This should be a really basic question, but I'm stuck on it.
The question. I see written everywhere (for example here, or in the article [DHI] Hypercovers and simplicial presheaves of Dugger, Hollander and Isaksen) that the model structure of Jardine (resp. Blander) is the left Bousfield localization of the global injective (resp. projective) structure on simplicial presheaves at the class of hypercovers. However, when I try to prove it by hands, I fail.
Let me fix the notations and the definitions. $(\mathcal C, J)$ is a Grothendieck site. To fix the ideas, I will write everything in terms of the global injective model structure. Feel free to answer using the global projective (but do not mix the two!).
Jardine model structure. It is the model structure on $\mathrm{sPSh}(\mathcal C)$ where:
- weak equivalences are maps inducing equivalences of sheaves of homotopy groups;
- cofibrations are objectwise cofibrations (i.e. monomorphisms);
- fibrations are the maps having the RLP with respect to trivial cofibrations.
I will denote by $W$ the class of local injective weak equivalences (i.e. the weak equivalences of the above model structure).
The class $\mathcal L$ of hypercovers. The maps in $\mathcal L$ are the local acyclic cofibrations $U \to X$ where $X$ is an object of $\mathcal C$ reviewed as constant simplicial presheaf and $U$ is an hypercover of $X$ (I won't recall the definition).
Remark. Below, I will sketch what I attempted in order to prove the statement. But I don't really care about it, so feel free not to read what follows, as long as you can provide a proof of the statement (or a reference to the proof).
What is clear to me. I feel comfortable with the following few facts:
$\mathcal L$-local objects in the sense of Hirschhorn are exactly the injective-fibrant objects satisfying descent for all the hypercovers in the sense of [DHI, Definition 4.3];
by the very definition of the left Bousfield localization it is sufficient to show that $\mathcal L$-local objects are precisely the fibrant objects in the Jardine model structure.
finally, if $F$ is fibrant in the Jardine model structure, then it has in particular the RLP with respect to every map in $\mathcal L$, $U \to X$, which means precisely that $F$ satisfies descent for all the hypercovers.
What I attempted. (I will simply sketch the idea of my proof, emphasizing the point where I'm stuck) Fix a $\mathcal L$-local object $F$ and a local acyclic cofibration $\alpha \colon A \to B$ for the Jardine model structure. We have to show that $\alpha$ has the extension property with respect to $F$. We can moreover assume from the outset that $A$ and $B$ are locally-fibrant (by the definition of weak equivalence via sheaves of homotopy groups). Given $\beta \colon A \to F$, my idea is to construct $\gamma \colon B \to F$ objectwise, in the following way: given $X \in \mathrm{Ob}(\mathcal C)$, an object in $B(X)$ is the same as a map $X \to B$. Using [DHI, Proposition 5.7], we can find a hypercover $U \to X$ such that $U \to X \to B$ lifts (up to homotopy) to a map $U \to A$. Since $F$ satisfies descent, the composite $U \to A \to F$ determines (up to homotopy) a map $X \to F$. Therefore I send $X \to B$ to the map $X \to F$ constructed as indicated. This is well defined up to homotopy (i.e. a change of the hypercover produces homotopically equivalent maps). Moreover, it should be straightforward to see that this induces a map $B \to F$ when we review those presheaves as elements of $\mathrm{Ho}(\mathbf{sSet})^{\mathcal C^{\mathrm{op}}}$. The best at this point would be to strictify this map, i.e. to realize it as a map in $\mathrm{Ho}(\mathrm{sPSh}(\mathcal C))$ (w.r.t. the global injective model structure); if this could be done, I think we should be more or less done (every object is cofibrant and we are working with locally-fibrant objects). However, at the moment, I am a bit sceptical about the possibility of operating such a strictification.
Edit. Ok, I guess that the core of the needed argument is contained in [HAG I, Theorem 3.4.1]. I will be reading the (rather technical) proof. If in the meanwhile someone has comments, I will read them with extreme interest.
