Hypercovers and simplicial presheaves, by Daniel Dugger, Sharon Hollander, and Daniel C. Isaksen

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition. These ideas are central to constructing realization functors on the homotopy theory of schemes. We give a few other applications for this new description of the homotopy theory of simplicial presheaves.

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