of a dual pair C, C' of categories, d in C and d' in C'. When C is equivalent to C' (self-duality), as with FinVect, CSLat, Chu(V,k), etc., this point of view seems mathematically appropriate. But when not, as with the Stone duality of Boolean algebras, the duality D: C --> C' doesn't even carry d to d', and moreover C(d,d) and C'(d',d') typically don't even have the same number of endomorphisms. Typically d and d' cogenerate and their respective images D(d) and D'(d') generate. In this case it would seem preferable to call D(d) the counterpart (up to isomorphism) in C' of d in C, and conversely for D'(d') (writing D' for the adjoint to D making it a duality). What's odd is that the term seems to be used precisely when it is mathematically inappropriate in the above sense (quite apart from medical or sensitivity issues). The real manifestation of the "same" entity is not with objects at all but with homsets, namely the homset C(D'(d'), d) in C and the homset C'(D(d), d') in C', which *do* have the same number of morphisms. If anything deserves the epithet in question it is that homset in each category. The two homsets are in bijection, but their targets don't correspond, having only in common that they are the dualizers in the respective categories. (I made the same point in the previous flurry on this topic a year or so ago, hopefully more clearly this time around.) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]