Strict algebraic models for rational parametrised spectra II

TL;DR

This paper develops algebraic models for the rational homotopy theory of parametrised spectra, extending Sullivan's PL de Rham theory to simplify calculations and provide a comprehensive algebraic-topological dictionary.

Contribution

It extends Sullivan's rational homotopy theory to parametrised spectra, establishing algebraic models and equivalences that facilitate direct computations.

Findings

Established an equivalence between rational homotopy categories and differential graded modules.

Provided algebraic models for key topological constructions like base change and fibrewise smash products.

Enabled algebraic calculations in parametrised stable homotopy theory.

Abstract

In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of arXiv:1910.14608, which are based on Quillen's rational homotopy theory. According to Sullivan, the rational homotopy type of a nilpotent space $X$ with finite Betti numbers is completely determined by a commutative differential graded algebra $A$ modelling the cup product on rational cohomology. In this article we extend this correspondence between topology and algebra to parametrised stable homotopy theory: for a space $X$ corresponding to the cdga $A$, we prove an equivalence between specific rational homotopy categories for parametrised spectra over $X$ and for differential graded $A$-modules. While not full, the rational homotopy categories we consider contain a large class of parametrised spectra. The simplicity of the approach that we develop enables direct calculations in parametrised stable homotopy theory using differential graded modules. To illustrate the usefulness of our approach, we build a comprehensive dictionary of algebraic translations of topological constructions; providing algebraic models for base change functors, fibrewise stabilisations, parametrised Postnikov sections, fibrewise smash products, and complexes of fibrewise stable maps.