{
  "id": "2103.04927",
  "version": "v1",
  "published": "2021-03-08T17:39:30.000Z",
  "updated": "2021-03-08T17:39:30.000Z",
  "title": "Realization of a Lie algebra and classifying space of crossed modules",
  "authors": [
    "Yves Félix",
    "Daniel Tanré"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-08T17:39:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P62",
      "17B55",
      "55U10"
    ],
    "keywords": [
      "crossed module",
      "complete differential graded lie algebra",
      "classifying space",
      "differential graded lie algebras appear",
      "geometric realization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}