A Comparison of two Models of Orbispaces

TL;DR

This paper demonstrates that two different models of orbispaces used in global homotopy theory are equivalent by constructing a zig-zag of Dwyer-Kan equivalences, and extends known results on homotopy quotients for free actions.

Contribution

It provides the first explicit zig-zag of equivalences between two models of orbispaces and generalizes homotopy quotient results to all free actions of compact Lie groups.

Findings

The two models of orbispaces are equivalent via a zig-zag of Dwyer-Kan equivalences.

Homotopy quotient by a compact Lie group action is weakly equivalent to the strict quotient for all free actions.

Extended known results on homotopy quotients from manifolds to all compactly generated Hausdorff spaces.

Abstract

This paper proves that the two homotopy theories for orbispaces given by Gepner and Henriques and by Schwede, respectively, agree by providing a zig-zag of Dwyer-Kan equivalences between the respective topologically enriched index categories. The aforementioned authors establish various models for unstable global homotopy theory with compact Lie group isotropy, and orbispaces serve as a common denominator for their particular approaches. Although the two flavors of orbispaces are expected to agree with each other, a concrete comparison zig-zag has not been known so far. We bridge this gap by providing such a zig-zag which asserts that all those models for unstable global homotopy theory with compact Lie group isotropy which have been described by the authors named above agree with each other. On our way, we provide a result which is of independent interest. For a large class of free actions of a compact Lie group, we prove that the homotopy quotient by the group action is weakly equivalent to the strict quotient. This is a known result under more restrictive conditions, e.g., for free actions on a manifold. We broadly extend these results to all free actions of a compact Lie group on a compactly generated Hausdorff space.