From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5262 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Fri, 13 Nov 2009 11:06:41 -0500 (EST) Message-ID: References: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1258141705 5787 80.91.229.12 (13 Nov 2009 19:48:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 19:48:25 +0000 (UTC) To: George Janelidze , Original-X-From: categories@mta.ca Fri Nov 13 20:48:07 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 1N927q-0003Lk-Eq for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 20:48:06 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N91l6-0004w7-Tp for categories-list@mta.ca; Fri, 13 Nov 2009 15:24:36 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5262 Archived-At: Actually, on further thought, I agree with you. I didn't originally want a slick proof but to understand and then I forgot why I really raised the question. After all, I had already proved it. So what I really wanted and still want to know is what conditions on a map between two three term sequences gives the 6 term exact sequence (with or without the end 0s). The situation of the snake lemma is so different from the situation I (and, obviously others) discovered that one wonders still what general conditions could possibly encompass the two cases. That really was my initial question and that question now comes back to me. Michael On Fri, 13 Nov 2009, George Janelidze wrote: > All right, then I shall better stop, unless there will be new unexpected > comments (because what Bill and others say, will take us too far...) > > George > > ----- Original Message ----- > From: "Michael Barr" > To: "George Janelidze" > Sent: Friday, November 13, 2009 4:33 AM > Subject: Re: categories: Re: Question on exact sequence > > >> Actually that diagram with the sums does really answer the question as I >> had understood it. There may be a deeper question, but I am not sure how >> to formulate it. >> >> Michael >> >> On Fri, 13 Nov 2009, George Janelidze wrote: >> >>> I have further comments to Marco and Steve (maybe tomorrow...), but now > I am >>> only answering >>> >>>> Since my curious sequence was an exercise in CWM, it is surprising that >>>> Saunders never raised the question in the form I did. The conclusion >>>> certainly looks like something out of the snake lemma, but I was unable > to >>>> formulate it as a cosequence. >>> >>> from Michael's message: >>> >>> Dear Michael, >>> >>> Does "formulate" mean "obtain/deduce"? Obtaining the curious sequence as > a >>> consequence of the snake lemma is actually easy, and Saunders surely > knew >>> it - which probably explains why did not he raise your question. Given > your >>> f : A ---> B, h : B ---> C and g = hf, just apply the snake lemma to >>> >>> <1,f> [f,-1] >>> A ---> A + B ---> B >>> | | | >>> | f | g+1 | h >>> v v v >>> B ---> C + B ---> C >>> [1,-h] >>> >>> where + denotes the direct sum, <...> "uses" it as product, and [...] > "uses" >>> it as coproduct (and use the fact that Ker(g) = Ker(g+1)). >>> >>> However, this does not answer your original question of course. >>> >>> George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]