The homotopy type of a topological stack

Abstract

The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X\to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space $X$. The homotopy type of $\mathfrak{X}$ should be the classifying space $B\mathbb{X}$. The choice of an atlas is not part of the data of a stack and hence it is not immediately clear why this construction of a homotopy type is well-defined, let alone functorial. The purpose of this note is to give an elementary construction of such a homotopy-type functor.

12 pages

References in corpus (3)

• Foundations of Topological Stacks I
• Homotopy Theory of Orbispaces
• Pontrjagin-Thom maps and the homology of the moduli stack of stable curves