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

Abstract

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