{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *congruence* $R$ on $\\mathcal{C}$ is a family of [[equivalence relations|0G8VEZ3WTK]] $\\sim_{X,Y}$, one on each set $\\Hom(X, Y)$ of [[morphisms|859F0TFVA3]], with this property.\nIf $f \\sim f'$ for $f, f' \\colon X \\to Y$, then $g \\circ f \\circ h \\sim g \\circ f' \\circ h$ for all $h \\colon W \\to X$ and $g \\colon Y \\to Z$.\nThe *quotient category* $\\mathcal{C}/R$ has the same objects as $\\mathcal{C}$.\nIts morphisms $X \\to Y$ are the equivalence classes $[f]$ of morphisms $f \\colon X \\to Y$, with composition $[g] \\circ [f] = [g \\circ f]$ and identities $[\\id_X]$.\n\n## Remarks\n\nThe compatibility condition makes the composition $[g] \\circ [f] = [g \\circ f]$ independent of the chosen representatives.\nThe quotient [[functor|28NQ0KRHD3]] $Q \\colon \\mathcal{C} \\to \\mathcal{C}/R$ is the identity on objects and sends $f$ to $[f]$.\nIt is universal: a functor $G \\colon \\mathcal{C} \\to \\mathcal{D}$ with $G(f) = G(f')$ whenever $f \\sim f'$ factors uniquely as $G = \\bar{G} \\circ Q$.\n\nExample: view the monoid $(\\Z, +)$ as a category with one object, whose morphisms are the integers and whose composition is addition.\nLet $m \\sim n$ when $m - n$ is even.\nIf $m \\sim m'$ then $(k + m + l) - (k + m' + l) = m - m'$ is even, so $\\sim$ is a congruence, and the quotient is the one-object category of the group $\\Z/2\\Z$.\n\nNon-example: on the same category let $m \\sim n$ when $m = n$ or $\\{m, n\\} = \\{0, 1\\}$.\nThis is an equivalence relation, but $0 \\sim 1$ while $0 + 1 = 1$ and $1 + 1 = 2$ are not related, so it is not compatible with composition and gives no quotient category.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "43MP9NWS6H",
    "kind": "definition",
    "msc": [
      "18A32",
      "18A05"
    ],
    "name": "Quotient category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quotient_category"
      }
    ],
    "tag": "231CSHFC5K",
    "uses": [
      "A2DE66937D",
      "859F0TFVA3",
      "0G8VEZ3WTK",
      "28NQ0KRHD3"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/231CSHFC5K/",
  "links": [
    {
      "field": "defines",
      "target": "43MP9NWS6H"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "859F0TFVA3"
    },
    {
      "field": "uses",
      "target": "0G8VEZ3WTK"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/231CSHFC5K.md",
  "msc": [
    "18A32",
    "18A05"
  ],
  "name": "Quotient category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/231/231CSHFC5K.md",
  "status": null,
  "tag": "231CSHFC5K",
  "uri": "m2://231CSHFC5K",
  "url": "https://tamnd.github.io/math2/wiki/quotient-category-definition/",
  "writing": {
    "tag": "231CSHFC5K",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}