From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7071 Path: news.gmane.org!not-for-mail From: tholen@mathstat.yorku.ca Newsgroups: gmane.science.mathematics.categories Subject: Re: Dualities arising via pairs of schizophrenic objects Date: Sat, 26 Nov 2011 11:45:33 -0500 Message-ID: References: Reply-To: tholen@mathstat.yorku.ca 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 1322404015 23166 80.91.229.12 (27 Nov 2011 14:26:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 27 Nov 2011 14:26:55 +0000 (UTC) Cc: Categories , tholen@mathstat.yorku.ca To: Sebastian Kerkhoff Original-X-From: majordomo@mlist.mta.ca Sun Nov 27 15:26:50 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 1RUfgt-0004LK-Rd for gsmc-categories@m.gmane.org; Sun, 27 Nov 2011 15:26:48 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:52958) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RUfg3-0005oP-Nb; Sun, 27 Nov 2011 10:25:55 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RUfg2-00028X-BJ for categories-list@mlist.mta.ca; Sun, 27 Nov 2011 10:25:54 -0400 In-Reply-To: Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7071 Archived-At: Dear Sebastian, Here is an elementary categorical introduction to dualities for the "working mathematician" which you may like to have a look at for the purpose of preparing your course: H.-E. Porst, W. Tholen: Concrete dualities. In: Research and Exposition in Mathematics 18 (Heldermann Verlag, Berlin 1991), pp 111-136. Best wishes, Walter Quoting Sebastian Kerkhoff : > Dear all, > > I have a short and probably very simple question (and I apologize for it > in advance): > > I believe it is a well-known fact that a potential duality arises when a > single object essentially lives in two different categories. Famous > examples for such objects and such dualities are the Gelfand-Duality > (where this object is the space of complex numbers, once as a > topological space and once as an algebraic structure) or the Stone > Duality (where this object is the two-element lattice, once as a Boolean > algebra and once as a bounded poset with discrete topology). > > As far as I know (correct me if am wrong), people started to call these > objects "schizophrenic objects" after this term was introduced by Harold > Simmons in 1982. What I would like to know is the following: Could > anybody provide me with a few lines about the historical development of > this principle? I know that John Isbell is often cited as a source > (however, my impression is that people are not entirely sure), and I > have also heard that Peter Freyd was supposedly the first who studied > these kind of dual adjunctions systematically (proving that such > constructions are often essentially the only way to create dual > adjunctions between two categories). > > In case you are interested, I can also provide you with the reason for > my question: I am giving a (small) course about duality theory in > Dresden, and since most of my students are very interested in universal > algebra, the course also covers the theory of natural dualities > developed by Brian Davey and his various co-authors (it is a theory that > tries to generalize the Stone duality to other algebraic structures). > However, I would like to point out to the students that the principle of > schizophrenic objects is not only a convenient ad-hoc construction for > such natural dualities, but actually a much more general principle that > gives rise to many other dualities (which will be covered in the course > in much less detail). For that, I would like to provide the students > with some historical development of this idea, which I obviously cannot > do as long as I am not at all sure about it myself. Plus, I am also > personally very interested in some background information about this > "schizophrenic" idea. > > Thank you very much. > > Best regards, > Sebastian Kerkhoff > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]