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Algebraic models for equivariant rational homotopy theory for discrete groups

José M. Moreno-Fernández, Bruno Stonek

TL;DR

The paper develops a systematic, category-theoretic framework to extend algebraic models of rational homotopy theory to genuine $G$-equivariant settings for discrete groups by using ${\mathcal{O}_G}$-indexed presheaves. It establishes that partial Quillen equivalences between spaces and algebraic models (cdga and cdgl) lift to functor categories, yielding concrete equivariant models via $G$-systems of cdgas and cdgls. Through Elmendorf’s orbit-category perspective, it provides a unified approach to modeling connected nilpotent finite-type rational spaces in the equivariant context, and it discusses alternative model structures and their limitations. The work thus offers a modular, rigorous path to equivariant rational homotopy theory, connecting with prior non-equivariant results and inspiring future refinements of minimal-model-type descriptions in the equivariant setting.

Abstract

We provide a framework which generalizes algebraic models of a homotopy theory of spaces to the genuine equivariant case for a discrete group. We explain how this applies to commutative differential graded algebra (cdga) models and complete differential graded Lie algebra models for rational spaces. We compare the cdga model to other model categories in the literature.

Algebraic models for equivariant rational homotopy theory for discrete groups

TL;DR

The paper develops a systematic, category-theoretic framework to extend algebraic models of rational homotopy theory to genuine -equivariant settings for discrete groups by using -indexed presheaves. It establishes that partial Quillen equivalences between spaces and algebraic models (cdga and cdgl) lift to functor categories, yielding concrete equivariant models via -systems of cdgas and cdgls. Through Elmendorf’s orbit-category perspective, it provides a unified approach to modeling connected nilpotent finite-type rational spaces in the equivariant context, and it discusses alternative model structures and their limitations. The work thus offers a modular, rigorous path to equivariant rational homotopy theory, connecting with prior non-equivariant results and inspiring future refinements of minimal-model-type descriptions in the equivariant setting.

Abstract

We provide a framework which generalizes algebraic models of a homotopy theory of spaces to the genuine equivariant case for a discrete group. We explain how this applies to commutative differential graded algebra (cdga) models and complete differential graded Lie algebra models for rational spaces. We compare the cdga model to other model categories in the literature.
Paper Structure (13 sections, 23 theorems, 8 equations)

This paper contains 13 sections, 23 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Acknowledgments
  4. Generalities
  5. Modeling connected nilpotent spaces of finite type
  6. The unpointed case
  7. Some comments on the pointed case
  8. Modeling connected nilpotent pointed spaces
  9. From partial Quillen equivalences to functor categories
  10. Algebraic models for equivariant rational homotopy theory
  11. The cdga model
  12. Other model structures
  13. The cdgl model

Key Result

Lemma 1

Let $I$ be a small category. A Quillen adjunction ${\mathcal{C}}\rightleftarrows {\mathcal{D}}$ which is a partial Quillen equivalence on the subcategories ${\mathcal{C}}_0$, ${\mathcal{D}}_0$ induces a partial Quillen equivalence ${\mathcal{C}}^I \rightleftarrows {\mathcal{D}}^I$ on the subcategori

Theorems & Definitions (44)

  • Lemma 1: \ref{['lem-sciarappa']}
  • Theorem 1: \ref{['cor-sciarappa-cdga']}
  • Theorem 2: \ref{['cor-cdgl']}
  • Definition 2.1
  • Proposition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Example 2.6
  • Theorem 2.7
  • Theorem 2.8
  • ...and 34 more