Orbispaces, orthogonal spaces, and 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.