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

Mehmet Akif Erdal

• 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.