From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5866 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: we do meet isomorphisms of categories Date: Thu, 27 May 2010 12:08:31 +0200 Message-ID: References: Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1275151364 25614 80.91.229.12 (29 May 2010 16:42:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 29 May 2010 16:42:44 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sat May 29 18:42:43 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OIP7S-0004Sy-NV for gsmc-categories@m.gmane.org; Sat, 29 May 2010 18:42:42 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OIOXs-0005UN-51 for categories-list@mta.ca; Sat, 29 May 2010 13:05:56 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5866 Archived-At: I should have mentioned another quite elementary example, that is perhaps more intriguing. Let us write Top for (topological spaces defined by open sets) and Top' for the isomorphic category (topological spaces defined by closed sets). Let (X, L) be a set X equipped with a complete sublattice L of its lattice of parts. Viewing it as on object of Top or Top' will interchange an Alexandrov topology for X with the opposite one, generally different. This says that - formally - we cannot think of these two isomorphic categories as being the same thing. Even if, of course, we do think that way, informally and in practice. I am not entirely convinced by a comment of Peter: "in practice (so far as I know) one never makes use of the fact that they are isomorphisms rather than mere equivalences". I am happy with the fact that, going from Top to Top' and back, we get the same space on the nose; this spares a lot of complications. Best regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]