Global model structures for $*$-modules

TL;DR

This paper develops a new global model structure for $*$-modules, extending previous work on orthogonal spaces, and establishes Quillen equivalences that connect different models for the homotopy theory of $A_in$-spaces.

Contribution

It introduces a global model structure for $*$-modules and proves its monoidal Quillen equivalence to orthogonal spaces, unifying models for $A_in$-spaces.

Findings

Global model structure for $*$-modules established.

Quillen equivalences between $*$-modules and orthogonal spaces.

Equivalent models for the homotopy theory of $A_in$-spaces identified.

Abstract

We extend Schwede's work on the unstable global homotopy theory of orthogonal spaces and $\mathcal{L}$-spaces to the category of $*$-modules (i.e., unstable $S$-modules). We prove a theorem which transports model structures and their properties from $\mathcal{L}$-spaces to $*$-modules and show that the resulting global model structure for $*$-modules is monoidally Quillen equivalent to that of orthogonal spaces. As a consequence, there are induced Quillen equivalences between the associated model categories of monoids, which identify equivalent models for the global homotopy theory of $A_\infty$-spaces.