From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/417 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: algebraic logic via arrows Date: Mon, 30 Jun 1997 23:42:44 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016953 25665 80.91.229.2 (29 Apr 2009 14:55:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:55:53 +0000 (UTC) To: categories Original-X-From: cat-dist Mon Jun 30 23:42:45 1997 Original-Received: by mailserv.mta.ca; id AA06928; Mon, 30 Jun 1997 23:42:44 -0300 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:417 Archived-At: Date: Mon, 30 Jun 1997 11:07:12 +0100 From: Marco Grandis This is a collateral remark, but I would be surprised if there were no connections. In an abelian category C, a square of epis and monos as considered by Zinovy Diskin * --m--> A | | e' e | | v v X --m'--> * is a pullback iff it is a pushout. Such a bicartesian square represents a "subquotient" X of A (a subobject m' of a quotient e, and a quotient e' of a subobject m); and it is a subobject X >-+-> A in the category of relations RelC. Subquotients are a crucial tool in homological algebra, where everything - from homology to the terms of spectral sequences - is a subquotient of some "main object" (or an induced morphism between subquotients). See MacLane, "Homology". A categorical study of subquotients in abelian categories and their extensions can be found in the following papers of mine. The last setting ("semiexact" and "homological" categories) is much more general than the classical abelian one M. Grandis, Sous-quotients et relations induites dans les categories exactes, Cahiers Top. Geom. Diff. 22 (1981), 231-238. -, On distributive homological algebra, I. RE-categories; II. Theories and models; III. Homological theories. Cahiers Top. Geom. Diff. 25 (1984), 259-301; 353-379; 26 (1985), 169-213. -, On the categorical foundations of homological and homotopical algebra, Cahiers Top. Geom. Diff. Categ. 33 (1992), 135-175. With best regards Marco Grandis