From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5257 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Fri, 13 Nov 2009 02:16:02 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1258079678 20517 80.91.229.12 (13 Nov 2009 02:34:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 02:34:38 +0000 (UTC) To: Original-X-From: categories@mta.ca Fri Nov 13 03:34:32 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 1N8lzb-0005by-3z for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 03:34:31 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8lYt-0006ZN-Eh for categories-list@mta.ca; Thu, 12 Nov 2009 22:06:55 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5257 Archived-At: 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 ----- Original Message ----- From: "Michael Barr" To: "George Janelidze" ; Sent: Thursday, November 12, 2009 2:41 PM Subject: categories: Re: Question on exact sequence > I do appreciate the example since I wondered if the "connecting > homomorphism" could be induced by a composite of relations as in the snake > lemma. I thought not and George has provided an example. Since Tuesday, > we have had house guests so I really have not had time to absorb all the > replies, but when I have time, I plan to collect them all and try to see > if there is a satisfactory general answer of which the two instances I > described are special cases. There is something going on here that I > don't quite comprehend (although maybe the answer is in the theorem Marco > mentioned. > > 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. > > Incidentally, the theorem on acyclic models, as it appears in my book, > can be described as a map induced by a composite of relations that, in > homology, becomes functional. > > Michael > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]