Local injective model structure for simplicial presheaves

Question

The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) injective model structure and has a mapping space functor $Hom$, given by $Hom(X, Y)_n = hom(X\times \Delta^n, Y)$.

Using the given topology, one can actually define 'local' weak equivalences and Jardine showed that the left Bousfield localization at the class of local weak equivalence exists. This model category structure is called local injective model structure.

Now, it's often mentioned that this is also the same as localizing the injective model structure at the class $S := \{X\rightarrow L^2X\}$ (where $L^2$ is the sheafification functor).

So if $W$ denotes the class of local weak equivalences, this amounts to say that $S$-local is the same as $W$-local, that is for an injective fibrant simplicial presheaf $A$ the following are equivalent:

1) $Hom(L^2X, A) \cong Hom(X, A)$ for all $X$

2) $Hom(Y, A) \cong Hom(X, Y)$ for all local weak equivalences $X\rightarrow Y$.

Here, $\cong$ stands for weak equivalence of simplicial sets.

2) implies 1), of course. But how does 1) imply 2)? Or am I mistaken?

Thanks!

Answer 1

Many statements in the question are incorrect, and so I want to put an answer here to prevent future visitors to this thread from being confused. First, it is NOT TRUE that localizing at the local weak equivalences is the same as localizing at S $= \lbrace X \to L^2X\rbrace$ where $L^2$ is sheafification. From the comments, this can be seen as Theorem A.6 here.

Let $W$ be the class of local weak equivalences. Then localization at $W$ exists and localization at $S$ exists. Furthermore, if an object is $W$-local then it is $S$-local, but the converse is false. Now suppose you apply fibrant replacement in the model categories $M_S$ and $M_W$ coming from Bousfield localizing at these two classes of maps. Then you have maps $X\to R_W(X)$ and $X\to R_S(X)$ which are trivial cofibrations in their respective categories and so are cofibrations in $M$. The object $R_W(X)$ is $S$-local, so the universal property of $S$-localization says we have $R_S(X)\to R_W(X)$. This is the most that can be said about relating $S$-locals and $W$-locals.

On the chain of Bousfield localizations, $S$-localization ``sees'' more than $W$-localization, i.e. it will be able to distinguish things that $W$ sees as equivalent. This means the class of $S$-local equivalences is contained in the class of $W$-local equivalences, so $S$ is contained in the class of $W$-local equivalences. Perhaps this led to the OP's confusion, since passage to $M_W$ will send the maps $S$ into the new weak equivalences.