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

We prove that the unit of the Quillen pair $\mathfrak{L}\text{\hspace{0.17em}⁣}:\mathbf{s}\mathbf{s}\mathbf{e}\mathbf{t}\rightleftarrows \mathbf{c}\mathbf{d}\mathbf{g}\mathbf{l}\text{\hspace{0.17em}⁣}:⟨\cdot ⟩$ given by the model and realization functor is, up to homotopy, the Bousfield-Kan $\mathbb{Q}$-completion.