A Lie characterization of the Bousfield-Kan ${\mathbb{Q}}$-completion and ${\mathbb{Q}}$-good spaces

TL;DR

This paper establishes a Lie algebraic characterization of the Bousfield-Kan rational completion of spaces, showing that a certain Quillen pair models this completion up to homotopy.

Contribution

It provides a Lie algebraic perspective on the Bousfield-Kan rational completion, linking model and realization functors to this classical construction.

Findings

The unit of the Quillen pair models the Bousfield-Kan ${f Q}$-completion.

The approach characterizes ${f Q}$-good spaces using Lie algebraic methods.

Homotopy equivalence is established between the functor pair and the ${f Q}$-completion.

Abstract

We prove that the unit of the Quillen pair ${\mathfrak{L}}\colon {\bf sset}\rightleftarrows {\bf cdgl}\colon {\langle\,\cdot\,\rangle}$ given by the model and realization functor is, up to homotopy, the Bousfield-Kan ${\mathbb{Q}}$-completion.