Numerable open covers and representability of topological stacks

TL;DR

This paper characterizes numerable open covers in topological spaces and applies this to establish homotopy descent criteria for topological stacks and sheaves, with special results for manifolds.

Contribution

It introduces a minimal class of covers for topological spaces and proves their sufficiency for homotopy descent, extending to stacks and chain complexes.

Findings

Characterization of numerable open covers as minimal class

Homotopy descent for stacks is determined by simple cover types

New criteria for chain complexes of sheaves to satisfy homotopy descent

Abstract

We prove that the class of numerable open covers of topological spaces is the smallest class that contains covers with pairwise disjoint elements and numerable covers with two elements, closed under composition and coarsening of covers. We apply this result to establish an analogue of the Brown--Gersten property for numerable open covers of topological spaces: a simplicial presheaf on the site of topological spaces satisfies the homotopy descent property for all numerable open covers if and only if it satisfies it for numerable covers with two elements and covers with pairwise disjoint elements. We also prove a strengthening of these results for manifolds, ensuring that covers with two elements can be taken to have a specific simple form. We apply these results to deduce a representability criterion for stacks on topological spaces similar to arXiv:1912.10544. We also use these results to establish new simple criteria for chain complexes of sheaves of abelian groups to satisfy the homotopy descent property.