Homology, Homotopy and Applications

Volume 26 (2024)

Number 2

Simplicial $*$-modules and mild actions

Pages: 229 – 258

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n2.a12

Authors

Tobias Lenz (Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany; and Mathematical Institute, University of Utrecht, The Netherlands)

Anna Marie Schröter (Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany)

Abstract

$\def\M{\mathcal{M}}$We develop an analogue of the theory of $*$-modules in the world of simplicial sets, based on actions of a certain simplicial monoid $E\M$ originally appearing in the construction of global algebraic $K$-theory.

As our main results, we show that strictly commutative monoids with respect to a certain box product on these simplicial $*$-modules yield models of equivariantly and globally coherently commutative monoids, and we give a characterization of simplicial $*$-modules in terms of a certain mildness condition on the $E\M$-action, relaxing the notion of tameness previously investigated by Sagave–Schwede and the first author.

Keywords

infinite loop space, $E_\infty$-monoid, $\mathcal{M}$-set, global homotopy theory, equivariant homotopy theory

2010 Mathematics Subject Classification

55P48

Received 21 August 2023

Received revised 18 December 2023

Accepted 19 December 2023

Published 2 October 2024