{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $X$ be an object of $\\mathcal{C}$.\nFor [[monomorphisms|Y2NP1JFR86]] $m \\colon A \\to X$ and $m' \\colon A' \\to X$, write $m \\le m'$ if there is a [[morphism|859F0TFVA3]] $k \\colon A \\to A'$ with $m = m' \\circ k$, and write $m \\equiv m'$ if $m \\le m'$ and $m' \\le m$.\nA *subobject* of $X$ is an equivalence class of monomorphisms into $X$ for the [[equivalence relation|0G8VEZ3WTK]] $\\equiv$.\n\n## Remarks\n\nSince $m'$ is a monomorphism, the factor $k$ in $m = m' \\circ k$ is unique when it exists.\nIf $m \\equiv m'$, the two factors are inverse isomorphisms, so $m \\equiv m'$ holds exactly when $m = m' \\circ k$ for an isomorphism $k$.\nThe relation $\\le$ induces a partial order on the subobjects of $X$.\n\nExample: in the category of sets, the monomorphisms are the injective functions.\nAn injective $m \\colon A \\to X$ is equivalent to the inclusion of its image $m(A) \\subseteq X$, since $m$ is a bijection onto $m(A)$.\nSo the subobjects of $X$ correspond to the [[subsets|0BJR9QR08E]] of $X$.\n\nNon-example: the map $\\{a, b\\} \\to \\{1\\}$ that sends both points to $1$ does not represent a subobject of $\\{1\\}$, since it is not injective and hence not a monomorphism of sets.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "FS1FKAE89H",
    "kind": "definition",
    "msc": [
      "18A20",
      "18A32"
    ],
    "name": "Subobject",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subobject"
      }
    ],
    "tag": "G0MYA3D9ZZ",
    "uses": [
      "A2DE66937D",
      "859F0TFVA3",
      "Y2NP1JFR86",
      "0G8VEZ3WTK",
      "0BJR9QR08E"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/G0MYA3D9ZZ/",
  "links": [
    {
      "field": "defines",
      "target": "FS1FKAE89H"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "859F0TFVA3"
    },
    {
      "field": "uses",
      "target": "Y2NP1JFR86"
    },
    {
      "field": "uses",
      "target": "0G8VEZ3WTK"
    },
    {
      "field": "uses",
      "target": "0BJR9QR08E"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/G0MYA3D9ZZ.md",
  "msc": [
    "18A20",
    "18A32"
  ],
  "name": "Subobject",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G0M/G0MYA3D9ZZ.md",
  "status": null,
  "tag": "G0MYA3D9ZZ",
  "uri": "m2://G0MYA3D9ZZ",
  "url": "https://tamnd.github.io/math2/wiki/subobject-definition/",
  "writing": {
    "tag": "G0MYA3D9ZZ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}