From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5107 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in REL Date: Fri, 21 Aug 2009 22:36:31 +0100 (BST) Message-ID: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1250947109 8601 80.91.229.12 (22 Aug 2009 13:18:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 22 Aug 2009 13:18:29 +0000 (UTC) To: Michael Barr , categories@mta.ca Original-X-From: categories@mta.ca Sat Aug 22 15:18:22 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 1MeqUA-0006rP-5P for gsmc-categories@m.gmane.org; Sat, 22 Aug 2009 15:18:22 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MepxK-0006z2-0Y for categories-list@mta.ca; Sat, 22 Aug 2009 09:44:26 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5107 Archived-At: On Thu, 20 Aug 2009, Michael Barr wrote: > 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. > > Sadly, that doesn't work. Let A = {0,1}, B = {0,1}, and let R and S be respectively the identity relation and the relation which relates each member of A to both members of B. Then Michael's proposed equalizer is empty, but the relation from C = {0} to A which relates 0 to both members of A has equal composites with R and S. (The pair (R,S) does have an equalizer in Rel, namely the relation C -+-> A just described, but with a little more ingenuity you can find parallel pairs in Rel having no equalizer.) Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]