{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "5YH6FTMVYP"
    },
    {
      "field": "uses",
      "target": "XYH30V13SE"
    }
  ],
  "body": "\n## Summary\n\nA closed monoidal category is a [[monoidal category|M3MSQ3NKAE]] in which morphisms between two objects form again an object, the internal hom.\nThe internal hom $[B, C]$ is characterised by a currying bijection between morphisms $A \\otimes B \\to C$ and morphisms $A \\to [B, C]$.\n[[Cartesian closed categories|BFBNQ36QS8]] are the case where $\\otimes$ is the product.\nVector spaces, abelian groups and modules over a commutative ring, with the tensor product, are closed monoidal.\n\n## Explanations\n\n### L1 human\n\nIn many categories, the set of maps from $B$ to $C$ carries the same kind of structure as $B$ and $C$.\nFor example, the linear maps between two vector spaces form a vector space.\nA closed monoidal category is one where this happens in a way that matches the tensor product.\nA linear map out of $A \\otimes B$ is the same thing as a linear map from $A$ into the space of linear maps from $B$ to $C$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18D15",
      "18M05"
    ],
    "name": "Closed monoidal category",
    "same_as": {
      "wikidata": "Q5135347",
      "wikipedia": "Closed monoidal category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Closed_monoidal_category"
      }
    ],
    "specializes": [
      "M3MSQ3NKAE"
    ],
    "tag": "EKJBDD07TY"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/EKJBDD07TY/",
  "links": [
    {
      "field": "specializes",
      "target": "M3MSQ3NKAE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EKJBDD07TY.md",
  "msc": [
    "18D15",
    "18M05"
  ],
  "name": "Closed monoidal category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EKJ/EKJBDD07TY.md",
  "status": null,
  "tag": "EKJBDD07TY",
  "uri": "m2://EKJBDD07TY",
  "url": "https://tamnd.github.io/math2/wiki/closed-monoidal-category/",
  "writing": {
    "tag": "EKJBDD07TY",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}