Singular chains on topological stacks

TL;DR

This paper extends the singular chain functor to topological stacks, demonstrating its compatibility with weak equivalences and fibrations, thus generalizing classical singular homology to a broader categorical context.

Contribution

It introduces a functor Sing for topological stacks, preserving key homotopical properties and extending classical singular chains to this new setting.

Findings

Sing respects weak equivalences in topological stacks.

Morphisms that are Serre and Reedy fibrations map to Kan fibrations.

Recovers classical singular functor on topological spaces.

Abstract

We extend the functor Sing of singular chains to the category of topological stacks and establish its main properties. We prove that Sing respects weak equivalences and takes a morphism of topological stacks that is both a Serre and a Reedy fibration to a Kan fibration of simplicial sets. When restricted to the category of topological spaces Sing coincides with the usual singular functor.