{
  "id": "1404.1852",
  "version": "v2",
  "published": "2014-04-07T17:25:56.000Z",
  "updated": "2014-09-23T14:27:42.000Z",
  "title": "The Grothendieck construction for model categories",
  "authors": [
    "Yonatan Harpaz",
    "Matan Prasma"
  ],
  "comment": "Includes a new subsection (3.1) which contains a comparison with Lurie's infinity-categorical Grothendieck construction",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by $\\mathcal{M}$ and a new notion of \\textbf{model fibrations} over $\\mathcal{M}$. When $\\mathcal{M}$ is a model category, our construction endows the Grothendieck construction with a model structure which gives a presentation of Lurie's $\\infty$-categorical Grothendieck construction and enjoys several good formal properties. We apply our construction to various examples, yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-07T17:25:56.000Z",
      "abstract": "The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing cat- egory. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by M and a new notion of model fibrations over M. When M is a model category, our construction endows the Grothendieck construction with a model struc- ture which enjoys several good formal properties. We apply our construc- tion to various examples, yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-23T14:27:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "model category",
      "grothendieck construction",
      "suitable monoidal model categories",
      "weak group actions",
      "yielding model structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.1852H"
    }
  }
}