Equivariant homotopy theory via simplicial coalgebras

Key Points

A G-equivariant version of homotopy equivalence is established using simplicial coalgebras.
The results generalize a famous theorem by Elmendorf related to fundamental groups.
Modeling of G-simplicial sets is discussed under linearized quasi-categorical equivalence.
The approach introduces a new notion of weak equivalence based on the localized Cobar functor.

Abstract

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.

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Cite This Study

Alberga et al. (Tue,) studied this question.

synapsesocial.com/papers/689e03e9d61984b91e13d19e https://doi.org/https://doi.org/10.1090/proc/17315

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