From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5274 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Sat, 14 Nov 2009 11:24:40 -0500 (EST) Message-ID: References: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1258294904 2173 80.91.229.12 (15 Nov 2009 14:21:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 15 Nov 2009 14:21:44 +0000 (UTC) To: George Janelidze , Original-X-From: categories@mta.ca Sun Nov 15 15:21:37 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 1N9fyy-0005iw-FY for gsmc-categories@m.gmane.org; Sun, 15 Nov 2009 15:21:36 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N9fVk-0005yB-A7 for categories-list@mta.ca; Sun, 15 Nov 2009 09:51:24 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5274 Archived-At: I think I have now come to understand this, at least in part. A couple things you said triggered this realization. First the point that this was taking place in the arrow category. Second that maybe sometimes you needed a map to get the connecting homomorphism and sometimes not. The first point made me think maybe rather than the arrow category I should perhaps be thinking about the category of chain complexes. The second somehow made me think about the difference between homology and homotopy equivalences. Perhaps this explanation amounts to shooting flies with elephant guns, but it satsifies me that it has fully explained things. Let me start by discribing an important difference between the two situations. Both diagrams: 0 --> A --> B --> C --> 0.......A --> A --> B ......|.....|.....|.............|.....|.....| ......|.....|.....|.............|.....|.....| ......|.....|.....|.............|.....|.....| ......v.....v.....v.............v.....v.....v 0 --> A'--> B'--> C'--> 0.......B --> C --> C with maps as in earlier posts, give 6 term exact sequences, but the second continues continues to do so when you apply a homfunctor Hom(D,-) or Hom(-,D), which is not generally true of the first. Now this, had I noticed it earlier, would have immediately put me in mind of the difference between homotopy and homology. It is a fact that a homology equivalence between two chain complexes is a homotopy equivalence iff it remains an equivalence if you apply any covariant (or any contravariant) homfunctor. This fact, along with a couple other things I will mention below, is proved somewhere in "Acyclic Models". Next, when I drew the diagram 0 ---> A ---> A + B ---> B ---> 0 .......|........|........|....... .......|........|........|....... .......|........|........|....... .......|........|........|....... .......v........v........v....... 0 ---> B ---> B + C ---> C ---> 0 which was marked by a peculiar appearance of - signs, I recognized that it looked like a mapping cone of something and, had I only worked out of what, I might have realized sooner what was going on. It is actually the mapping cone of the map 0 ---> A |......| |......| |......| v......v B ---> B |......| |......| |......| v......v C ---> 0 Finally, the observation that the chain complex 0 ---> A ---> C ---> 0 is homotopic to 0 ---> A + B ---> B + C ---> 0 is a complete triviality. So what is the general situation. Let me raise the question in this form. Suppose f: C' ---> C and g: C ---> C'' is a map of chain complexes. When can we expect an exact triangle H(C') ------> H(C) ...^............./ ....\.........../ .....\........./ ......\......./ .......\...../ ........\.../ .........\.v .........H(C'') Clearly we need something to induce the map H(C'') ---> H(C'). The obvious thing would be a map S(C'') ---> C' (S is the suspension functor). The example of two-length sequences shows that this is too much to hope for. It looks like what you need is a relation from S(C'') --> C' that induces, somehow a map on homology. This assumption holds for the case 0 --> C' --> C --> C' --> 0 and also holds for the original curious sequence (the suspension is what saves the day here, as seen above). Is there a better way to express this? I don't see one. Perhaps the last word on this hasn't been said yet. Another question I haven't answered but should be doable is whether the existence of a relation S(C'') - - - -> C' that you assume induces a morphism on homology is sufficient to make the triangle exact. Beyond this, I have some inchoate ideas that I will explore. Clearly there has to be some compatibility between R and f and g. As I said, this explanation is perhaps a little heavy but I know no simpler one. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]