From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4421 Path: news.gmane.org!not-for-mail From: Sam Staton Newsgroups: gmane.science.mathematics.categories Subject: Re: General notions of equivalence and exactness Date: Thu, 5 Jun 2008 19:54:57 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v924) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019937 13219 80.91.229.2 (29 Apr 2009 15:45:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:37 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jun 6 09:57:21 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 06 Jun 2008 09:57:21 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1K4bGB-0003DU-8C for categories-list@mta.ca; Fri, 06 Jun 2008 09:41:35 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 2 Original-Lines: 71 Xref: news.gmane.org gmane.science.mathematics.categories:4421 Archived-At: Many thanks to all who replied to my message, in private and publicly. I notice that these concerns also arose on this mailing list back in 1992: http://www.mta.ca/~cat-dist/archive/1992/92-06.txt At that time, Michael Barr was asking, amongst other things, about the exactness property that "every Mal'cev [=difunctional=z-closed] relation is a pullback", which holds both in toposes and in abelian categories (as I mentioned below). I wonder if anything more came out of that. He mentioned a possible connection with "effective unions", but I haven't been able to get anything to work there. By the way, following the comments about Mal'cev operators, and Peter Freyd's "Mal'cev allegories", I note that (exact) categories in which every relation is difunctional have been called "Mal'cev categories" [see e.g. the book by Bourn and Borceux on the topic (pointed out by Peter Lumsdaine), or Carboni, Lambek, Pedicchio, Diagram chasing in Mal'cev categories, JPAA 69]. Sam On 29 May 2008, at 10:01, Sam Staton wrote: > Hello. In a category with pullbacks, say that a binary relation > X <- R -> Y > is "z-closed" if it satisfies the following axiom (interpreted as > usual): > > If x R y and x' R y and x' R y' then x R y'. > > (The "z" in "z-closed" refers to the pattern of variables in the > premise.) > > Z-closedness seems to be a sensible generalization of "equivalence" > to relations between two different objects. (e.g. In computer > science, it is common to relate the state spaces of two different > systems.) Note that an endorelation is an equivalence relation if and > only if it is z-closed and reflexive. Also note that, in an abelian > category, every relation is z-closed. > > The [z-closed v. equivalence] connection seems to extend to > [pullbacks v. kernel pairs]. Every span that arises from a pullback > is a z-closed relation. Say that a category is "z-effective" if every > z-closed relation arises as a pullback. > > - every abelian category is straightforwardly z-effective; > - in a topos, every z-closed relation arises as a pullback span. > Indeed, an extensive regular category has effective equivalence > relations if and only if it is z-effective. > > These notions and ideas seem quite elementary, even fundamental, and > I would be surprised if no-one had thought of them before. I borrowed > the terminology "z-closed" from a paper by Erik de Vink and Jan > Rutten (Theoret Comput Sci, 221:271-293, 1999) but I couldn't find > any other references. > > Have I missed something? I'd be grateful for any observations or > suggestions. > > Sam > > PS. I'd like to take the opportunity to acknowledge the helpful > replies (public and private) to my question about W-types, a few > months ago. > >