Orbispaces, orthogonal spaces, and the universal compact Lie group

TL;DR

This paper connects the homotopy theories of orbispaces and stacks with global homotopy theory via the universal compact Lie group, providing new tools for studying cohomology theories on stacks and orbispaces.

Contribution

It introduces a novel perspective by interpreting orbispaces as spaces with an action of the universal compact Lie group and establishes a global model structure for these spaces.

Findings

Established a global model structure on the category of al Lspaces.

Proved the al Lspaces model category is Quillen equivalent to orthogonal spaces.

Connected orbispaces with unstable global homotopy theory through the universal compact Lie group.

Abstract

This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of `spaces with an action of the universal compact Lie group'. The upshot is a novel way to construct and study genuine cohomology theories on stacks, orbifolds, and orbispaces, defined from stable global homotopy types represented by orthogonal spectra. The universal compact Lie group (which is neither compact nor a Lie group) is a well known object, namely the topological monoid $\mathcal L$ of linear isometric self-embeddings of $\mathbb R^\infty$. The underlying space of $\mathcal L$ is contractible, and the homotopy theory of $\mathcal L$-spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces. However, the monoid $\mathcal L$ contains copies of all compact Lie groups in a specific way, and we define global equivalences of $\mathcal L$-spaces by testing on corresponding fixed points. We establish a global model structure on the category of $\mathcal L$-spaces and prove it to be Quillen equivalent to the global model category of orthogonal spaces, and to the category of orbispaces, i.e., presheaves of spaces on the global orbit category.