Equivariant Homotopy Theory via Simplicial Coalgebras

TL;DR

This paper develops an equivariant homotopy theory framework using simplicial coalgebras, extending previous models to include group actions and linearized quasi-categorical equivalences.

Contribution

It generalizes the model of homotopy theory via simplicial coalgebras to the equivariant setting and introduces a linearized approach for $G$-simplicial sets.

Findings

Established a $G$-equivariant analog of the simplicial coalgebra model.

Proved a generalization of Elmendorf's theorem for equivariant cases.

Modeled $G$-simplicial sets under linearized quasi-categorical equivalences.

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.