From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5222 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Tue, 10 Nov 2009 14:22:57 +1100 Message-ID: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1257859618 17569 80.91.229.12 (10 Nov 2009 13:26:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 10 Nov 2009 13:26:58 +0000 (UTC) To: Michael Barr , categories Original-X-From: categories@mta.ca Tue Nov 10 14:26:51 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 1N7qiK-0000k4-O0 for gsmc-categories@m.gmane.org; Tue, 10 Nov 2009 14:24:52 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N7q9a-0004rZ-QS for categories-list@mta.ca; Tue, 10 Nov 2009 08:48:58 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5222 Archived-At: Dear Michael, This is the sort of thing that Enrico Vitale has been working on with various people for a number of years. I'm sure he'll provide more precise references, but the idea is that you think of the vertical morphisms in your diagrams as internal categories: A A+A' | | | | f <---> | | v v v A' A' (an internal category in Ab amounts to just a morphism - I'll abbreviate this to just (A,A').) and then an exact sequence of internal categories, in a suitably defined sense of exactness, induces a long exact sequence involving the pi_0's and pi_1's. (pi_0 of an internal category is the cokernel of the corresponding morphism, while pi_1 is the kernel.) In your diagram (the "curious" one), the morphism 1:C-->C is saying that the corresponding internal functor (A,C)-->(B,C) is (not just essentially surjective but) the identity on objects. This is the relevant notion of "epi". The morphism 1:A-->A says that the corresponding internal functor (A,B)-->(A,C) is (among other things) faithful. This is the relevant notion of "mono". There is also an exactness condition at (A,C). Vitale, with various coauthors, has studied such exactness conditions at varying levels of generality, but the simplest of these is just internal categories in Ab. Steve. On 10/11/09 9:57 AM, "Michael Barr" wrote: > I have recently discovered a curious fact about abelian categories. > First, let me briefly describe the well-known snake lemma. If we have a > commutative diagram with exact rows (there are variations without the 0 > at the left end of the top and without the 0 at the right end of the > bottom, but here is the strongest form) > > 0 ---> A ----> B ----> C ----> 0 > | | | > | | | > |f |g |h > | | | > v v v > 0 ---> A' ---> B' ---> C' ---> 0 > > then there is an exact sequence > 0 --> ker f --> ker g --> ker h --> cok f --> cok g --> cok h --> 0 > > The curious discovery is that you have any pair of composable maps f: A > --> B and h: B --> C and you form the diagram (with g = hf) > 1 f > A ----> A ----> B > | | | > | | | > |f |g |h > | | | > v v v > B ----> C ----> C > h 1 > you get the same exact sequence. So I would imagine that there must be > a "master theorem" of which these are two cases. Does anyone know what > it says? The connecting map here is just the inclusion of ker h into B > followed by the projection on cok f. > > Michael > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]