The leray spectral sequence

AbstractThe purpose of this note is to construct a Leray-type spectral sequence for homotopy classes of maps of simplicial presheaves, both stably and unstably, for any morphism of Grothendieck sites. This spectral sequence specializes to the ordinary Leray spectral sequence in sheaf cohomology theory, but may also be used for generalized étale cohomology theories such as étale K-theory