Sunday, 10 September 2006

at.algebraic topology - How to localize a model category with respect to a class of maps created by a left Quillen functor

I suspect that you already have one, but here is a proof. I will assume that MM and N are combinatorial and that M is left proper (otherwise, I don't think that the literature contains a general construction of the left Bousfield localizations of M by any small set of maps). Everything needed for a quick proof is available in Appendix A of



J. Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, 2009.



First, there exists a cofibrant resolution functor Q in M which is accessible: the one obtained by the small object argument (as accessible functors are closed under colimits, it is sufficient to know that HomM(X,) is an accessible functor for any object X in M, which is true, as N is combinatorial). Let W be the class of maps f of M such that L(Q(f)) is a weak equivalence in N. As N is combinatorial, the class of weak equivalences of N is accessible see Corollary A.2.6.9 in loc. cit. Therefore, by virtue of Corollary A.2.6.5 in loc. cit, the class W is accessible. To Prove what you want, it is sufficient to check that M, W and C={cofibrations of M} satisfy the conditions of Proposition A.2.6.8 in loc. cit. The only non trivial part is the fact that the class CcapW satisfies all the usual stability properties for a class of trivial cofibrations (namely: stability by pushout, transfinite composition). That is where we use the left properness. For instance, if AtoB is in W and if AtoA is a cofibration of M, we would like the map AtoB=AamalgAB to be in W as well. This is clear, by definition, if A, A and B are cofibrant. For the general case, as M is left proper, B is (weakly equivalent to) the homotopy pushout AamalghAB, and as left derived functors of left Quillen functors preserve homotopy pushouts, we may assume after all that A, A and B are cofibrant (by considering the adequate cofibrant resolution to construct the homotopy pushout in a canonical way), and we are done. The case of transfinite composition is similar.

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