#### Volume 21, issue 2 (2021)

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Strict algebraic models for rational parametrised spectra, I

### Vincent Braunack-Mayer

Algebraic & Geometric Topology 21 (2021) 917–1019
##### 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. 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 parameter space $X\phantom{\rule{-0.17em}{0ex}}$.

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 novel algebraic descriptions of rational homotopy classes of fibrewise stable maps, providing new tools for the study of section spaces.

##### Keywords
rational homotopy theory, parametrised spectra, Whitehead products, Koszul duality
##### Mathematical Subject Classification 2010
Primary: 55P62
Secondary: 16T15, 55N99, 55Q15
##### Publication
Revised: 14 April 2020
Accepted: 15 June 2020
Published: 25 April 2021
##### Authors
 Vincent Braunack-Mayer Mathematics, Division of Science New York University Abu Dhabi Abu Dhabi United Arab Emirates