{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] with binary [[products|S5744QZV4N]], and let $Y$ and $Z$ be objects of $\\mathcal{C}$.\nAn *exponential object* of $Z$ by $Y$ is an object $Z^Y$ with a morphism $\\mathrm{ev} \\colon Z^Y \\times Y \\to Z$ with this [[universal property|EN4DD2FAJH]].\nFor every object $X$ and every morphism $g \\colon X \\times Y \\to Z$, there is a unique morphism $\\lambda g \\colon X \\to Z^Y$ with $\\mathrm{ev} \\circ (\\lambda g \\times \\id_Y) = g$.\n\n## Remarks\n\nThe morphism $\\mathrm{ev}$ is the *evaluation*, and $\\lambda g$ is the *transpose* or *curried form* of $g$.\nAn exponential object is unique up to a unique isomorphism compatible with $\\mathrm{ev}$.\nIf $Z^Y$ exists for all $Z$, then $Z \\mapsto Z^Y$ is a functor right adjoint to $- \\times Y$.\n\nExample: in the category of sets, let $Z^Y$ be the set of maps $Y \\to Z$ and $\\mathrm{ev}(h, y) = h(y)$.\nFor $g \\colon X \\times Y \\to Z$, the map $\\lambda g$ sends $x$ to $y \\mapsto g(x, y)$.\nIt is the only map with $\\lambda g(x)(y) = g(x, y)$ for all $x, y$.\n\nNon-example: in the category of topological spaces, not every exponential exists.\nFor the space $\\Q$ of rational numbers, the functor $- \\times \\Q$ has no right adjoint, so some $Z^{\\Q}$ do not exist.\nFor $Y$ locally compact Hausdorff, $Z^Y$ exists and is the space of maps with the [[compact-open topology|RG15F0TWPW]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "RHWD8DY41R",
    "kind": "definition",
    "msc": [
      "18D15"
    ],
    "name": "Exponential object",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Exponential_object"
      }
    ],
    "tag": "GD9JMCZHNY",
    "uses": [
      "A2DE66937D",
      "S5744QZV4N",
      "EN4DD2FAJH"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/GD9JMCZHNY/",
  "links": [
    {
      "field": "defines",
      "target": "RHWD8DY41R"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "S5744QZV4N"
    },
    {
      "field": "uses",
      "target": "EN4DD2FAJH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GD9JMCZHNY.md",
  "msc": [
    "18D15"
  ],
  "name": "Exponential object",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GD9/GD9JMCZHNY.md",
  "status": null,
  "tag": "GD9JMCZHNY",
  "uri": "m2://GD9JMCZHNY",
  "url": "https://tamnd.github.io/math2/wiki/exponential-object-definition/",
  "writing": {
    "tag": "GD9JMCZHNY",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}