A derived Milnor-Moore theorem

TL;DR

This paper develops a new derived version of the Milnor-Moore theorem using Koszul duality, constructing an enveloping Hopf algebra functor for Lie algebras in stable monoidal categories and analyzing its properties in rational and chromatic homotopy theory.

Contribution

It introduces a derived enveloping Hopf algebra functor for Lie algebras in stable monoidal categories and studies its properties, extending classical results to a homotopical setting.

Findings

The functor is fully faithful in rational stable categories.

The unit of the adjunction relates to the Goodwillie tower in v_n-periodic homotopy types.

The construction generalizes the classical Milnor-Moore theorem to a derived setting.

Abstract

For every stable presentably symmetric monoidal $\infty$-category $\mathcal{C}$ we use the Koszul duality between the spectral Lie operad and the cocommutative cooperad to construct an enveloping Hopf algebra functor $\mathcal{U}: \mathrm{Alg}_{\mathrm{Lie}}(\mathcal{C}) \to \mathrm{Hopf}(\mathcal{C})$ from Lie algebras in $\mathcal{C}$ to cocommutative Hopf algebras in $\mathcal{C}$ left adjoint to a functor of derived primitive elements $\mathrm{Prim}$. We study the unit of this adjunction in rational and chromatic homotopy theory: we prove that if $\mathcal{C}$ is a rational stable presentably symmetric monoidal $\infty$-category, the enveloping Hopf algebra functor $\mathcal{U}: \mathrm{Alg}_{\mathrm{Lie}}(\mathcal{C}) \to \mathrm{Hopf}(\mathcal{C})$ is fully faithful reproving a result of Gaitsgory-Rozenblyum. Let $n \geq 1 $ be a natural and $\Phi[-1]: \mathcal{S}_{v_n} \to \mathrm{Alg}_{\mathrm{Lie}}(\mathrm{Sp}_{T_n})$ the shifted Bousfield-Kuhn functor from $v_n$-periodic homotopy types to spectral Lie algebras in $T_n$-local spectra. We prove that for every $v_n$-periodic homotopy type $X$ the unit $\Phi(X)[-1] \to Prim \mathcal{U}(\Phi(X)[-1])$ identifies with the Goodwillie completion $ \Phi \to \lim_{n \geq 0} P_n(\Phi)$ evaluated at the loop space of $X.$