The paper presents a theorem that generalizes Elmendorf’s theorem to actions of monoids. The author constructs a model structure on the category of \(M\)-spaces and \(M\)-equivariant maps, where \(M\) is a monoid. The construction is based on a collection \(\mathcal{X}\) of submonoids of \(M\) and for each \(N\) in \(\mathcal{X}\), a collection \(\mathcal{Y}_N\) of subgroups of \(G(N)\), where \(G\) is the group completion functor from monoids to groups (i.e. the left adjoint of the forgetful functor). The weak equivalences and fibrations in this model structure are determined by the standard \(\mathcal{Y}_N\)-model structures on \(G(N)\)-spaces for all \(N\) in \(\mathcal{X}\). The paper also demonstrates that there exists a small category \(\mathbf{O}_{(\mathcal{X},\mathcal{Y})}\) whose objects are \(M\)-spaces \(M\times_N G(N)/H\) for each \(N \in \mathcal{X}\) and \(H\in \mathcal{Y}_N\) and morphisms are \(M\)-equivariant maps, such that, under mild conditions on \(M\) and \(\mathcal{X}\) and \(\mathcal{Y}\), the projective model structure on the category of contravariant \(\mathbf{O}_{(\mathcal{X},\mathcal{Y})}\)-diagrams of spaces is Quillen equivalent to the constructed model structure.