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