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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!

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Can you provide a reference for that "often mentioned fact" that localizing at S is the same as localizing at W? –  Alberto García-Raboso Dec 9 '10 at 18:08
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Ok, sorry. Now I see, that I mixed some things up. Localizing at $S$ gives not the usual local weak equivalences. This is Theorem A.6. in math.uiuc.edu/K-theory/0563 . –  ruediger Dec 9 '10 at 18:45
    
Yup, that's the paper I was thinking about. Well, you answered your own question! –  Alberto García-Raboso Dec 9 '10 at 20:11
    
@ruediger: it seems you've answered your own question in the comment. Perhaps you can make that an official answer and accept it, so that the software won't go bumping this question to the front page (as it sometimes does for questions with no answer) –  David White Jan 12 '13 at 23:26
    
There's a typo in (2): it should be $Hom(Y,A)\to Hom(X,A)$ –  David White Mar 16 '13 at 22:35
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1 Answer 1

up vote 2 down vote accepted

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.

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