Strict algebraic models for rational parametrised spectra, I

Braunack-Mayer, Vincent

doi:10.2140/agt.2021.21.917

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Strict algebraic models for rational parametrised spectra, I

Vincent Braunack-Mayer

Algebraic & Geometric Topology 21 (2021) 917–1019

DOI: 10.2140/agt.2021.21.917

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 $Λ_{X}$ 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 $Λ_{X}$ –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 $X$ .

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.

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Keywords

rational homotopy theory, parametrised spectra, Whitehead products, Koszul duality

Mathematical Subject Classification 2010

Primary: 55P62

Secondary: 16T15, 55N99, 55Q15

References

Publication

Received: 3 November 2019

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

Volume 21, issue 2 (2021)