{
  "id": "2207.14608",
  "version": "v1",
  "published": "2022-07-29T11:04:15.000Z",
  "updated": "2022-07-29T11:04:15.000Z",
  "title": "An $\\infty$-categorical localisation functor for diagrams of simplicial sets",
  "authors": [
    "Severin Bunk"
  ],
  "comment": "27 pages; comments welcome",
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the $\\infty$-categorical localisation of the former at the objectwise weak homotopy equivalences. This builds on a Quillen equivalence between the projective and covariant model structures associated to $C$ due to Heuts-Moerdijk, as well as Cisinski's theory of $\\infty$-categorical localisations. We use the localisation functor to give simplified proofs that the left (resp. right) homotopy Kan extension of diagrams of simplicial sets presents the $\\infty$-categorical left (resp. right) Kan extension of coherent diagrams of spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-29T11:04:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simplicial sets",
      "categorical localisation functor",
      "shaped homotopy coherent diagrams",
      "homotopy kan extension",
      "objectwise weak homotopy equivalences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}