Global model categories and topological Andr\'e-Quillen cohomology

TL;DR

This paper develops a comprehensive framework called global model categories to unify phenomena in global equivariant homotopy theory, introduces their stabilizations, and defines a global version of topological Andr"e-Quillen cohomology.

Contribution

It introduces global model categories, constructs their stabilizations, and defines global topological Andr"e-Quillen cohomology, extending classical concepts to a global equivariant setting.

Findings

Global model categories effectively capture phenomena in global equivariant homotopy theory.

Genuine stabilizations generalize the transition from unstable to stable global homotopy theory.

Global topological Andr"e-Quillen cohomology is expressed via genuine stabilization, paralleling classical non-equivariant results.

Abstract

We introduce global model categories as a general framework to capture several phenomena in global equivariant homotopy theory. We then construct genuine stabilizations of these, generalizing the usual passage from unstable to stable global homotopy theory. Finally, we define the global topological Andr\'e-Quillen cohomology of an ultra-commutative ring spectrum and express it in terms of a genuine stabilization in our framework in analogy with the classical non-equivariant description obtained by Basterra and Mandell.