Equivariant Homotopy Theory via Simplicial Coalgebras

Abstract: Given a commutative ring $R$, a $\pi_1$-$R$-equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers. When $R$ is an algebraically closed field, Raptis and Rivera described a full and faithful model for the homotopy theory of spaces up to $\pi_1$-$R$-equivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by a localized version of the Cobar functor. In this article, we prove a $G$-equivariant analog of this statement using a generalization of a celebrated theorem of Elmendorf. We also prove a more general result about modeling $G$-simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras.