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 NN are combinatorial and that MM is left proper (otherwise, I don't think that the literature contains a general construction of the left Bousfield localizations of MM 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 QQ in MM 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,)HomM(X,) is an accessible functor for any object XX in MM, which is true, as NN is combinatorial). Let WW be the class of maps ff of MM such that L(Q(f))L(Q(f)) is a weak equivalence in NN. As NN is combinatorial, the class of weak equivalences of NN 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 WW is accessible. To Prove what you want, it is sufficient to check that MM, WW and C=C={cofibrations of MM} satisfy the conditions of Proposition A.2.6.8 in loc. cit. The only non trivial part is the fact that the class CcapWCcapW 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 AtoBAtoB is in WW 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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