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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.

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    $\begingroup$ As Tyler Lawson points out in the answers, I misunderstood (probably a lot about) injective versus Reedy structures; the "complicated" link doesn't illustrate just how complicated things are. Like, I think, many users of model categories, I in fact only case about Bousfield localizations of presheaf categories, and only for pretty well-behaved bases. $\endgroup$
    – Theo Johnson-Freyd
    Commented Nov 3, 2014 at 18:45

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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.)

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  • $\begingroup$ Great. I agree that what I linked to does not define cofibrations levelwise. But I think that for $\Delta$, the Reedy and injective model structures agree? I would have to think a lot about how to prove that directly, but my impression was that it was reasonably known. The reference I know is to arxiv.org/abs/1110.1066, although the claim about $\Delta$ seems to be older. $\endgroup$
    – Theo Johnson-Freyd
    Commented Nov 3, 2014 at 18:41
  • $\begingroup$ Or, no, I've misunderstood. For $M = \mathrm{sSet}$, they agree, not in general. My apologies. $\endgroup$
    – Theo Johnson-Freyd
    Commented Nov 3, 2014 at 18:43
  • $\begingroup$ @TheoJohnson-Freyd Right, that's a particular nicety of simplicial sets (essentially, cofibrations are levelwise monomorphisms) that doesn't have an analogue in general. $\endgroup$
    – Tyler Lawson
    Commented Nov 3, 2014 at 18:53
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    $\begingroup$ It should be also pointed out that $\Delta$ being an elegant Reedy category means that in the category of simplicial diagrams over $\Delta^{\mathrm{op}}$ the Reedy and injective model structures coincide. However, $\Delta^{\mathrm{op}}$ is not elegant and the same is not true for diagrams over $\Delta$. $\endgroup$
    – Karol Szumiło
    Commented Nov 3, 2014 at 19:05
  • $\begingroup$ @KarolSzumiło: Great. Incidentally, I advocate the following notation: $X^Y$ means $X$-valued presheaves on $Y$, and not functors. I would write $^YX$ for the category of $X$-valued copresheaves on $Y$. The reason is that I think of right modules as contravariant functors. $\endgroup$
    – Theo Johnson-Freyd
    Commented Nov 3, 2014 at 19:15

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