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Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean combinatorial model category. Consider a Quillen pair $$ L: M\rightleftarrows N: R.$$ I want the following result:

There exists a set of maps $S$ in $M$, such that $L$ and $R$ descend to a Quillen pair $$ L: S^{-1}M \rightleftarrows N: R,$$ with the property that a map $f:X\to Y$ between cofibrant objects in $M$ is a weak equivalence in $S^{-1}M$ if and only if $L(f)$ is a weak equivalence in $N$.

Here $S^{-1}M$ denotes the model category with the same underlying category $M$ obtained by localizing $M$ with respect to the set of maps $S$.

This result seems to encode a standard technique; in fact, the very first example of a localized model category (Bousfield's localization of spaces with respect to a homology theory) can be viewed (in retrospect) of a special case of this.

I think I could prove this result if I need to. But I would rather have a reference. I've looked in the usual places, but I can't seem to find anything exactly like it. I expected to find something like this in one of Dugger's papers on presentable model categories, but I don't find it there; he proves that under an additional condition, you can get a Quillen equivalence $S^{-1}M\rightleftarrows N$, but his construction of the set $S$ does not apply in my case.

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Hello! I am interested in the case where one gets a Quillen equivalence $S^{-1} M\rightleftarrows N$ - could you give a reference? Thank you! –  Hanno Becker Feb 6 '12 at 12:32
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up vote 10 down vote accepted

I suspect that you already have one, but here is a proof. I will assume that $M$ 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 $Hom_M(X,-)$ 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 $C\cap W$ 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 $A\to B$ is in $W$ and if $A\to A'$ is a cofibration of $M$, we would like the map $A'\to B'=A'\amalg_A B$ 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 $A'\amalg^h_A B$, 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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Thanks, that is pretty much the proof I had in mind, though the use of the language of accessible functors (as in Lurie) makes it much cleaner that I could have said it. –  Charles Rezk Oct 20 '10 at 21:43
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