Simplicial $*$-modules and mild actions

TL;DR

This paper develops a new theory of simplicial $*$-modules based on actions of a simplicial monoid, providing models for equivariantly and globally coherently commutative monoids and introducing a mildness condition that relaxes previous notions.

Contribution

It introduces a simplicial $*$-module framework with an $E extbf{M}$-action, connecting to global algebraic $K$-theory and relaxing tameness conditions.

Findings

Strictly commutative monoids form models of equivariant and global coherence.

Characterization of simplicial $*$-modules via a mildness condition on $E extbf{M}$-actions.

Provides a relaxation of the tameness condition in the theory of simplicial modules.

Abstract

We develop an analogue of the theory of $*$-modules in the world of simplicial sets, based on actions of a certain simplicial monoid $E\mathcal M$ 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 $E\mathcal M$-action, relaxing the notion of tameness previously investigated by Sagave-Schwede and the first author.