From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8035 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits and colimits in Rel? Date: Thu, 27 Feb 2014 10:56:07 +0100 Message-ID: References: Reply-To: Marco Grandis NNTP-Posting-Host: plane.gmane.org Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1393506790 17730 80.91.229.3 (27 Feb 2014 13:13:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 27 Feb 2014 13:13:10 +0000 (UTC) Cc: categories@mta.ca To: Uwe.Wolter@ii.uib.no Original-X-From: majordomo@mlist.mta.ca Thu Feb 27 14:13:20 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 1WJ0lw-00060k-Ri for gsmc-categories@m.gmane.org; Thu, 27 Feb 2014 14:13:09 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59441) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WJ0lI-0006hT-0E; Thu, 27 Feb 2014 09:12:28 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WJ0lG-0003GD-GR for categories-list@mlist.mta.ca; Thu, 27 Feb 2014 09:12:26 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8035 Archived-At: As Peter J. is saying, categories of relations have poor (co)limits. For abelian groups, Rel(Ab) does not even have products (sums). However, if you insert the 2-category Rel into the double category =20 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 =20 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. oSee [GP] for definitions and discussion of these aspects. Similarly, many bicategories of spans, cospans, relations, =20 profunctors... have poor (co)limits, but can be usefully embedded in weak double categories (with the same objects, "strict morphisms", =20 "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. =20= G=E9om. Diff=E9r. Cat=E9g. 40 (1999), 162-220. [GP2] M. Grandis - R. Par=E9, Adjoint for double categories, Cah. =20 Topol. G=E9om. Diff=E9r. Cat=E9g. 45 (2004), 193-240. both downloadable at: http://ehres.pagesperso-orange.fr/Cahiers/=20 Ctgdc.htm On 24 Feb 2014, at 23:36, Uwe.Wolter@ii.uib.no wrote: > Dear all, > > I remember that there was some time ago a discussion on this list > about limits and colimits in the category Rel of binary relations. > Unfortunately, I can not remember or trace the final answer. But, if I > remember right there are, besides initial and terminal objects, in > general no limits or colimits in Rel. > > So my questions are: > > 1. Is there a characterization of monomorphisms and epimorphisms in =20= > Rel? > 2. Is it true that there are, in general, no products and equalizer > (sums and coequalizer) in Rel? > 3. Are there some general results about what limits/colimits exist or > don't exist? > 4. Is the presumable non-existence related to the fact that the > formation of converse relations establishes an isomorphism between Rel > and its opposite Rel^op? > > Any reply or reference is well-come. > > Best regards > > Uwe Wolter > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]