An Elmendorf-Piacenza type Theorem for Actions of Monoids

Let $M$ be a monoid and $G:\mathbf{Mon} \to \mathbf{Grp}$ be the group completion functor from monoids to groups. Given a collection $\mathcal{Z}$ of submonoids of $M$ and for each $N\in \mathcal{Z}$ a collection $\mathcal{Y}_N$ of subgroups of $G(N)$, we construct a model structure on the category of $M$-spaces and $M$-equivariant maps, in which weak equivalences and fibrations are determined by the standard $\mathcal{Y}_N$-model structures on $G(N)$-spaces for all $N\in \mathcal{Z}$. We also show that there is a small category $\mathsf{O}_(\mathcal{Z},\mathcal{Y})$ such that, under mild conditions on $\mathcal{Z}$ and $\mathcal{Y}_N$'s, the projective model structure on the category of contravariant $\mathsf{O}_(\mathcal{Z},\mathcal{Y})$-diagrams of spaces is Quillen equivalent to our model structure. In particular, we prove a theorem generalizing Elmendorf-Piacenza Theorem to actions of monoids.