Strict algebraic models for rational parametrised spectra I

TL;DR

This paper develops algebraic models for the rational homotopy theory of parametrised spectra, extending Quillen's approach, and establishes a dictionary translating topological constructions into algebraic data.

Contribution

It introduces dg Lie algebra and coalgebra models for rational parametrised spectra, linking topological and algebraic perspectives via symmetric monoidal equivalences.

Findings

Rational parametrised spectra are modeled by dg Lie algebra representations and dg coalgebra comodules.

The fibrewise smash product corresponds to derived tensor and cotensor products in algebra.

Provides algebraic descriptions of rational homotopy classes of fibrewise stable maps.

Abstract

Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply-connected space $X$ there is a dg Lie algebra $\Lambda_X$ and a (coassociative cocommutative) dg coalgebra $C_X$ that model the rational homotopy type. In this article, we prove that the rational homotopy type of an $X$-parametrised spectrum is completely encoded by a $\Lambda_X$-representation and also by a $C_X$-comodule. The correspondence between rational parametrised spectra and algebraic data is obtained by means of symmetric monoidal equivalences of homotopy categories that vary pseudofunctorially in the simply-connected parameter space $X$. Our results establish a comprehensive dictionary enabling the translation of topological constructions into homological algebra using Lie representations and comodules, and conversely. For example, the fibrewise smash product of parametrised spectra is encoded by the derived tensor product of dg Lie representations and also by the derived cotensor product of dg comodules. As an application, we obtain algebraic descriptions of rational homotopy classes of fibrewise stable maps, providing new tools for the study of spaces of sections.