{
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  "body": "\n## Summary\n\nA subcategory of a [[category|A2DE66937D]] $\\mathcal{C}$ is a choice of some objects and some [[morphisms|859F0TFVA3]] of $\\mathcal{C}$.\nThe choice must be closed under composition and contain the identities of its objects.\nIt is a category in its own right, and its inclusion into $\\mathcal{C}$ is a [[faithful functor|GRG8WEBSH5]].\nImportant special cases are the [[full subcategories|V3BFGKBJF7]], which keep all morphisms between the chosen objects, and the wide subcategories, which keep all objects.\nThe [[category of abelian groups|082GHNPJ53]] inside the category of groups is a standard example.\n\n## Explanations\n\n### L1 human\n\nStart from a category and keep only part of it: some of the objects, and some of the arrows between them.\nTo still have a category, the identity arrow of each kept object must be kept, and a chain of kept arrows must compose to a kept arrow.\nFor example, among all sets and functions, keep all sets but only the injective functions.\nIdentities are injective and composites of injective functions are injective, so this is a subcategory.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A05"
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    "name": "Subcategory",
    "same_as": {
      "wikidata": "Q541563",
      "wikipedia": "Subcategory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subcategory"
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    ],
    "specializes": [
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    "tag": "D8C0EYD9ED"
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  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/D8C0EYD9ED/",
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  "markdown": "https://tamnd.github.io/math2/node/D8C0EYD9ED.md",
  "msc": [
    "18A05"
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  "name": "Subcategory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D8C/D8C0EYD9ED.md",
  "status": null,
  "tag": "D8C0EYD9ED",
  "uri": "m2://D8C0EYD9ED",
  "url": "https://tamnd.github.io/math2/wiki/subcategory/",
  "writing": {
    "tag": "D8C0EYD9ED",
    "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
      }
    ]
  }
}