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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 $\operatorname{Hom}$, given by $\operatorname{Hom}(X, Y)_n =\operatorname{hom}(X\times \Delta^n, Y)$.

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 presheav $A$ the following are equivalent.

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

2) $\operatorname{Hom}(Y, A) \cong \operatorname{Hom}(X, Y)$ 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?


Edit: In general 1) does not imply 2), and this is not stated anywhere. I mixed things up.

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I think this question would probably be acceptable over at MO. – Akhil Mathew Dec 9 '10 at 13:16
I agree with Akhil. – Martin Brandenburg Dec 9 '10 at 13:54
Ok, thanks. I'll try my luck there. – ruediger Dec 9 '10 at 17:42
x-posted:… – Grigory M Jun 28 '11 at 11:27

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