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# Is the simplicial completion ofalocalizer always a left bousfield localization of the injective model structure?

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## Background

Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following axioms:

1.)

• The class $W$ satisfies 2-for-3

2.)

• The class $W$ contains $\mathrm{rlp}(\mathrm{Mono}(A))$, where $\mathrm{Mono}(A)$ denotes the class of all monomorphisms in $Psh(A)$.

3.)

• The class $W\cap \mathrm{Mono}(A)$ is closed under transfinite composition and pushouts.

Given a class $C$ of morphisms in $\mathrm{Psh}(A)$, define $W(C)$, the localizer generated by $C$, to be the intersection of all $A$-localizers containing $C$. We say that a localizer $W$ is accessible if there exists a small set $S$ such that $W=W(S)$.

Let $(A,W)$ be a pair comprising a small category $A$ and an $A$-localizer $W$. We define the simplicial completion of $(A,W)$ to be the pair $(A\times \Delta, W_\Delta)$, where we define the $A\times \Delta$-localizer $W_\Delta$ as follows: The class $W_\Delta$ is the localizer generated by $W^{\Delta^{op}}$, the class of levelwise $W$-equivalences (viewing $\mathrm{Psh}(A\times \Delta)$ as $\mathrm{Psh}(A)^{\Delta^{op}}$, so the levels are indexed by the objects of $\Delta$) together with the class of all maps $T\otimes \Delta^1\to T$ where $T$ is an object of $\mathrm{Psh}(A\times \Delta)$ (given an object $T$ and a simplicial set $X$, we define $(T\otimes X)(a,s)=T(a,s)\times X(s)$ for a pair of objects $a$ in $A$ and $s$ in $\Delta$).

It is a theorem of Cisinski that the functor $pr_1^\ast:\mathrm{Psh}(A)\to \mathrm{Psh}(A\times \Delta)$ obtained from the projection onto the first factor induces an equivalence of categories $W^{-1}Psh(A) \to W_\Delta^{-1}\mathrm{Psh}(A\times \Delta)$. Further, if the $A$-localizer $W$ is accessible, the functor $pr_1^\ast$ is the left Quillen functor of a Quillen equivalence between the model categories $(\mathrm{Psh}(A), \mathrm{Mono}(A), W)$ and $(\mathrm{Psh}(A\times \Delta),\mathrm{Mono}(A\times \Delta), W_\Delta)$.

Recall that the injective (with respect to $\Delta$) model structure on $\mathrm{Psh}(A\times\Delta)\cong \mathrm{Psh}(\Delta)^{A^{op}}=\mathrm{sSet}^{A^{op}}$ is the model structure for which the cofibrations are exactly the monomorphisms and the class of weak equivalences $W_{\mathrm{inj}}$ comprises the levelwise (indexed by the objects of $A$) weak homotopy equivalences.

We will say that an $A$-localizer $W$ is a Bousfield localization of another $A$-localizer $W^\prime$ if $W^\prime \subseteq W$ (because if $W$ and $W^\prime$ are accessible, the model structure associated with $W$ is a left Bousfield localization of the model structure associated with $W'$.

## Question:

Given a small category $A$ and an $A$-localizer $W$, is the $A\times \Delta$-localizer $W_\Delta$ a Bousfield localization of the class of levelwise weak homotopy equivalences $W_{\mathrm{inj}}$?

## Remarks

I can prove that this is true in the special case that $A$ has the following property:

The category $A$ is a Reedy category such that the Reedy model structure on $sSet^{A^{\mathrm{op}}}$ coincides with the injective model structure. Somewhat surprisingly, this special case happens to include some of the most interesting examples. Cisinski has given an axiomatic characterization of a wide swath of these categories using his formalism of "catégories squelletiques" (Astérisque 308 ch. 8). Also see this MO Question.

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