Homotopy theory of monoid actions via group actions and an Elmendorf style theorem

Let M be a monoid and \(G:\mathbf {Mon} \rightarrow \mathbf {Grp}\) the group completion functor from monoids to groups. Given collection \(\mathcal {X}\) of submonoids for each \(N\in \mathcal {Y}_N\) subgroups G(N), we construct model structure on category M-spaces M-equivariant maps, called \((\mathcal {X},\mathcal {Y})\)-model structure, in which weak equivalences fibrations are induced standard {Y}_N\)-model structures G(N)-spaces all {X}\). We also show that pair collections {Y})\) there is small \({{\mathbf {O}}}_{(\mathcal {Y})}\) whose objects \(M\times _NG(N)/H\) \(H\in morphisms such Quillen equivalent projective contravariant {Y})}\)-diagrams spaces.