{
  "id": "1510.03150",
  "version": "v1",
  "published": "2015-10-12T06:06:34.000Z",
  "updated": "2015-10-12T06:06:34.000Z",
  "title": "The universality of the Rezk nerve",
  "authors": [
    "Aaron Mazel-Gee"
  ],
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "We functorially associate to each relative $\\infty$-category $(R,W)$ a simplicial space $N^R_\\infty(R,W)$, called its Rezk nerve (a straightforward generalization of Rezk's \"classification diagram\" construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve $N^R_\\infty(R,W)$ is precisely the one corresponding to the localization $R[[W^{-1}]]$; and (ii) that the Rezk nerve functor defines an equivalence $RelCat_\\infty [[ W_{BK}^{-1} ]] \\xrightarrow{\\sim} Cat_\\infty$ from a localization of the $\\infty$-category of relative $\\infty$-categories to the $\\infty$-category of $\\infty$-categories.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-12T06:06:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universality",
      "rezk nerve functor defines",
      "complete segal space",
      "global universal properties",
      "localization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003150M"
    }
  }
}