From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5287 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Mon, 16 Nov 2009 17:43:54 +0100 Message-ID: References: Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1258417204 8500 80.91.229.12 (17 Nov 2009 00:20:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 17 Nov 2009 00:20:04 +0000 (UTC) To: "George Janelidze" , Michael Barr , categories@mta.ca Original-X-From: categories@mta.ca Tue Nov 17 01:19:57 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 1NABnV-0003L6-SF for gsmc-categories@m.gmane.org; Tue, 17 Nov 2009 01:19:54 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NABP9-0006nQ-3e for categories-list@mta.ca; Mon, 16 Nov 2009 19:54:43 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5287 Archived-At: Dear George (and others), Continuing what you and others have already said, there are (at least) two ways of extending those two lemmas. 1. Within homological algebra, there is the 'general form' of my first msg on this point. As I was saying, I do not like very much this extension; unless one can find some 'meaning' for the 'generalised homologies' H'(f,g,h) and H"(f,g,h) that appear there. 2. Within homotopical algebra, one can find a deeper solution, if more involved. The question is now about a triple (f, g, alpha) where f, g are consecutive arrows and alpha is a nullhomotopy of their composite gf. The hypothesis that we want to express is that this 'h-differential sequence' is 'h-exact' (h for homotopically, or homotopical). This is studied in my paper [*] (Como 2000), in a general setting and more particularly for a category of morphisms (Section 4); the latter is the case we are interested in for our extension. I will only sketch what is of interest here; the interested reader can look at [*]. Let D be a category with pullbacks and pushouts, and D' its category of morphisms. Think of an object as a truncated chain complex A = (d_A: A1 --> A0). (In [*], D is also assumed to be additive, but this is not necessary here.) There are obvious nullhomotopies of morphisms: inserting a diagonal A0 --> B1 in a square f: A = B. Nullhomotopies can be whiskered with morphisms. Every morphism has an h-kernel (= homotopy kernel) and an h- cokernel, defined by universal properties and constructed with the pullback and pushout 'inside' the square. An object is *contractible* if its identity is nullhomotopic; ie if its differential is iso. Given two consecutive morphisms f: X --> A, g: A --> Y, and a nullhomotopy alpha of gf, there is a *homotopical homology* object H(f, g, alpha) =(w: B --> Z), constructed via h- kernels and h-cokernels ([*], thm 4.5). Think of B as h-boundaries and of Z as h-cycles. Say that the h-differential sequence (f, g, alpha) is *h- exact* if H(f, g, alpha) is contractible, ie w is iso. This yields a nullhomotopy from the h-kernel of g to the h-cokernel of f. Now (this is not written in [*]), assuming that D is pointed, there is a composed nullhomotopy (by whiskering) Omega(Y) --> hKer(g) ==> hCok(f) --> Sigma(X), where: Omega(Y) = hKer(0 --> Y (sic!) ) = (0 --> Ker(d_Y)), Sigma(X) = hCok(X --> 0) = (Cok(d_X) --> 0). Thus our nullhomotopy gives the connecting morphism H_1(Y)) = Ker(d_Y)) --> H_0(X) = Cok(d_X). Exactness (of the three Omegas followed by the three Sigmas) has to be studied. Best Marco (PS. Notice that, if D is not additive, we have nullhomotopies without homotopies!) [*] M. Grandis, A note on exactness and stability in homotopical algebra, Theory Appl. Categ. 9 (2001), No. 2, 17-42 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]