From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8189 Path: news.gmane.org!not-for-mail From: Uday S Reddy Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits in REL Date: Fri, 4 Jul 2014 12:55:31 +0100 Message-ID: References: Reply-To: Uday S Reddy NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1404540788 30054 80.91.229.3 (5 Jul 2014 06:13:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 5 Jul 2014 06:13:08 +0000 (UTC) Cc: "categories\@mta.ca" To: Ondrej Rypacek Original-X-From: majordomo@mlist.mta.ca Sat Jul 05 08:12:58 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 1X3JDT-0002s8-TV for gsmc-categories@m.gmane.org; Sat, 05 Jul 2014 08:12:56 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46611) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X3JD7-0004dU-Do; Sat, 05 Jul 2014 03:12:33 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X3JD8-0004TN-Mh for categories-list@mlist.mta.ca; Sat, 05 Jul 2014 03:12:34 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8189 Archived-At: [Note from moderator: Apologies for forgetting how recently this was asked and extensivly answered. Some short responses are not being posted.] Ondrej Rypacek writes: > Hi all > > What is known about limits in REL , the (bi)category of sets and relations? > I know there are biproducts; are there equalisers? Professor Johnstone answered this question a few months ago as below. Since REL is the category of *free algebras* for the powerset monad, there is very little chance of a limit of such algebras being free again. To get decent limits, you need to move to the Eilenberg-Moore category of the powerset monad, viz., the category of complete semilattices. Rel does have products and coproducts; they coincide (by self-duality) and are just disjoint unions of sets. If's not hard to see that a relation R \subseteq A \times B is a monomorphism A \to B iff the map PA \to PB sending a subset of A to the set of all R-relatives of its members is injective; dually for epimorphisms. Rel has very few (co)limits other than (co)products; it doesn't even have splittings of all idempotents. (All symmetric idempotents have splittings, but the order-relation \leq \subseteq {0,1} \times {0,1} can't be split.) However, I don't think that the self-duality is in any sense responsible for the lack of (co)limits in Rel. The category of complete join-semilattices is self-dual, and is complete and cocomplete. Cheers, Uday [For admin and other information see: http://www.mta.ca/~cat-dist/ ]