{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "EC6C1MR0C1"
    }
  ],
  "body": "\n## Summary\n\nAn extensive category is a category with finite [[coproducts|4FJ0S71P58]] that behave like disjoint unions of sets.\nCoproducts are disjoint and stable under pullback.\nSo an object over $A + B$ splits uniquely into a part over $A$ and a part over $B$.\nSets, topological spaces, small categories and every topos are extensive, while categories of modules are not.\nExtensivity is the categorical form of the idea that a space can be cut into disjoint pieces.\n\n## Explanations\n\n### L1 human\n\nIf a set $X$ maps into a disjoint union $A \\sqcup B$, then $X$ breaks into the part that lands in $A$ and the part that lands in $B$.\nThese two parts do not overlap, and their union is $X$.\nAn extensive category is one where every object over a sum breaks up in this way.\nIn vector spaces this fails: the diagonal line in $\\R \\oplus \\R$ meets both axes only at zero, yet it is not the sum of those two meetings.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18B50"
    ],
    "name": "Extensive category",
    "same_as": {
      "wikidata": "Q55611299",
      "wikipedia": "Extensive category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Extensive_category"
      }
    ],
    "tag": "5H16H7FY5D"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/5H16H7FY5D/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/5H16H7FY5D.md",
  "msc": [
    "18B50"
  ],
  "name": "Extensive category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5H1/5H16H7FY5D.md",
  "status": null,
  "tag": "5H16H7FY5D",
  "uri": "m2://5H16H7FY5D",
  "url": "https://tamnd.github.io/math2/wiki/extensive-category/",
  "writing": {
    "tag": "5H16H7FY5D",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}