Global model structures for $\ast$-modules

We extend Schwede’s work on the unstable global homotopy theory of orthogonal spaces and $\mathcal{L}$-spaces to the category of $\ast$-modules (i.e., unstable S-modules). We prove a theorem which transports model structures and their properties from $\mathcal{L}$-spaces to $\ast$-modules and show that the resulting global model structure for $\ast$-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.

Keywords

global homotopy theory, equivariant homotopy theory, model category, orthogonal space, $\ast$-module

2010 Mathematics Subject Classification

18G55, 55P91

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This research was partly supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

Received 6 September 2016

Received revised 1 March 2018

Published 27 February 2019