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.
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