From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5111 Path: news.gmane.org!not-for-mail From: Dusko Pavlovic Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in REL Date: Fri, 21 Aug 2009 16:45:14 -0700 Message-ID: Reply-To: Dusko Pavlovic NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1250947469 9421 80.91.229.12 (22 Aug 2009 13:24:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 22 Aug 2009 13:24:29 +0000 (UTC) To: Michael Barr , soloviev@irit.fr, categories@mta.ca Original-X-From: categories@mta.ca Sat Aug 22 15:24:22 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 1MeqZx-0000Mc-SS for gsmc-categories@m.gmane.org; Sat, 22 Aug 2009 15:24:22 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Meq1M-0007EB-KC for categories-list@mta.ca; Sat, 22 Aug 2009 09:48:36 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5111 Archived-At: On Aug 20, 2009, at 2:38 PM, 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. i must be missing something. what if A = {0,1}, B = {#}, R = {<0,#>} and S = {<1,#>}? then A_0 = empty. (you don't say how to quantify over b; but since there is just one b here, it doesn't matter) on the other hand take the relation E \inc BxA where E = {<#,0>,<#,1>} now E;R = E;S = id_B but E does not factor through A_0. what am i missing? -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]