{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1MJW4YPQW6"
    },
    {
      "field": "uses",
      "target": "1S9JGHT5HJ"
    },
    {
      "field": "uses",
      "target": "3GZ99NCX22"
    },
    {
      "field": "uses",
      "target": "6EWEMZ6FX1"
    },
    {
      "field": "uses",
      "target": "AYQ3VQ4B0V"
    },
    {
      "field": "uses",
      "target": "JQAB8D3FFN"
    },
    {
      "field": "uses",
      "target": "XYZWTQD6NN"
    },
    {
      "field": "specializes",
      "target": "CSD6W0JV0P"
    },
    {
      "field": "defines",
      "target": "Y1RAX7VVBX"
    }
  ],
  "body": "\n## Summary\n\nA full subcategory of a [[category|A2DE66937D]] $\\mathcal{C}$ is a [[subcategory|D8C0EYD9ED]] that keeps every morphism of $\\mathcal{C}$ between any two of its objects.\nSo it is fixed by its class of objects alone.\nIts inclusion functor is [[fully faithful|GRG8WEBSH5]].\nMost categories of \"special\" objects, such as abelian groups among groups or Hausdorff spaces among topological spaces, are full subcategories.\n\n## Explanations\n\n### L1 human\n\nTo form a full subcategory, choose some of the objects and keep all arrows between them.\nNothing about the arrows is decided: once the objects are chosen, the arrows come along.\nFor example, the finite sets with all functions between them form a full subcategory of the category of sets.\nKeeping all sets but only the injective functions gives a subcategory that is not full.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A05"
    ],
    "name": "Full subcategory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subcategory"
      }
    ],
    "specializes": [
      "D8C0EYD9ED"
    ],
    "tag": "V3BFGKBJF7"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/V3BFGKBJF7/",
  "links": [
    {
      "field": "specializes",
      "target": "D8C0EYD9ED"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/V3BFGKBJF7.md",
  "msc": [
    "18A05"
  ],
  "name": "Full subcategory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V3B/V3BFGKBJF7.md",
  "status": null,
  "tag": "V3BFGKBJF7",
  "uri": "m2://V3BFGKBJF7",
  "url": "https://tamnd.github.io/math2/wiki/full-subcategory/",
  "writing": {
    "tag": "V3BFGKBJF7",
    "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
      }
    ]
  }
}