Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean [combinatorial model category](http://ncatlab.org/nlab/show/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.



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 $M$ 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.


