Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the [left Bousfield localization](http://en.wikipedia.org/wiki/Bousfield_localization) $L\_SM$ of $M$ with respect to $S$ is the model category structure on the same underlying category as $M$, with the same cofibrations, more weak equivalences, and correspondingly fewer fibrations; it is the universal model category structure with these properties in which the elements of $S$ are weak equivalences.


Let $M^\Delta$ denote the category of simplicial objects in $M$. It has two related model category structures: the *injective* one, in which cofibrations and weak equivalences are "levelwise" (and hence fibrations are [complicated](http://ncatlab.org/nlab/show/Reedy+model+structure)), and the *projective* one, in which fibrations and weak equivalences are levelwise (and cofibrations are [more complicated](http://arxiv.org/abs/1403.6873)). I am interested in both structures, so in some sense this question is really two questions.



> 
> **Question:** Is $(L\_SM)^\Delta$ a left Bousfield localization of $M^\Delta$? What is an explicit description of the maps at which you localize?
> 
> 
> 


I expect the answer is "yes", and that you can localize at the set of maps in $M^\Delta$ that are levelwise in $S$. These seems like the type of thing that should be provable by definition unpacking. But I got stuck somewhere, suggesting that maybe there is a subtlety. The worry is that perhaps there just aren't a lot of levelwise-$S$-maps in $M^\Delta$, so that testing just against the levelwise-$S$-maps might give false positives when you are trying to find out if an object is levelwise-$S$-local.



Does "simplicial" commute with "Bousfield localization"?
Under the injective model structure or the Reedy model structure, the answer is yes: this is a left Bousfield localization.


For any $X \in M$, define $\Delta[n] \otimes X$ to be the element of $M^\Delta$ given by
$$
(\Delta[n] \otimes X)\_k = \coprod\_{(\Delta[n])\_k} X,
$$
with maps induced by the natural maps on coproducts. This is a functor which is a left adjoint:
$$
Hom\_{M^\Delta}(\Delta[n] \otimes X, Y) = Hom\_M(X,Y\_n).
$$
Moreover, $\Delta[n] \otimes (-)$ preserves cofibrations in both the injective and Reedy model structures, and so $[\Delta[n] \otimes X,Y] = [X,Y\_n]$ for fibrant $Y$. Thus, in these cases localizing with respect to maps of the form $\Delta[n] \otimes f$ for $f \in S$ gives you $(L\_S M)^\Delta$.


However, $\Delta[n] \otimes f$ is (except in the case $n=0$) unlikely to be levelwise in $S$ unless $S$ is closed under finite coproducts.


(Quick comment: the Reedy model structure you linked to does not have its cofibrations defined levelwise.)


