{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PWHVYMN5T3"
    }
  ],
  "body": "\n## Summary\n\nA Grothendieck universe is a set $U$ that is closed under the usual operations of set theory.\nThese are taking elements, pairs, [[power sets|00E0YZCERH]] and unions of families indexed by elements of $U$.\nInside a universe one can do ordinary mathematics without leaving $U$.\nCategory theorists use universes to speak of the category of all \"small\" sets or groups as a set-sized object.\nUncountable Grothendieck universes are exactly the sets $V_\\kappa$ for [[inaccessible cardinals|NW4DREQ5DQ]] $\\kappa$, so their existence goes beyond ZFC.\n\n## Explanations\n\n### L1 human\n\nIn category theory one wants to talk about \"the category of all groups\", but the collection of all groups is too big to be a set.\nA universe is a large set that is closed under every construction one normally uses.\nOne then calls a set small when it lies in the universe, and the category of all small groups is a legitimate object.\nThe axiom that every set lies in some universe was used by Grothendieck in algebraic geometry.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "03E55",
      "18A15"
    ],
    "name": "Grothendieck universe",
    "same_as": {
      "wikidata": "Q1465090",
      "wikipedia": "Grothendieck universe"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Grothendieck_universe"
      }
    ],
    "tag": "SMMK9E0MTM"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/SMMK9E0MTM/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/SMMK9E0MTM.md",
  "msc": [
    "03E55",
    "18A15"
  ],
  "name": "Grothendieck universe",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SMM/SMMK9E0MTM.md",
  "status": null,
  "tag": "SMMK9E0MTM",
  "uri": "m2://SMMK9E0MTM",
  "url": "https://math2.tamnd.com/wiki/grothendieck-universe/",
  "writing": {
    "tag": "SMMK9E0MTM",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}