Local Weak Equivalences

Abstract

Local weak equivalences and the theory of local fibrations are the subjects of this chapter. Most naively, a local weak equivalence is a map of simplicial presheaves which induces isomorphisms in all possible sheaves of homotopy groups. A local fibration is a map which has a suitably defined local right lifting property with respect to all inclusions of horns in simplices. It is fundamental result that a map is both a local weak equivalence and a local fibration if and only if it is a hypercover in the sense that it has the local right lifting property with respect all inclusions of boundaries in simplices. Hypercovers first appeared in the literature, with the birth of etale homotopy theory, as high order generalizations of Cech resolutions.