Homotopy types of topological stacks

TL;DR

This paper introduces the concept of classifying spaces for topological stacks, establishing their existence and uniqueness up to homotopy under certain conditions, and extends these results to diagrams of stacks.

Contribution

It defines classifying spaces for topological stacks and proves their existence and uniqueness up to homotopy, extending to diagrams of stacks.

Findings

Every topological stack has a classifying space well-defined up to weak homotopy equivalence.

Under paracompactness, the classifying space is well-defined up to homotopy equivalence.

Results are formulated via functors to the (weak) homotopy category of spaces.

Abstract

We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain paracompactness condition on \X, we show that X is actually well-defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks.