{
  "id": "math/0611728",
  "version": "v2",
  "published": "2006-11-23T14:57:14.000Z",
  "updated": "2007-01-23T16:15:03.000Z",
  "title": "Normalisation for the fundamental crossed complex of a simplicial set",
  "authors": [
    "Ronald Brown",
    "Rafael Sivera"
  ],
  "comment": "29 pages, accepted for JHRS Mac Lane memorial volume Revised Jan 2007. Minor points. More references",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the {\\it fundamental crossed complex} of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, {and give a survey of the required basic facts on crossed complexes.}",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-23T16:15:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18D10",
      "18G30",
      "18G50",
      "20L05",
      "55N10",
      "55N25"
    ],
    "keywords": [
      "fundamental crossed complex",
      "simplicial set",
      "crossed complexes",
      "simplicial abelian groups",
      "homotopy addition lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11728B"
    }
  }
}