Equivariant homotopy theory via simplicial coalgebras

Given a commutative ring, R R , a π 1 π _1 - R R -equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an R R -homology equivalence between universal covers. When R R is an algebraically closed field, Raptis and Rivera [Int. Math. Res. Not. IMRN 16 (2024), pp. 11766–11811] described a full and faithful model for the homotopy theory of spaces up to π 1 π _1 - R R -equivalence. They use 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 G -equivariant analog of this statement using a generalization of a celebrated theorem of Elmendorf [Trans. Amer. Math. Soc. 277 (1983), pp. 275–284]. We also prove a more general result about modeling G G -simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras.