Realization of Lie algebras and classifying spaces of crossed modules

8 Mar 2021  ·  Yves Félix, Daniel Tanré ·

The category of complete differential graded Lie algebras appears 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 concentrated in degrees 0 and 1. We first establish that it has the structure of a crossed module and prove that its geometric realization is isomorphic to the simplicial classifying space of this crossed module. read more

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Algebraic Topology 55P62, 17B55, 55U10