DANIEL DUGGER a1, SHARON HOLLANDER a2andDANIEL C. ISAKSEN a3 a1 Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A. e-mail: email@example.com a2 Department of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A. e-mail: firstname.lastname@example.org a3 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. e-mail: email@example.com
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, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.