TITLE: Does "simplicial" commute with "Bousfield localization"?
QUESTION [6 upvotes]: 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 $L_SM$ 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), and the projective one, in which fibrations and weak equivalences are levelwise (and cofibrations are more complicated).  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.

REPLY [6 votes]: 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.)