From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5104 Path: news.gmane.org!not-for-mail From: Jamie Vicary Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in REL Date: Fri, 21 Aug 2009 21:38:07 +0100 Message-ID: Reply-To: Jamie Vicary NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1250946914 8195 80.91.229.12 (22 Aug 2009 13:15:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 22 Aug 2009 13:15:14 +0000 (UTC) To: Michael Barr , categories@mta.ca Original-X-From: categories@mta.ca Sat Aug 22 15:15:07 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 1MeqR1-0005oi-5s for gsmc-categories@m.gmane.org; Sat, 22 Aug 2009 15:15:07 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MepwO-0006xL-PZ for categories-list@mta.ca; Sat, 22 Aug 2009 09:43:28 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5104 Archived-At: Rel does not have all finite equalizers. I believe the smallest example is the parallel pair (1 1 0 0) and (0 0 1 1), seen as relations from the 4-element set to the 1-element set. Cheers, Jamie. 2009/8/20 Michael Barr : > On Thu, 20 Aug 2009, soloviev@irit.fr wrote: > >> To the list: >> >> I am actually travelling (in Russia) and I need more or >> less =A0urgently 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. =A0Rel is self dual and each > object is too. =A0So limits are colimits (of the dual diagram). =A0Rel ha= s > arbitrary sums and products--they are disjoint unions. =A0The empty set i= s > initial and terminal. =A0So to have pullbacks you need equalizers. =A0So = let A > and B be sets and R,S \inc A x B. =A0Let A_0 be the subset of A consistin= g > of all a such that (a,b) \in R iff (a,b) \in S. =A0Then it seems to me th= at > 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/ ]