Numerable open covers and representability of topological stacks

Abstract

In this note, we establish an analogue of the Brown-Gersten property for numerable open covers of topological spaces: a simplicial presheaf on the site of topological spaces with numerable open covers satisfies the homotopy descent property for all open covers if and only if it satisfies it for covers with two elements and covers with pairwise disjoint elements. We apply this result to deduce a representability criterion for stacks on topological spaces similar to arXiv:1912.10544. We also use this result to establish new simple criteria for chain complexes of sheaves of abelian groups to satisfy the homotopy descent property.