{
  "id": "1511.03098",
  "version": "v1",
  "published": "2015-11-10T13:18:39.000Z",
  "updated": "2015-11-10T13:18:39.000Z",
  "title": "Extensions between functors from groups",
  "authors": [
    "Christine Vespa"
  ],
  "categories": [
    "math.AT",
    "math.CT",
    "math.KT"
  ],
  "abstract": "We compute Ext-groups between tensor powers composed by the abelianization functor. More precisely, we compute the groups $Ext^*\\_{\\mathcal{F}(\\textbf{gr})}(T^n \\circ \\mathfrak{a}, T^m \\circ \\mathfrak{a})$ where $T^n$ is the n-th tensor power functor and $\\mathfrak{a}$ is the abelianization functor from the category of free groups to abelian groups. These groups are shown to be non-zero if and only if $*=m-n \\geq 0$ and $Ext^{m-n}\\_{\\mathcal{F}(\\textbf{gr})}(T^n \\circ \\mathfrak{a}, T^m \\circ \\mathfrak{a})=\\mathbb{Z}[\\Omega(m,n)]$ where $\\Omega(m,n)$ is the set of surjections from the set having $m$ elements to the one having $n$ elements. We make explicit the action of symmetric groups on these groups and the Yoneda and external products. We use this computation in order to obtain other computations of Ext-groups between functors from groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-10T13:18:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extensions",
      "abelianization functor",
      "n-th tensor power functor",
      "free groups",
      "ext-groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151103098V"
    }
  }
}