2019/10/31 by Vincent Braunack-Mayer, Vincent Braunack‐Mayer
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Algebraic structure #Algebraic structures and combinatorial models #Coalgebra #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Mathematical analysis #Mathematics #Product (mathematics) #Pure mathematics #Tensor product #math.AT #msc:16T15 #msc:55N99 #msc:55P62 #msc:55Q15
paper · pdf · doi:10.2140/agt.2021.21.917
published as Algebr. Geom. Topol. 21 (2021) 917-1019 · 76 pages. V2: background material added, typos fixed
arxiv created 2020/07/21 · openalex publication_date 2021/04/25 · arxiv updated 2021/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 ΛX and a (coassociative cocommutative) dg coalgebra CX 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 ΛX-representation and also by a CX-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.