From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5099 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in REL Date: Thu, 20 Aug 2009 17:38:29 -0400 (EDT) Message-ID: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1250881604 30308 80.91.229.12 (21 Aug 2009 19:06:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 21 Aug 2009 19:06:44 +0000 (UTC) To: soloviev@irit.fr, categories@mta.ca Original-X-From: categories@mta.ca Fri Aug 21 21:06:36 2009 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 4.50) id 1MeZRb-00042B-Lf for gsmc-categories@m.gmane.org; Fri, 21 Aug 2009 21:06:35 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MeYtX-0000QU-92 for categories-list@mta.ca; Fri, 21 Aug 2009 15:31:23 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5099 Archived-At: On Thu, 20 Aug 2009, soloviev@irit.fr wrote: > To the list: > > I am actually travelling (in Russia) and I need more or > less urgently a reference concerning pullbacks and pushouts > in the category of sets and relations - do they always exist > etc - I would not adress it to the list if it would be > not urgent and I would not have some difficulty with > search from here - > > Best to all - > > Sergei Soloviev > You should check this, but it seems right. Rel is self dual and each object is too. So limits are colimits (of the dual diagram). Rel has arbitrary sums and products--they are disjoint unions. The empty set is initial and terminal. So to have pullbacks you need equalizers. So let A and B be sets and R,S \inc A x B. Let A_0 be the subset of A consisting of all a such that (a,b) \in R iff (a,b) \in S. Then it seems to me that the inclusion function of A_0 into A is the equalizer of R and S. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]