A derived Milnor-Moore theorem

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 $X.$