Singular chains on Lie groups and the Cartan relations II

TL;DR

This paper extends the equivalence between smooth modules over the DG Hopf algebra of singular chains on a compact Lie group and representations of a universal DG Lie algebra to an $ ext{A}_$-quasi-equivalence of DG categories, using advanced algebraic tools.

Contribution

It demonstrates an $ ext{A}_$-quasi-equivalence of DG categories for compact Lie groups, extending previous categorical equivalences to a homotopical setting.

Findings

Constructed an $ ext{A}_$-quasi-isomorphism between Bott-Shulman-Stasheff algebra and Hochschild cochains.

Extended categorical equivalence to an $ ext{A}_$-quasi-equivalence for compact groups.

Utilized Van Est map and Gugenheim's $ ext{A}_$-De Rham theorem in the proof.

Abstract

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and denote by $\mathrm{C}_{\bullet}(G)$ the DG Hopf algebra of smooth singular chains on $G$. In a companion paper it was shown that the category of sufficiently smooth modules over $\mathrm{C}_{\bullet}(G)$ is equivalent to the category of representations of $\mathbb{T} \mathfrak{g}$, the DG Lie algebra which is universal for the Cartan relations. In this paper we show that, if $G$ is compact, this equivalence of categories can be extended to an $\mathsf{A}_{\infty}$-quasi-equivalence of the corresponding DG categories. As an intermediate step we construct an $\mathsf{A}_{\infty}$-quasi-isomorphism between the Bott-Shulman-Stasheff DG algebra associated to $G$ and the DG algebra of Hochschild cochains on $\mathrm{C}_{\bullet}(G)$. The main ingredients in the proof are the Van Est map and Gugenheim's $\mathsf{A}_{\infty}$ version of De Rham's theorem.