From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5237 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Wed, 11 Nov 2009 18:29:35 +0100 Message-ID: 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 1257994198 29661 80.91.229.12 (12 Nov 2009 02:49:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 02:49:58 +0000 (UTC) To: Clemens.BERGER@unice.fr, categories@mta.ca Original-X-From: categories@mta.ca Thu Nov 12 03:49:51 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 1N8Pkt-0003Gj-L5 for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 03:49:51 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8PK8-00004S-Hb for categories-list@mta.ca; Wed, 11 Nov 2009 22:22:12 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5237 Archived-At: Dear Clemens, Thank you for your comments. > Are there generalizations to n composable arrows ? The lemma I was proposing works for any sequence of composable arrows. It can be rewritten in a notation similar to yours, but based on three consecutive arrows f, g, h. One would use, alternatively, two kind of 'generalised homologies' H'(f,g,h) = Ker(hg) / Im(f), H"(f,g,h) = Ker(h) / Im(gf), where, again, H/K is meant as in my previous msg. > What about generalizations to non-abelian categories ? The proof I was mentioning works for Puppe-exact categories, and is obvious (AFTER one has constructed the universal model of a sequence of consecutive arrows). This is the natural setting of distributive homological algebra, that cannot be developed under the assumption of products. It would be too long to explain this here; please see my three papers on this subject, in Cahiers 1984-85 (if interested, of course.) Best wishes Marco [For admin and other information see: http://www.mta.ca/~cat-dist/ ]