Almost Global Homotopy Theory

TL;DR

This paper introduces a new framework for global orthogonal spectra, establishing their properties and relations to equivariant cohomology theories, enabling the construction of numerous global ring spectra and exploring their underlying structures.

Contribution

It defines global orthogonal spectra and their unitary versions, relates them via model structures, and shows how to globalize various equivariant cohomology theories, including quasi-elliptic cohomology.

Findings

Global spectra form a model category Quillen equivalent to existing structures.

Many equivariant cohomology theories can be globalized within this framework.

Starting from one global ring spectrum, infinitely many can be constructed.

Abstract

In this paper we develop the definition of a global orthogonal spectrum and its unitary version. It relates $G-$equivariant spectra by equivariant weak equivalence in a coherent way. This category of global spectra has a model structure Quillen equivalent to the global model structure on orthogonal spectra. We also show that there is a large family of equivariant cohomology theories, including quasi-elliptic cohomology, that can be globalized in the new context. Starting from one global ring spectrum, we can construct infinitely many distinct global ring spectra. Moreover, in light of the results in this paper, we ask whether we have the conjecture that the globalness of a cohomology theory is completely determined by the formal component of its divisible group and when the $\acute{e}$tale component of it varies the globalness does not change.