Realization of Lie algebras and classifying spaces of crossed modules

Abstract: The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $\langle -\rangle$, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, $L_{0}$, concentrated in degree 0 and proved that $\langle L_{0}\rangle$ is isomorphic to the usual bar construction on the Malcev group associated to $L_{0}$. Here we consider the case of a complete differential graded Lie algebra, $L=L_{0}\oplus L_{1}$, concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Malcev groups. In particular, there is a crossed module $\mathcal{C}(L)$ associated to $L$. We prove that $\mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to the simplicial pair $(\langle L\rangle, \langle L_{0}\rangle)$. Our main result is the identification of $\langle L\rangle$ with the classifying space of $\mathcal{C}(L)$.