{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]], let $\\mathbf{Cat}$ be the category of small categories, and let $F \\colon \\mathcal{C} \\to \\mathbf{Cat}$ be a [[functor|28NQ0KRHD3]].\nThe *Grothendieck construction* of $F$ is the category $\\int F$ given as follows.\n\n1. An object is a pair $(c, x)$ with $c$ an object of $\\mathcal{C}$ and $x$ an object of $F(c)$.\n2. A morphism $(c, x) \\to (c', x')$ is a pair $(f, \\varphi)$ with $f \\colon c \\to c'$ in $\\mathcal{C}$ and $\\varphi \\colon F(f)(x) \\to x'$ in $F(c')$.\n3. The composite of $(f, \\varphi) \\colon (c, x) \\to (c', x')$ and $(g, \\psi) \\colon (c', x') \\to (c'', x'')$ is $(g \\circ f, \\ \\psi \\circ F(g)(\\varphi))$.\n4. The identity of $(c, x)$ is $(\\id_c, \\id_x)$.\n\nThe projection functor $p \\colon \\int F \\to \\mathcal{C}$ sends $(c, x)$ to $c$ and $(f, \\varphi)$ to $f$.\n\n## Remarks\n\nExample: if $F$ is the constant functor with value a small category $\\mathcal{D}$, then $F(f) = \\id_{\\mathcal{D}}$ for every $f$.\nA morphism $(c, x) \\to (c', x')$ is then a pair $(f, \\varphi)$ with $\\varphi \\colon x \\to x'$ in $\\mathcal{D}$, so $\\int F$ is the [[product category|AFNE9C4NH6]] $\\mathcal{C} \\times \\mathcal{D}$.\n\nNon-example: the disjoint union of the categories $F(c)$, with no morphisms between different fibres, is not $\\int F$ when $\\mathcal{C}$ has a non-identity morphism $f \\colon c \\to c'$ and $F(c)$ is non-empty.\nClause 2 gives the morphism $(f, \\id_{F(f)(x)}) \\colon (c, x) \\to (c', F(f)(x))$ between two fibres.\n\nWhen each $F(c)$ is a set seen as a category with only identity morphisms, $\\int F$ is the category of elements of $F$.\nFor a functor $F \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Cat}$ the dual construction takes morphisms $(f, \\varphi)$ with $f \\colon c \\to c'$ and $\\varphi \\colon x \\to F(f)(x')$ in $F(c)$; its projection is a fibration.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "272T16NWZ9",
    "kind": "definition",
    "msc": [
      "18D30",
      "18A25"
    ],
    "name": "Grothendieck construction",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Category_of_elements"
      }
    ],
    "tag": "9XYYWS5AFJ",
    "uses": [
      "A2DE66937D",
      "28NQ0KRHD3",
      "AFNE9C4NH6"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/9XYYWS5AFJ/",
  "links": [
    {
      "field": "defines",
      "target": "272T16NWZ9"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    },
    {
      "field": "uses",
      "target": "AFNE9C4NH6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9XYYWS5AFJ.md",
  "msc": [
    "18D30",
    "18A25"
  ],
  "name": "Grothendieck construction",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9XY/9XYYWS5AFJ.md",
  "status": null,
  "tag": "9XYYWS5AFJ",
  "uri": "m2://9XYYWS5AFJ",
  "url": "https://tamnd.github.io/math2/wiki/grothendieck-construction-definition/",
  "writing": {
    "tag": "9XYYWS5AFJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}