{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "57BXX32WF0"
    },
    {
      "field": "uses",
      "target": "DTNDR45FTH"
    },
    {
      "field": "uses",
      "target": "JZTCEHFVS2"
    },
    {
      "field": "uses",
      "target": "WMNEQ6A8FB"
    },
    {
      "field": "specializes",
      "target": "FWGPKXCPB7"
    },
    {
      "field": "defines",
      "target": "RNCFXWTPFK"
    }
  ],
  "body": "\n## Summary\n\nA finitely complete category is a [[category|A2DE66937D]] in which every finite diagram has a [[limit|CHS5AFJGWE]].\nEquivalently, it has a terminal object and all [[pullbacks|Q05K04QJ71]].\nFinite limits are enough for much of category theory, such as internal groups, regular categories and elementary toposes.\nEvery [[complete category|FWGPKXCPB7]] is finitely complete.\n\n## Explanations\n\n### L1 human\n\nA finite diagram is a picture with finitely many objects and arrows.\nA category is finitely complete when every such picture has a limit, such as a product of two objects or a solution set of an equation.\nThe category of finite sets is finitely complete, since finite products and subsets of finite sets are finite.\nIt is not complete, because infinite products of finite sets are not finite.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A35"
    ],
    "name": "Finitely complete category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complete_category"
      }
    ],
    "tag": "PYSBNWCHCF"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PYSBNWCHCF/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/PYSBNWCHCF.md",
  "msc": [
    "18A35"
  ],
  "name": "Finitely complete category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PYS/PYSBNWCHCF.md",
  "status": null,
  "tag": "PYSBNWCHCF",
  "uri": "m2://PYSBNWCHCF",
  "url": "https://tamnd.github.io/math2/wiki/finitely-complete-category/",
  "writing": {
    "tag": "PYSBNWCHCF",
    "kind": "concept",
    "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
      }
    ]
  }
}