local model structure on simplicial presheaves

Question

Hello,

Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.

Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its projective model structure (fib. and w.e. are level-wise).

Then one defines the class $S$ of local w.e. to be that of some maps of simplicial presheaves which induce isomorphisms on homotopy groups etc...

Then one takes the left Bousfield localization of the projective model structure along $S$, to get the projective local model structure (that which models "homotopy" sheaves).

I don't understand much in this things, so I have two questions:

1) In general, given a set $S$ of maps, we define the set of $S$-local equivalences (those which satisfy some left property w.r.t. $S$-local objects, which are those which satisfy some right property w.r.t. $S$...). For our $S$, will $S$-local equivalences coincide with $S$?

2) If I take $T$ to be the set of hypercovers, will $T$-local equivalences be $S$?

Thank you very much, Sasha

Answer 1

The answers to both of your questions is yes.

The original description by Jardine (Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87) of the local injective model structure on simplicial presheaves defines the weak equivalences as the class you refer in your question as $S$. This means that Jardine checked explicitly that the category of simplicial presheaves with the class $S$ of weak equivalences and the class of global cofibrations does indeed satisfy all the axioms in the definition of a model category.

Notice that given two model category structures $(W, cof, fib)$ and $(W', cof', fib')$ on the same underlying category $C$, you say that the second is a left Bousfield localization of the first if $cof = cof'$ and $W \subseteq W'$, and that the above picture fits this definition.

Now, given a model category structure $(W, cof, fib)$ and an arbitrary class $M$ of morphisms, you can try to define a new model category structure $(W', cof', fib')$ which is a left Bousfield localization of the first and which is minimal subject to the condition that $M \subseteq W'$. Then $W'$, the class of so-called $M$-local equivalences, is a kind of saturation of $M$, in the sense that it is the smallest class of morphisms containing $M$ and satisfying the necessary conditions for it to be the class of weak equivalences of a model category structure on your category (with the cofibrations fixed, of course). Since your class $S$ is already saturated in this sense, it is itself the class of $S$-local equivalences.

Your second question is precisely the beautiful paper Hypercovers and simplicial presheaves (Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51) by Dugger, Hollander and Isaksen.

A nice guide to everything that is going on is at the nLab page model structure on simplicial presheaves.