{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be [[categories|A2DE66937D]], and let $S \\colon \\mathcal{C}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]] on the [[product|AFNE9C4NH6]] of the [[opposite category|DAMK8XE1FR]] with $\\mathcal{C}$.\n\nA *wedge* to $S$ is an object $w$ of $\\mathcal{D}$ with morphisms $e_c \\colon w \\to S(c, c)$ for all objects $c$.\nThey must satisfy $S(c, f) \\circ e_c = S(f, c') \\circ e_{c'}$ for every morphism $f \\colon c \\to c'$ of $\\mathcal{C}$.\nAn *end* of $S$ is a wedge $(E, e)$ such that every wedge $(w, e')$ factors as $e'_c = e_c \\circ u$ for a unique $u \\colon w \\to E$.\n\nA *cowedge* from $S$ is an object $w$ with morphisms $\\omega_c \\colon S(c, c) \\to w$.\nThey must satisfy $\\omega_c \\circ S(f, c) = \\omega_{c'} \\circ S(c', f)$ for every $f \\colon c \\to c'$.\nA *coend* of $S$ is a cowedge $(E, \\omega)$ such that every cowedge $(w, \\omega')$ factors as $\\omega'_c = u \\circ \\omega_c$ for a unique $u \\colon E \\to w$.\n\n## Remarks\n\nThe end is written $\\int_{c} S(c, c)$ and the coend $\\int^{c} S(c, c)$.\nBoth are unique up to a unique isomorphism.\nFor $\\mathcal{C}$ small and $\\mathcal{D}$ complete, the end is the equalizer of two maps $\\prod_c S(c, c) \\rightrightarrows \\prod_{f \\colon c \\to c'} S(c, c')$.\n\nExample: let $F, G \\colon \\mathcal{C} \\to \\mathcal{E}$ be functors with $\\mathcal{C}$ small and $\\mathcal{E}$ locally small, and let $S(c, c') = \\Hom_{\\mathcal{E}}(F c, G c')$ in $\\mathbf{Set}$.\nA wedge from a one-point set is a family $\\alpha_c \\colon F c \\to G c$ with $G f \\circ \\alpha_c = \\alpha_{c'} \\circ F f$, that is, a natural transformation.\nSo $\\int_c \\Hom(F c, G c)$ is the set of natural transformations $F \\Rightarrow G$.\n\nNon-example: in the same setting, the set $\\prod_c \\Hom(F c, G c)$ of all families, with its projections, is in general not a wedge.\nThe wedge condition would require every family $(\\alpha_c)$ to be natural.\nIt fails as soon as some family has $G f \\circ \\alpha_c \\neq \\alpha_{c'} \\circ F f$ for some $f \\colon c \\to c'$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "W9QTFV0Z8D",
    "kind": "definition",
    "msc": [
      "18A30",
      "18A23"
    ],
    "name": "End and coend",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/End_(category_theory)"
      }
    ],
    "tag": "JKXTT55WCV",
    "uses": [
      "A2DE66937D",
      "DAMK8XE1FR",
      "AFNE9C4NH6",
      "28NQ0KRHD3"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/JKXTT55WCV/",
  "links": [
    {
      "field": "defines",
      "target": "W9QTFV0Z8D"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "DAMK8XE1FR"
    },
    {
      "field": "uses",
      "target": "AFNE9C4NH6"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/JKXTT55WCV.md",
  "msc": [
    "18A30",
    "18A23"
  ],
  "name": "End and coend",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JKX/JKXTT55WCV.md",
  "status": null,
  "tag": "JKXTT55WCV",
  "uri": "m2://JKXTT55WCV",
  "url": "https://tamnd.github.io/math2/wiki/end-and-coend-definition/",
  "writing": {
    "tag": "JKXTT55WCV",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}