turkmath.org

For a group $G$ and a collection of subgroups $Y$ of $G$, the orbit category $O_{Y}$ is the category whose objects are $G$-orbits $G/H$ for each $H\in Y$ and whose morphisms are  $G$-equivariant maps in between. Due to Elmendorf's Theorem that the category $G$-spaces and the category of contravariant  $O_{Y}$-diagrams of spaces have equivalent homotopy theories. This provides a great convenience when studying $G$-equivariant homotopy theory since one can reduce it to non-equivariant homotopy theory of associated fixed point systems. In this talk, we describe a non-trivial extension of  this idea to the actions of monoids. Let $M$ be a monoid and $G(M)$ be its group completion, $Z$ be a collection of submonoids of $M$ and for each $N\in Z$, $Y_N$ be a collection of subgroups of $G(N)$. First we will show that the category of $M$-spaces and $M$-equivariant maps admits a model structure in which weak equivalences and fibrations are determined by the standard equivariant homotopy theory of $G(N)$-spaces for each $N \in Z$. Then, we will show that this model structure is Quillen equivalent to the projective model structure
on the category of contravariant $O_{Z,Y}$-diagrams of spaces, where $ O_{Z,Y}$ is the category whose objects are induced orbits $M\times_N G(N)/H$ for each $N\in Z$ and $H \in Y_N$ and morphisms are $M$-equivariant maps. Finally, if time permits, we will state some applications.