From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5228 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Wed, 11 Nov 2009 22:05:46 +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 1257953502 29301 80.91.229.12 (11 Nov 2009 15:31:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 11 Nov 2009 15:31:42 +0000 (UTC) To: Michael Barr , Original-X-From: categories@mta.ca Wed Nov 11 16:31:34 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 1N8FAR-0008Bi-8z for gsmc-categories@m.gmane.org; Wed, 11 Nov 2009 16:31:31 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8Ehb-0005cT-2y for categories-list@mta.ca; Wed, 11 Nov 2009 11:01:43 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5228 Archived-At: Dear Michael, I worked out what exactly Vitale's exactness condition says in this case. A commutative diagram i' p' A'-->B' --> C' f| g| h| v v v A --> B --> C i p induces a long exact sequence in the way you suggested, when there is a morphism z:A-->C' for which the induced 0--> A' --> A+B' --> B+C'-->C-->0 is exact. Steve. On 10/11/09 2:22 PM, "Steve Lack" wrote: > 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/ ]