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

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.