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

TL;DR

This paper develops a homotopy theory framework for monoid actions by relating them to group actions through a new model structure, and establishes an Elmendorf-style theorem connecting these models.

Contribution

It introduces a novel model structure for monoid actions based on submonoids and subgroups, and proves a Quillen equivalence to a diagram category, extending homotopy theory tools to monoid actions.

Findings

Constructed the $(\\mathcal{X},\\mathcal{Y})$-model structure on $M$-spaces.

Established a Quillen equivalence with a diagram category of spaces.

Extended homotopy theoretic methods to monoid actions via group completion.

Abstract

Let $M$ be a monoid and $G:\mathbf{Mon} \to \mathbf{Grp}$ 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.