From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5241 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence (by G.J.) Date: Thu, 12 Nov 2009 09:12:47 +0100 Message-ID: References: <020701ca62ed$1faf6500$0b00000a@C3> <027601ca62f9$ace359c0$0b00000a@C3> Reply-To: Marco Grandis NNTP-Posting-Host: lo.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 1258047025 20517 80.91.229.12 (12 Nov 2009 17:30:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 17:30:25 +0000 (UTC) To: George Janelidze , categories@mta.ca Original-X-From: categories@mta.ca Thu Nov 12 18:30:18 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N8dUs-0002iX-E0 for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 18:30:14 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8cyu-0000CN-VF for categories-list@mta.ca; Thu, 12 Nov 2009 12:57:13 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5241 Archived-At: Dear George, I receive now your question. > The Barr's case (=Exercise VIII.4.6 in Mac Lane's book) and the > Snake Lemma > seem to have very different canonical connecting morphisms; how > does your > (beautiful!) general theorem solve this problem? All these connecting morphisms are canonically induced on subquotients, there is no need of using relations (even though you can, in both cases: a subquotient is the same as a subobject in the cat. of relations, and induced morphisms can always be computed that way: this is already in Mac Lane's Homology.) In the Snake Lemma, with Barr's notation: - Ker h is a subquotient of B (being a subobject of C), - Cok f is a subquotient of B' (being a quotient of A'), - the connecting morphism is induced by g: B --> B'. In the other lemma (Mac Lane, Barr), Ker h and Cok h are both subquotients of the middle object, and the (obvious) connecting morphism is (trivially) induced by the identity of the latter. Subquotients are characterised by a pullback-pushout square with two monos and two epis (in abelian categories; more generally in the Puppe-exact ones; more generally in 'my' homological categories, where you do not have relations). 'Regular' induction just means that there is a commutative cube from the first square to the second. Best wishes Marco [For admin and other information see: http://www.mta.ca/~cat-dist/ ]