An Equivariant Generalization of McDuff's Theorem

In 1976, Kan and Thurston proved the theorem that any path-connected space $X$ is homology equivalent to the classifying space of some discrete group $G$. In 1979, McDuff proved a homotopy version of it: any path-connected space $X$ has the same weak homotopy type as the classifying space of some discrete monoid $M$. In 1984, Fiedorowicz reproved McDuff’s theorem using a largely categorical construction. In this paper we will generalize Fiedorowicz’s proof of McDuff’s theorem to the equivariant case. Precisely, we will prove that any $G$-connected space $X$ with a $G$-fixed basepoint $x_0$ has the same weak homotopy type as the classifying space of some discrete $G$-monoid.