Abstract: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 $\widehat{UL}$ of a Lie model $L$ of the base simplicial set $B$. Explicitly, there is a sequence of Quillen adjunctions $$ \mathbf{Sp}_B \leftrightarrows \mathbf{Sp}_L \leftrightarrows \mathbf{Sp}_{\widehat{UL}}^0 \leftrightarrows
\mathbf{cdgm}_{\widehat{UL}} $$ which induces a natural, strong monoidal equivalence of categories $$
{\rm Ho}\,\mathbf{Sp}_B^{\Bbb Q}\cong {\rm Ho}\, \mathbf{cdgm}_{\widehat{UL}}.
$$ 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 $\widehat{UL}$-modules. Here $\mathbf{Sp}_B$ denotes the stable model category of spectra of retractive simplicial sets over $B$, $\mathbf{Sp}_L$ denotes the stable model category of spectra of retractive complete differential graded Lie algebras over $L$, $\mathbf{Sp}_{\widehat{UL}}^0$ denotes the stable model category of connected $\widehat{UL}$-module spectra, and $\mathbf{cdgm}_{\widehat{UL}}$ denotes the category of complete differential graded $\widehat{UL}$-modules.
From: Aniceto Murillo [view email]
[v1]
Sun, 14 Sep 2025 19:02:30 UTC (68 KB)
[v2]
Wed, 24 Jun 2026 07:14:33 UTC (62 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。