{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *subcategory* $\\mathcal{D}$ of $\\mathcal{C}$ consists of a subclass $\\mathrm{Ob}(\\mathcal{D}) \\subseteq \\mathrm{Ob}(\\mathcal{C})$ and, for all $A, B \\in \\mathrm{Ob}(\\mathcal{D})$, a subclass $\\Hom_{\\mathcal{D}}(A, B) \\subseteq \\Hom_{\\mathcal{C}}(A, B)$ of [[morphisms|859F0TFVA3]], such that:\n\n1. $1_A \\in \\Hom_{\\mathcal{D}}(A, A)$ for every $A \\in \\mathrm{Ob}(\\mathcal{D})$;\n2. $g \\circ f \\in \\Hom_{\\mathcal{D}}(A, C)$ for all $f \\in \\Hom_{\\mathcal{D}}(A, B)$ and $g \\in \\Hom_{\\mathcal{D}}(B, C)$.\n\n## Remarks\n\nWith the composition and identities of $\\mathcal{C}$, a subcategory is a category.\nIt is *wide* if $\\mathrm{Ob}(\\mathcal{D}) = \\mathrm{Ob}(\\mathcal{C})$, and *full* if $\\Hom_{\\mathcal{D}}(A, B) = \\Hom_{\\mathcal{C}}(A, B)$ for all its objects $A$, $B$.\nExample: inside the category of [[groups|G5S1M9]], the [[abelian groups|082GHNPJ53]] with all homomorphisms between them form a subcategory.\nThe identity of an abelian group is a homomorphism, and a composite of homomorphisms is a homomorphism.\nNon-example: in the category of sets, keep all sets but only the functions whose image has at most one point.\nComposites of such functions are again of this kind, so clause 2 holds, but the identity of $\\set{0, 1}$ is missing, so clause 1 fails.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "D8C0EYD9ED",
    "kind": "definition",
    "msc": [
      "18A05"
    ],
    "name": "Subcategory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subcategory"
      }
    ],
    "tag": "AT8K76KJ5R",
    "uses": [
      "A2DE66937D",
      "859F0TFVA3",
      "082GHNPJ53",
      "G5S1M9"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/AT8K76KJ5R/",
  "links": [
    {
      "field": "defines",
      "target": "D8C0EYD9ED"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "859F0TFVA3"
    },
    {
      "field": "uses",
      "target": "082GHNPJ53"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AT8K76KJ5R.md",
  "msc": [
    "18A05"
  ],
  "name": "Subcategory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AT8/AT8K76KJ5R.md",
  "status": null,
  "tag": "AT8K76KJ5R",
  "uri": "m2://AT8K76KJ5R",
  "url": "https://tamnd.github.io/math2/wiki/subcategory-definition/",
  "writing": {
    "tag": "AT8K76KJ5R",
    "kind": "definition",
    "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
      }
    ]
  }
}