Realization of Lie algebras and classifying spaces of crossed modules

TL;DR

This paper connects differential graded Lie algebras with classifying spaces of crossed modules, extending algebraic models for rational homotopy types to include non-simply connected spaces.

Contribution

It establishes an equivalence between two-stage differential graded Lie algebras and crossed modules, and identifies their realization with classifying spaces.

Findings

The category of two-stage Lie algebras is equivalent to certain crossed modules.

The realization functor of these Lie algebras corresponds to classifying spaces of associated crossed modules.

The work extends the algebraic modeling of rational homotopy types to non-simply connected spaces.

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 $\mathcal{C}(L)$.