{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2W2X7EBG2S"
    },
    {
      "field": "uses",
      "target": "QGPAB6AS9F"
    },
    {
      "field": "uses",
      "target": "QNMWK3GCF2"
    },
    {
      "field": "defines",
      "target": "QGG9GJN73C"
    }
  ],
  "body": "\n## Summary\n\nA monoidal functor is a [[functor|28NQ0KRHD3]] between [[monoidal categories|M3MSQ3NKAE]] that is compatible with the tensor products and units.\nIt comes with maps $F(A) \\otimes F(B) \\to F(A \\otimes B)$ and $I \\to F(I)$ that respect the associators and unitors.\nWhen these maps are isomorphisms the functor is called strong, and when they are identities it is called strict.\nMonoidal functors send monoid objects to monoid objects, and they are the morphisms of the category of monoidal categories.\n\n## Explanations\n\n### L1 human\n\nA monoidal functor translates one \"side-by-side\" world into another.\nIt gives a way to combine the images of $A$ and $B$ into the image of $A \\otimes B$, in a way that ignores how the brackets are placed.\nFor example, the free vector space functor sends a product of sets $X \\times Y$ to a space isomorphic to $k[X] \\otimes k[Y]$.\nThe forgetful functor from abelian groups to sets is monoidal in the weaker sense: there is a map $(a, b) \\mapsto a \\otimes b$ from $A \\times B$ to $A \\otimes B$, but it is not a bijection.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18M05",
      "18A22"
    ],
    "name": "Monoidal functor",
    "same_as": {
      "wikidata": "Q6901642",
      "wikipedia": "Monoidal functor"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Monoidal_functor"
      }
    ],
    "specializes": [
      "28NQ0KRHD3"
    ],
    "tag": "PY430TXMG0"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PY430TXMG0/",
  "links": [
    {
      "field": "specializes",
      "target": "28NQ0KRHD3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PY430TXMG0.md",
  "msc": [
    "18M05",
    "18A22"
  ],
  "name": "Monoidal functor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PY4/PY430TXMG0.md",
  "status": null,
  "tag": "PY430TXMG0",
  "uri": "m2://PY430TXMG0",
  "url": "https://tamnd.github.io/math2/wiki/monoidal-functor/",
  "writing": {
    "tag": "PY430TXMG0",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}