Equivariant Homotopy Theory via Simplicial Coalgebras
Sofía Martínez Alberga, Manuel Rivera
TL;DR
The paper advances the program of modeling $G$-spaces at the chain level by extending the RR22 coalgebraic framework to the $G$-equivariant setting, with coefficients in an algebraically closed field and then in perfect fields. It introduces $G$-fixed-point and orbit-diagram model structures, and proves Elmendorf-type Quillen equivalences that relate $G$-spaces to $G$-connected simplicial cocommutative coalgebras via the equivariant chains functor. The main contributions include a $G$-equivariant full and faithful embedding of $G$-spaces up to $G$-$\pi_1$-$\mathbb{F}$-equivalence into $G$-coalgebras, a parallel embedding for a linearized quasi-categorical notion, and a rationalization interpretation in the perfect-field setting. These results provide a robust algebraic framework for extracting invariants of $G$-spaces and set the stage for practical computations of topological and geometric data in equivariant contexts, including extensions to rational homotopy theory via fixed points.
Abstract
Given a commutative ring $R$, a $π_1$-$R$-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 $π_1$-$R$-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.