Realization of a Lie algebra and classifying space of crossed modules

Yves Félix, Daniel Tanré

Published 2021-03-08Version 1

Complete differential graded Lie algebras appear to be a wonderful tool for giving nice algebraic models for the rational homotopy type of non-simply connected spaces. In particular, there is a realization functor of any Lie algebra as a simplicial set. In a previous work, we considered the particular case of a complete graded Lie algebra, $L$, concentrated in degree 0 and proved that its geometric realization is isomorphic to the usual bar construction on the Malcev group associated to $L$. Here we consider the case of a complete differential graded Lie algebra, $L$, concentrated in degrees 0 and 1. We first establish that $L$ has the structure of a crossed module and prove that the geometric realization of $L$ is isomorphic to the simplicial classifying space of this crossed module.