{
  "id": "1604.06551",
  "version": "v1",
  "published": "2016-04-22T06:49:02.000Z",
  "updated": "2016-04-22T06:49:02.000Z",
  "title": "Crossed modules as maps between connected components of topological groups",
  "authors": [
    "Emmanuel D. Farjoun",
    "Yoav Segev"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AT",
    "math.GR"
  ],
  "abstract": "The purpose of this note is to observe that a homomorphism of discrete groups $f:\\Gamma\\to G$ arises as the induced map $\\pi_0(\\mathfrak{M})\\to \\pi_0(\\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\\mathfrak{M}\\subseteq \\mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\\mathfrak{M}\\subseteq \\mathfrak{X}$ exists by homotopically discrete topological groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-22T06:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18G30",
      "55U10",
      "57T30"
    ],
    "keywords": [
      "connected components",
      "discrete groups",
      "homotopically discrete topological groups",
      "crossed module structure",
      "path components"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160406551F"
    }
  }
}