From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4416 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: General notions of equivalence and exactness Date: Thu, 29 May 2008 15:24:59 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019933 13188 80.91.229.2 (29 Apr 2009 15:45:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:33 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu May 29 15:59:40 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 May 2008 15:59:40 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1K1nBO-0004hO-Je for categories-list@mta.ca; Thu, 29 May 2008 15:49:02 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 39 Original-Lines: 96 Xref: news.gmane.org gmane.science.mathematics.categories:4416 Archived-At: In Genoa, in the late 60's and after, our group was studying categories of relations. I was interested in relations on abelian categories, for homological algebra, while Gabriele Darbo, Franco Parodi and others were more interested - after the general construction - in relations on sets, and even more in "corelations on sets" (relations on Set^op), called "transductors". (It is the dual construction, based on equivalence classes of cospans of sets, i.e. quotients of the sum of domain and codomain; used to simulate electric connections between two sets of terminals [see how they compose], and as a formal basis for a general "theory of devices".) At that time, a category with involution u |--> u* (typically, a category of relations in some sense) was called "von Neumann regular" if the condition u.u*.u = u holds for every arrow (plainly related to von Neumann regularity of semigroups and rings). The category of relations of sets is not vN-regular, the simplest counterexample being likely the Z-shaped relation which transgresses your condition: R: {x, y} --> {x', y'} x R y, x' R y, x' R y' This relation was precisely called "Z" in the paper [Pa] F. Parodi, Simmetrizzazioni di una categoria II, Sem. Mat. Univ. Padova, 44 (1970), 223-262. http://archive.numdam.org/ARCHIVE/RSMUP/RSMUP_1970__44_/ RSMUP_1970__44__223_0/RSMUP_1970__44__223_0.pdf On the other hand, as you say, category of relations on abelian categories are von Neumann regular (which is a crucial fact in studying subquotients, see Mac Lane's text on Homology). But, interestingly, the category of CORELATIONS on sets is also von Neumann regular, see the paper above [Pa]. Best regards Marco Grandis On 29 May 2008, at 11:01, 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. > > >