Realization of Lie algebras and classifying spaces of crossed modules

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 this category 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 $(langle Lrangle, langle L_{0}rangle)$. Our main result is the identification of $langle Lrangle$ with the classifying space of $mathcal{C}(L)$.