Given a commutative ring, R R, a π 1 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 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.
Alberga et al. (Tue,) studied this question.