{
  "id": "1510.03961",
  "version": "v1",
  "published": "2015-10-14T04:16:30.000Z",
  "updated": "2015-10-14T04:16:30.000Z",
  "title": "Hammocks and fractions in relative $\\infty$-categories",
  "authors": [
    "Aaron Mazel-Gee"
  ],
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "We study the *homotopy theory* of $\\infty$-categories enriched in the $\\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\\infty$-categories as presentations of ordinary $\\infty$-categories by means of a \"local\" geometric realization functor $Cat_{sS} \\to Cat_\\infty$, and we prove that their homotopy theory presents the $\\infty$-category of $\\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \\xrightarrow{\\sim} Cat_\\infty$ from a localization of the $\\infty$-category of $sS$-enriched $\\infty$-categories. Following Dwyer--Kan, we define a *hammock localization* functor from relative $\\infty$-categories to $sS$-enriched $\\infty$-categories, thus providing a rich source of examples of $sS$-enriched $\\infty$-categories. Simultaneously unpacking and generalizing one of their key results, we prove that given a relative $\\infty$-category admitting a *homotopical three-arrow calculus*, one can explicitly describe the hom-spaces in the $\\infty$-category presented by its hammock localization in a much more explicit and accessible way. As an application of this framework, we give sufficient conditions for the Rezk nerve of a relative $\\infty$-category to be a (complete) Segal space, generalizing joint work with Low.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-14T04:16:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric realization functor",
      "simplicial spaces",
      "homotopy theory",
      "functor induces",
      "hammock localization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003961M"
    }
  }
}