Simplicial $\ast$-modules and mild actions

We develop an analogue of the theory of $\ast$-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 $\ast$-modules yield models of equivariantly and globally coherently commutative monoids, and we give a characterization of simplicial $\ast$-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.