The Verdier Hypercovering Theorem

Abstract

AbstractThis note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms [X, Y] in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where Y is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object X to be locally fibrant.