{
  "id": "0801.3843",
  "version": "v2",
  "published": "2008-01-24T20:57:03.000Z",
  "updated": "2009-07-27T14:04:27.000Z",
  "title": "The Classifying Space of a Topological 2-Group",
  "authors": [
    "John C. Baez",
    "Danny Stevenson"
  ],
  "comment": "31 pages LaTeX, 2 eps figures, a few errors fixed",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-07-27T14:04:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18D05",
      "18D10",
      "55R35",
      "55R40"
    ],
    "keywords": [
      "classifying space",
      "first cech cohomology",
      "simply-connected compact simple lie group",
      "mild conditions",
      "coefficients"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.3843B"
    }
  }
}