{
  "id": "1912.08696",
  "version": "v1",
  "published": "2019-12-18T16:21:15.000Z",
  "updated": "2019-12-18T16:21:15.000Z",
  "title": "The long exact sequence of homotopy $n$-groups",
  "authors": [
    "Ulrik Buchholtz",
    "Egbert Rijke"
  ],
  "comment": "9 pages. Comments welcome",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "We introduce the notion of $n$-exactness for a short sequence $F\\to E\\to B$ of pointed types, and show that any fiber sequence $F\\hookrightarrow E \\twoheadrightarrow B$ of arbitrary types induces a short sequence $\\|F\\|_{n-1} \\to \\|E\\|_{n-1} \\to \\|B\\|_{n-1}$ that is $n$-exact at $\\|E\\|_{n-1}$. We explain how the indexing makes sense when interpreted in terms of $n$-groups, and we compare our definition to the existing definitions of an exact sequence of $n$-groups for $n=1,2$. In conclusion, we obtain the long $n$-exact sequence of homotopy $n$-groups of a fiber sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-18T16:21:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55U35",
      "55R65",
      "03B15"
    ],
    "keywords": [
      "long exact sequence",
      "short sequence",
      "fiber sequence",
      "arbitrary types induces",
      "definition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}