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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 $mathcal{C}(L)$.
We introduce a notion of $n$-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to $n$-ary case. This notion is also an algebraic analogue of $n$-Lie algebroids. We develop representation theory and describe a cohomology complex of $n
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that the
We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples
We prove that the set of concordance classes of sections of an infinity-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss. This is reminiscent of an h-principle in which the role of isotopy is played by concordance. As an
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admit