Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$(\langle L\rangle, \langle L_{0}\rangle)$. Our main result is the
identification of $\langle L\rangle$ with the classifying space of
$\mathcal{C}(L)$.

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The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: physics