{
  "id": "2303.10822",
  "version": "v1",
  "published": "2023-03-20T01:37:01.000Z",
  "updated": "2023-03-20T01:37:01.000Z",
  "title": "Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups",
  "authors": [
    "Ahmet A. Husainov"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "The paper considers non-Abelian homology groups for a diagram of groups introduced as homotopy groups of a simplicial replacement. It is proved that the non-Abelian homology groups of the group diagram are isomorphic to the homotopy groups of the homotopy colimit of the diagram of classifying spaces, with a dimension shift of 1. As an application, a method is developed for finding a nonzero homotopy group of least dimension for a homotopy colimit of classifying spaces. For a group diagram over a free category with a zero colimit, a criterion for the isomorphism of the first non-Abelian and first Abelian homology groups is obtained. It is established that the non-Abelian homology groups are isomorphic to the cotriple derived functors of the colimit functor defined on the category of group diagrams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-20T01:37:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55U10",
      "18C15",
      "18G10",
      "55P10",
      "55R35"
    ],
    "keywords": [
      "homotopy colimit",
      "classifying spaces",
      "non-abelian homology groups",
      "group diagram",
      "first abelian homology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}