A new approach to rational stable parametrized homotopy theory

This work develops a comprehensive algebraic model for rational stable parametrized homotopy theory over arbitrary base spaces. Building on the simplicial analogue of the foundational framework of May-Sigurdsson for parametrized spectra, and the homotopy theory of complete differential graded Lie algebras, we construct an explicit sequence of Quillen equivalences that translate the homotopy theory of rational spectra of retractive simplicial sets into the purely algebraic framework of complete differential graded modules over the completed universal enveloping algebra UL of a Lie model L of the base simplicial set B. Explicitly, there is a sequence of Quillen adjunctions SpB SpL Sp_UL⁰ cdgm_UL which induces a natural, strong monoidal equivalence of categories Ho\,SpBQ Ho\, cdgm_UL. This equivalence is highly effective in practice as it provides direct computational access to invariants of simplicial spectra by translating them into homotopy invariants of UL-modules. Here SpB denotes the stable model category of spectra of retractive simplicial sets over B, SpL denotes the stable model category of spectra of retractive complete differential graded Lie algebras over L, Sp_UL⁰ denotes the stable model category of connected UL-module spectra, and cdgm_UL denotes the category of complete differential graded UL-modules.