{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nFor $n \\ge 0$ let $\\Delta^n$ be the standard $n$-simplex, as a [[simplicial set|WW81ARB8K9]].\nFor $0 \\le k \\le n$ let $\\Lambda^n_k \\subset \\Delta^n$ be the $k$-th horn: the simplicial subset generated by all faces of $\\Delta^n$ except the face opposite the vertex $k$.\nThe horn is *inner* if $0 \u003c k \u003c n$.\nA *quasi-category* is a simplicial set $X$ with the following property.\nFor every $n \\ge 2$, every $0 \u003c k \u003c n$, and every map $\\sigma \\colon \\Lambda^n_k \\to X$, there is a map $\\tau \\colon \\Delta^n \\to X$ whose restriction to $\\Lambda^n_k$ is $\\sigma$.\n\n## Remarks\n\nThe vertices of $X$ are its objects and the $1$-simplices its morphisms.\nFor $n = 2$, $k = 1$, the condition says that two morphisms $f \\colon x \\to y$ and $g \\colon y \\to z$ have a composite: a $2$-simplex with edges $f$, $g$ and some $h \\colon x \\to z$.\nExample: the [[nerve|RV0MKJ3CQ6]] $N(\\mathcal{C})$ of a category $\\mathcal{C}$ is a quasi-category, and in it every inner horn has exactly one filler.\nExample: every [[Kan complex|FDFYCM3YYC]] is a quasi-category, since it has fillers for all horns, inner or not; the singular simplicial set of a topological space is one.\nNon-example: the horn $\\Lambda^2_1$ itself is not a quasi-category.\nIts identity map $\\Lambda^2_1 \\to \\Lambda^2_1$ does not extend to $\\Delta^2$, because every $2$-simplex of $\\Lambda^2_1$ is degenerate.\nA degenerate $2$-simplex cannot have the two distinct nondegenerate edges $0 \\to 1$ and $1 \\to 2$ as faces.\nQuasi-categories are also called weak Kan complexes or $\\infty$-categories.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9328WEVJEK",
    "kind": "definition",
    "msc": [
      "18N60",
      "18N50"
    ],
    "name": "Quasi-category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quasi-category"
      }
    ],
    "tag": "G87CWTXAWC",
    "uses": [
      "WW81ARB8K9",
      "RV0MKJ3CQ6",
      "FDFYCM3YYC"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/G87CWTXAWC/",
  "links": [
    {
      "field": "defines",
      "target": "9328WEVJEK"
    },
    {
      "field": "uses",
      "target": "WW81ARB8K9"
    },
    {
      "field": "uses",
      "target": "RV0MKJ3CQ6"
    },
    {
      "field": "uses",
      "target": "FDFYCM3YYC"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/G87CWTXAWC.md",
  "msc": [
    "18N60",
    "18N50"
  ],
  "name": "Quasi-category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G87/G87CWTXAWC.md",
  "status": null,
  "tag": "G87CWTXAWC",
  "uri": "m2://G87CWTXAWC",
  "url": "https://tamnd.github.io/math2/wiki/quasi-category-definition/",
  "writing": {
    "tag": "G87CWTXAWC",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}