Building on Quillen’s rational homotopy theory, we obtain algebraic models for the
rational homotopy theory of parametrised spectra. For any simply connected
space there is a dg Lie
algebra
and a (coassociative
cocommutative) dg coalgebra
that model the rational homotopy type. We prove that the rational homotopy type of an
–parametrised spectrum is completely
encoded by a
–representation
and also by a
–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
.
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.
