From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5267 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 20:15:36 +0200 Message-ID: References: <00a001ca63f6$80936b50$0b00000a@C3> <000f01ca644d$065eb590$0b00000a@C3> 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 1258142043 6962 80.91.229.12 (13 Nov 2009 19:54:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 19:54:03 +0000 (UTC) To: "Michael Barr" , Original-X-From: categories@mta.ca Fri Nov 13 20:53:56 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 1N92DT-00068c-0D for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 20:53:55 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N91nm-0005F8-Qi for categories-list@mta.ca; Fri, 13 Nov 2009 15:27:22 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5267 Archived-At: Yes, this is how I understood your original question. In order to make some comments, let me mention/recall: You say "a map between two three term sequences" and such a diagram in a category C is of course the same as a composable pair of morphisms in the arrow category Arr(C). Therefore let me write it as f ---> g ---> h (1) - and so your question is: when - for an abelian C - does (1) induce an exact sequence (0 -->) ker f --> ker g --> ker h --> cok f --> cok g --> cok h (--> 0) (2) My comments are: 1. I think Steve made a very good comment. Let me repeat it in my words: When C is abelian, the category Cat(C) of internal categories in C is equivalent to Arr(C). Writing cat(f) for the internal category corresponding to f, we have: (a) ker f = the internal group of automorphisms of 0 in cat(f) = the fundamental group of cat(f). Therefore, let me write ker f = pi_1(f). (b) cok f = the object of connected components of cat(f). Therefore, let me write cok f = pi_0(f). After that the sequence (2) becomes (0 -->) pi_1(f) --> pi_1(g) --> pi_1(h) --> pi_0(f) --> pi_0(g) --> pi_0(h) (--> 0), and your question becomes a classical question of abstract homotopy theory! And Steve gave (a partial?) answer that comes out of Enrico's work. 2. As we both agree, Marco also made a very good comment. Namely, Marco has a general theorem that applies to (1) in the two cases of main interest: to the standard snake lemma and to the case you called curious. . 3. My own comment (although I am slightly changing it here) was/is that the morphism ker h --> cok f (in (2)) should probably be considered as an extra structure on (1) rather than something "induced". In Marco's approach it is an induced morphism on subquotients, but it becomes "induced" in the two cases of interest for different reasons. And those two reasons ONLY BECOME the same when we present the two cases of interest as special cases of Marco's theorem in Marco's way. But using Marco's way we also automatically equip our diagram with an extra structure. Note also, that an extra structure appears in Steve's second message as z : A --> C' (which surely helps to create ker h --> cok f, although I have not checked the details). 4. It is amazing how many non-abelian versions of your question might be asked! Since Dominique has a non-abelian snake lemma it can be about semi-abelian/homological/protomodular categories - I would call it "Bourn-nonabelian" although there are older versions with "old" axioms. Or it can be Grandis-nonabelian - since Marco has own homological categories, where he develops abstract homological algebra. Or it can be replacing (all?) morphisms with pairs of parallel morphisms in a category enriched in commutative monoids - as I understand this is what Bill had in mind asking his question. Or it can be replacing f, g, h with internal categories (or groupoids) in a rather large class of categories - as follows from Steve's suggestion. And there are related questions that came out of other interesting messages... George ----- Original Message ----- From: "Michael Barr" To: "George Janelidze" Cc: "Categories list" Sent: Friday, November 13, 2009 6:06 PM Subject: Re: categories: Re: Question on exact sequence > 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 > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]