Subject: RE: three postings From: "Radulescu-Banu, Andrei" Date: Fri, 8 Jun 2007 10:48:22 -0400 Dear Philippe, I also spent some time thinking for which class of small categories D is the functor Ho (M^D) -> (Ho M)^D essentially surjective - and also full, for a model category M. I know this is true if D is direct and free (arxiv:math/0610009v2, Thm. 8.8.5), as an application of the theory of ABC cofibration categories. I do not know of a counterexample for D direct but not necessarily free. For a while I tried to prove the same for D free, not necessarily direct, but could not prove it or disprove it. Alex Heller also has some interesting remarks about this in his 'Homotopy Theories' memoir - see his axiom H2 and also his remarks on weak limits at the beginning of chap. III. You may also want to review a classic theorem of Vogt, explained in "Abstract Homotopy and Simple Homotopy Theory" by K. H. Kamps and T. Porter at pag. 333, where (at least in the case M = Top) it is shown, for any small D, that Ho(M^D) is equivalent to the category of homotopy coherent D-diagrams in M. Best, Andrei Andrei Radulescu-Banu andrei@alum.mit.edu __________________________________________________________ > Subject: about rectification of homotopy commutative diagram > From: Gaucher Philippe > Date: Fri, 8 Jun 2007 12:18:17 +0200 > Dear all, > Consider the small category I: 0->1->2->... Let M be a model category (good > enough, proper, etc...). A morphism of M^I looks like a ladder, i.e. a > diagram over another small category L(I) having the shape of a ladder. I > know how to rectify a homotopy commutative diagram of Ho(M)^L(I), that is how to > construct an object of Ho(M^L(I)) sent by the map > Ho(M^L(I))->Ho(M)^L(I) to > the morphism of M^I we are considering. Is it realistic to think that the > same result holds if I is any direct Reedy category ? > Thanks in advance. pg. ------------------------------------------------------------------------------------------------------------------------------ Service Quality Matters. Test the performance and quality of your VoIP or IP video service at: http://www.TestYourVoIP.com http://www.TestYourIPVideo.com