{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *monoidal category* is a [[category|A2DE66937D]] $\\mathcal{C}$ together with the following data:\n\n1. a [[functor|28NQ0KRHD3]] $\\otimes \\colon \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}$ on the [[product category|AFNE9C4NH6]], called the tensor product;\n2. an object $I$, called the unit object;\n3. a natural isomorphism $\\alpha$ with components $\\alpha_{A,B,C} \\colon (A \\otimes B) \\otimes C \\to A \\otimes (B \\otimes C)$, called the associator;\n4. natural isomorphisms $\\lambda$ and $\\rho$ with components $\\lambda_A \\colon I \\otimes A \\to A$ and $\\rho_A \\colon A \\otimes I \\to A$, called the left and right unitors.\n\nHere natural means natural as a [[natural transformation|KYCPCC57FT]] in each object variable.\nThese data must satisfy two axioms for all objects $A, B, C, D$:\n\n1. pentagon: $(\\id_A \\otimes \\alpha_{B,C,D}) \\circ \\alpha_{A, B \\otimes C, D} \\circ (\\alpha_{A,B,C} \\otimes \\id_D) = \\alpha_{A,B,C \\otimes D} \\circ \\alpha_{A \\otimes B, C, D}$ as morphisms $((A \\otimes B) \\otimes C) \\otimes D \\to A \\otimes (B \\otimes (C \\otimes D))$;\n2. triangle: $(\\id_A \\otimes \\lambda_B) \\circ \\alpha_{A,I,B} = \\rho_A \\otimes \\id_B$ as morphisms $(A \\otimes I) \\otimes B \\to A \\otimes B$.\n\n## Remarks\n\nExample: $\\mathbf{Set}$ with the Cartesian product $A \\times B$, a one-point set $I = \\set{*}$, $\\alpha((a,b),c) = (a,(b,c))$, $\\lambda(*,a) = a$ and $\\rho(a,*) = a$.\nBoth sides of the pentagon send $(((a,b),c),d)$ to $(a,(b,(c,d)))$, and both sides of the triangle send $((a,*),b)$ to $(a,b)$.\nNon-example: on the category of $\\R$-vector spaces with $\\otimes$ the usual tensor product, take $\\alpha$ to be $-1$ times the usual associator and the usual unitors.\nFor nonzero spaces the pentagon then fails: its left side contains three copies of $\\alpha$ and its right side two, so the two sides differ by the sign $-1$.\nA monoidal category is *strict* when $\\alpha$, $\\lambda$ and $\\rho$ are identities.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "M3MSQ3NKAE",
    "kind": "definition",
    "msc": [
      "18M05"
    ],
    "name": "Monoidal category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Monoidal_category"
      }
    ],
    "tag": "5R76HPDB8W",
    "uses": [
      "A2DE66937D",
      "28NQ0KRHD3",
      "AFNE9C4NH6",
      "KYCPCC57FT"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/5R76HPDB8W/",
  "links": [
    {
      "field": "defines",
      "target": "M3MSQ3NKAE"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    },
    {
      "field": "uses",
      "target": "AFNE9C4NH6"
    },
    {
      "field": "uses",
      "target": "KYCPCC57FT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5R76HPDB8W.md",
  "msc": [
    "18M05"
  ],
  "name": "Monoidal category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5R7/5R76HPDB8W.md",
  "status": null,
  "tag": "5R76HPDB8W",
  "uri": "m2://5R76HPDB8W",
  "url": "https://tamnd.github.io/math2/wiki/monoidal-category-definition/",
  "writing": {
    "tag": "5R76HPDB8W",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}