Parametrized stability and the universal property of global spectra

TL;DR

This paper develops a new higher categorical framework for parametrized stability and semiadditivity, leading to a universal property of global spectra that generalizes classical equivariant stable homotopy theory.

Contribution

It introduces global -categories and a universal property characterizing global spectra as free equivariantly stable global -categories.

Findings

Identification of the free presentable equivariantly stable global -category

Extension of semiadditivity and stability concepts to parametrized and global contexts

Connection to classical global spectra for finite groups

Abstract

We develop a framework of parametrized semiadditivity and stability with respect to so-called atomic orbital subcategories of an indexing $\infty$-category $T$, extending work of Nardin. Specializing this framework, we introduce global $\infty$-categories and the notions of equivariant semiadditivity and stability, yielding a higher categorical version of the notion of a Mackey 2-functor studied by Balmer-Dell'Ambrogio. As our main result, we identify the free presentable equivariantly stable global $\infty$-category with a natural global $\infty$-category of global spectra for finite groups, in the sense of Schwede and Hausmann.