Abstract:
Given a Grothendieck site C, Jardine has constructed a model category structure on simplicial presheaves over C which in some sense captures the `homotopical sheaf theory' of the site. One element missing from Jardine's work is a description of the fibrant objects in this model category, and providing this is one of the goals of the present paper: we show that they are essentially the simplicial presheaves which satisfy descent for all hypercovers. Another way of looking at this result is that we are giving a presentation for the homotopy theory of simplicial presheaves: it is the universal homotopy theory constructed from C subject to the relations saying that the homotopy colimit of a hypercover should be weakly equivalent to the original object. This is the key observation which lets one construct realization functors relating the homotopy theory of simplicial presheaves to other model categories (done elsewhere). In the present paper we also study the relation between various kinds of descent conditions on simplicial presheaves.