From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5112 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in REL Date: Sat, 22 Aug 2009 12:08:34 -0700 Message-ID: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1251561766 3493 80.91.229.12 (29 Aug 2009 16:02:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 29 Aug 2009 16:02:46 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sat Aug 29 18:02:39 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 1MhQNz-0007UB-21 for gsmc-categories@m.gmane.org; Sat, 29 Aug 2009 18:02:39 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MhPsP-00058w-NY for categories-list@mta.ca; Sat, 29 Aug 2009 12:30:01 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5112 Archived-At: Peter Freyd gave a talk at a small conference circa 1990 (at CSLI perhaps?) where he drew attention to the lack of (co)equalizers in Rel as a warm-up (I forget how) to an eloquent tribute to the benefits of toposes for intuitionistic logic. Oddly there didn't seem to be anything about this in Cats and Alligators. However 1.429 therein makes the point that "if a category has enough equalizers then all idempotents split." So a witness to missing equalizers is any nonsplitting idempotent, for example Dominic's z below. Vaughan Pratt Dominic Hughes wrote: > Rel doesn't have equalizers (pullbacks, pushouts, or co-equalizers). It's > rather surprising, and apparently not very well known. > > Counter-example. Let A = { a, b }, write id for the identity A -> A, and > define z : A -> A by adding an edge to id: > > a --- a > / > / > / > b --- b > > Then id and z do not have an equalizer in Rel. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]