{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\n$\\mathcal{C}$ is *complete* if every [[diagram|HJK75HDWFE]] $F \\colon J \\to \\mathcal{C}$ with $J$ a small category has a [[limit|CHS5AFJGWE]] in $\\mathcal{C}$.\nHere a category $J$ is small when its objects and its morphisms form sets.\n\n## Remarks\n\nDually, $\\mathcal{C}$ is *cocomplete* if every small diagram has a colimit.\nA category is complete if and only if it has all small products and all equalizers.\n\nExample: the category of sets is complete.\nThe limit of $F \\colon J \\to \\mathbf{Set}$ is the set of families $(x_j)_{j \\in J}$ with $x_j \\in F(j)$ and $F(u)(x_j) = x_k$ for every morphism $u \\colon j \\to k$ of $J$.\n\nNon-example: the category of finite sets is not complete.\nSuppose $P$ were a product of countably many copies of $\\set{0, 1}$ there.\nMaps from a one-point set to $P$ would match the elements of $P$ with all sequences in $\\set{0, 1}$, which form an uncountable set.\n\nA small category that is complete is a preorder, by a theorem of Freyd.\nSo completeness is a property of large categories.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "FWGPKXCPB7",
    "kind": "definition",
    "msc": [
      "18A35"
    ],
    "name": "Complete category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complete_category"
      }
    ],
    "tag": "FBGXAXAEJ1",
    "uses": [
      "A2DE66937D",
      "HJK75HDWFE",
      "CHS5AFJGWE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FBGXAXAEJ1/",
  "links": [
    {
      "field": "defines",
      "target": "FWGPKXCPB7"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "HJK75HDWFE"
    },
    {
      "field": "uses",
      "target": "CHS5AFJGWE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FBGXAXAEJ1.md",
  "msc": [
    "18A35"
  ],
  "name": "Complete category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FBG/FBGXAXAEJ1.md",
  "status": null,
  "tag": "FBGXAXAEJ1",
  "uri": "m2://FBGXAXAEJ1",
  "url": "https://tamnd.github.io/math2/wiki/complete-category-definition/",
  "writing": {
    "tag": "FBGXAXAEJ1",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}