{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "GD9JMCZHNY"
    },
    {
      "field": "uses",
      "target": "KJ2NZCB7TH"
    },
    {
      "field": "defines",
      "target": "VBS4A8G5XA"
    }
  ],
  "body": "\n## Summary\n\nA universal property describes an object with a morphism by a factorisation rule.\nEvery other morphism of the same type factors through it in exactly one way.\nProducts, quotients, free groups, tensor products, limits and colimits are all defined by universal properties.\nAn object with a universal property is unique up to a unique isomorphism, so the property pins it down without choosing a construction.\n\n## Explanations\n\n### L1 human\n\nInstead of saying what an object is made of, a universal property says how it relates to everything else.\nFor example, the [[free group|0KTW26C80J]] on a set $S$ is a group that contains $S$.\nAny function from $S$ to any group extends to exactly one group homomorphism on it.\nAny two objects with the same universal property are matched by a unique isomorphism, so it does not matter how the object was built.\nThis is why mathematicians often say \"the\" product or \"the\" free group.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A05",
      "18A40"
    ],
    "name": "Universal property",
    "same_as": {
      "wikidata": "Q1417809",
      "wikipedia": "Universal property"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Universal_property"
      }
    ],
    "tag": "EN4DD2FAJH"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/EN4DD2FAJH/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/EN4DD2FAJH.md",
  "msc": [
    "18A05",
    "18A40"
  ],
  "name": "Universal property",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EN4/EN4DD2FAJH.md",
  "status": null,
  "tag": "EN4DD2FAJH",
  "uri": "m2://EN4DD2FAJH",
  "url": "https://tamnd.github.io/math2/wiki/universal-property/",
  "writing": {
    "tag": "EN4DD2FAJH",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}