Global homotopy theory via partially lax limits

Abstract

Global equivariant homotopy theory is often motivated as the study of compatible collections of equivariant objects for some family of compact Lie groups. In this thesis we make this heuristic precise, by exhibiting the infinity categories of global spaces and global spectra as a partially lax limit of a diagram of equivariant spaces and spectra respectively. An object of such a partially lax limit is precisely a compatible collection of equivariant objects. We in fact present two approaches to this result. The first is of a direct and calculational nature, and works for arbitrary families of compact Lie groups. This method has the advantage of working in related situations, for example we also obtain a description of proper equivariant homotopy theory as a limit. It is the content of joint work of the author with Denis Nardin and Luca Pol, reproduced in this thesis. The second is of a more categorical nature, but only works for families of finite groups. In this generality it provides an interpretation of the partially lax limits above as a cocompletion procedure for infinity categories parametrized over the global indexing infinity category. We then identify a parametrized enhancement of global spaces and spectra with cocompletions of parametrized categories of equivariant spaces and spectra, using results of Bastiaan Cnossen, Tobias Lenz and the author. Additionally, we deduce a new universal property for Fin-global spectra, as the "representation stabilization" of global spaces at the representation spheres.