From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5234 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Wed, 11 Nov 2009 17:04:51 +0200 Message-ID: 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 1257994031 29393 80.91.229.12 (12 Nov 2009 02:47:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 02:47:11 +0000 (UTC) To: "Stephen Lack" , ,"Michael Barr" Original-X-From: categories@mta.ca Thu Nov 12 03:46:54 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 1N8Pi1-0002M2-Pq for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 03:46:54 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8PC3-0007CL-1m for categories-list@mta.ca; Wed, 11 Nov 2009 22:13:51 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5234 Archived-At: The "curious discovery" is Exercise 6 at the end of Chapter VIII ("Abelian Categories") of Mac Lane's "Categories for the Working Mathematician"... However, I think it is an interesting question, and: When for the standard snake lemma Michael says "...there is an exact sequence 0 --> ker f --> ker g --> ker h --> cok f --> cok g --> cok h --> 0", what does "there is" mean? There are two well known answers: ANSWER 1. ker f --> ker g --> ker h and cok f --> cok g --> cok h are the obvious induced morphisms and there exists a "connecting morphism" d : ker h ---> cok f making the sequence above exact. Such a d is not unique: for instance if d is such, then so is -d. However, since the snake lemma holds in functor categories, the unnaturality of d does not make big problems in concrete situations. ANSWER 2. ker f --> ker g --> ker h and cok f --> cok g --> cok h are the obvious induced morphisms as before, while THE "connecting morphism" d : ker h ---> ker f is the composite of the zigzag ker h ---> C <--- B ---> B' <---A' ---> cok f (where the arrows are considered as internal relations). This "canonical connecting morphism" d works even in the non-abelian case of Dominique Bourn as I learned from my daughter Tamar who developed the "relative version". Note also, that the desire to have such a canonical d (in the abelian case) was a big original reason for developing what we call today "calculus of relations" (at the beginning with great participation of Saunders himself). And... in the "curious case = Exercise 6" the "canonical d" does not work! For, consider the simplest case of the composite 0 ---> B ---> 0: the exact ker-cok sequence will become 0 --> 0 --> 0 --> B --> B --> 0 --> 0 --> 0, where B --> B must be an isomorphism, while it is easy to check that the "canonical d" will become the relation opposite to the zero morphism B --> B. A possible conclusion is that the "master theorem" should involve some kind of "d" as an extra structure. To Steve's message: does Enrico really generalize the standard snake lemma and the "curious case" simultaneously? George ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Tuesday, November 10, 2009 12:57 AM Subject: categories: Question on exact sequence > 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/ ]