From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8190 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits in REL Date: Fri, 4 Jul 2014 15:09:49 +0200 Message-ID: References: Reply-To: Marco Grandis NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1085) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1404540844 30460 80.91.229.3 (5 Jul 2014 06:14:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 5 Jul 2014 06:14:04 +0000 (UTC) To: Ondrej Rypacek , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Jul 05 08:13:59 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1X3JEU-0003lu-G4 for gsmc-categories@m.gmane.org; Sat, 05 Jul 2014 08:13:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46618) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X3JE7-0004mX-VU; Sat, 05 Jul 2014 03:13:35 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X3JE9-0004Vr-7n for categories-list@mlist.mta.ca; Sat, 05 Jul 2014 03:13:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8190 Archived-At: There was a similar question "Limits and colimits in Rel?" by = UweWolter, on 24 Feb 2014, which had various replies. I copy my own (with an addition in [[...]]). Regards MG ----COPY---- As Peter J. is saying, categories of relations have poor (co)limits. [[ = Eg no equalisers nor coequalisers.]] For abelian groups, Rel(Ab) does not even have products (sums). However, if you insert the 2-category Rel into the double category RRel = of sets, mappings and relations [GP1] you have a double category with all double limits and colimits. For instance: the obvious cartesian product a x b: XxY --> X' x Y' = (resp. sum a + b: X+Y --> X' + Y') of two relations a, b is indeed a product (resp. a sum) in the double category. See [GP1] for definitions and discussion of these aspects. Similarly, many bicategories of spans, cospans, relations, = profunctors... have poor (co)limits, but can be usefully embedded in weak double categories (with the same objects, "strict morphisms", "same = morphisms", suitable double cells) that have all limits and colimits. Also adjoints work well in the extended settings: see [GP2]. Best regards Marco [GP1] M. Grandis - R. Par=E9, Limits in double categories, Cah. Topol. = G=E9om. Diff=E9r. Cat=E9g. 40 (1999), 162-220. [GP2] M. Grandis - R. Par=E9, Adjoint for double categories, Cah. = Topol. G=E9om. Diff=E9r. Cat=E9g. 45 (2004), 193-240. both downloadable at: = http://ehres.pagesperso-orange.fr/Cahiers/Ctgdc.htm [For admin and other information see: http://www.mta.ca/~cat-dist/ ]