{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "231CSHFC5K"
    },
    {
      "field": "uses",
      "target": "26RWFX4H2H"
    },
    {
      "field": "uses",
      "target": "2V8FATXF50"
    },
    {
      "field": "uses",
      "target": "3GZ99NCX22"
    },
    {
      "field": "uses",
      "target": "494YXJH790"
    },
    {
      "field": "uses",
      "target": "5R76HPDB8W"
    },
    {
      "field": "uses",
      "target": "5S1R5AVKZ6"
    },
    {
      "field": "uses",
      "target": "5T6HCEK1M8"
    },
    {
      "field": "uses",
      "target": "779ZSEF36C"
    },
    {
      "field": "uses",
      "target": "88XF0NVBVG"
    },
    {
      "field": "uses",
      "target": "8QTP2EFA92"
    },
    {
      "field": "uses",
      "target": "8TPR96GV7F"
    },
    {
      "field": "uses",
      "target": "8V296XR2M3"
    },
    {
      "field": "uses",
      "target": "98F5NBDR0V"
    },
    {
      "field": "uses",
      "target": "9P83YFRWDJ"
    },
    {
      "field": "uses",
      "target": "9V800QGBQH"
    },
    {
      "field": "uses",
      "target": "9XYYWS5AFJ"
    },
    {
      "field": "uses",
      "target": "A4SS2SN0MB"
    },
    {
      "field": "uses",
      "target": "AYQ3VQ4B0V"
    },
    {
      "field": "uses",
      "target": "C3JHS6W73G"
    },
    {
      "field": "uses",
      "target": "DGZ8RS5KMZ"
    },
    {
      "field": "uses",
      "target": "DS6TC7FSDQ"
    },
    {
      "field": "uses",
      "target": "DV3PWY8NVW"
    },
    {
      "field": "uses",
      "target": "ECDKZ8G998"
    },
    {
      "field": "uses",
      "target": "EDP5426NNF"
    },
    {
      "field": "uses",
      "target": "F426Q5R0RY"
    },
    {
      "field": "uses",
      "target": "F9RQZ6EPS7"
    },
    {
      "field": "uses",
      "target": "FN6P3CTPTS"
    },
    {
      "field": "uses",
      "target": "G3TZAZ3XDR"
    },
    {
      "field": "uses",
      "target": "H4NTJ12J88"
    },
    {
      "field": "uses",
      "target": "H5QDJACKMS"
    },
    {
      "field": "uses",
      "target": "HSQJSK4E23"
    },
    {
      "field": "uses",
      "target": "JBVKKDXETM"
    },
    {
      "field": "uses",
      "target": "JKXTT55WCV"
    },
    {
      "field": "uses",
      "target": "KSN3AJ6EJD"
    },
    {
      "field": "uses",
      "target": "KVX3R8KC84"
    },
    {
      "field": "uses",
      "target": "NS98MJFMYR"
    },
    {
      "field": "uses",
      "target": "Q0KB9QDH8V"
    },
    {
      "field": "uses",
      "target": "QD9H1WHHSW"
    },
    {
      "field": "uses",
      "target": "QGG9GJN73C"
    },
    {
      "field": "uses",
      "target": "QNMWK3GCF2"
    },
    {
      "field": "uses",
      "target": "S47R8TG0T3"
    },
    {
      "field": "uses",
      "target": "S9FVESJ0CS"
    },
    {
      "field": "uses",
      "target": "SM2KFBGJTH"
    },
    {
      "field": "uses",
      "target": "TCWV185JKY"
    },
    {
      "field": "uses",
      "target": "TZFSE2VNAQ"
    },
    {
      "field": "uses",
      "target": "VBS4A8G5XA"
    },
    {
      "field": "uses",
      "target": "VWYXK7E49K"
    },
    {
      "field": "uses",
      "target": "WA60RW9E04"
    },
    {
      "field": "uses",
      "target": "Y77K0TEKN7"
    },
    {
      "field": "uses",
      "target": "ZE780BQS42"
    },
    {
      "field": "uses",
      "target": "ZFT23VCN7Z"
    },
    {
      "field": "uses",
      "target": "ZY6R4H65P8"
    },
    {
      "field": "specializes",
      "target": "4DKB6AQ8J7"
    },
    {
      "field": "specializes",
      "target": "AND2WHZC8P"
    },
    {
      "field": "specializes",
      "target": "HJK75HDWFE"
    },
    {
      "field": "specializes",
      "target": "KAN07YC9T2"
    },
    {
      "field": "specializes",
      "target": "KCCHH87H6E"
    },
    {
      "field": "specializes",
      "target": "N353JF8RBK"
    },
    {
      "field": "specializes",
      "target": "PY430TXMG0"
    },
    {
      "field": "specializes",
      "target": "RECR96V97B"
    },
    {
      "field": "specializes",
      "target": "VJXS8AK67E"
    },
    {
      "field": "specializes",
      "target": "WV3BPYCHVG"
    },
    {
      "field": "defines",
      "target": "KJB47D7Z05"
    },
    {
      "field": "about",
      "target": "S9FVESJ0CS"
    }
  ],
  "body": "\n## Summary\n\nA functor is a map between two [[categories|A2DE66937D]]: it sends objects to objects and [[morphisms|859F0TFVA3]] to morphisms, and it respects composition and identities.\nFunctors are how one field of mathematics is translated into another, as when the [[fundamental group|S1XK3HXHFX]] turns spaces and continuous maps into groups and homomorphisms.\nBecause functors preserve composition, they send isomorphisms to isomorphisms, which makes them a source of invariants.\nFunctors between two categories are compared by [[natural transformations|KYCPCC57FT]].\n\n## Explanations\n\n### L1 human\n\nA functor takes every object of one category to an object of another, and every arrow to an arrow between the images.\nIt must send a chain of arrows to the chain of the images, and an identity arrow to an identity arrow.\nA simple example forgets structure: send each group to its underlying set and each homomorphism to the same map seen as a function.\nA deeper one sends each space to its fundamental group.\nIf two spaces are homeomorphic, their fundamental groups are isomorphic, so different groups prove that two spaces are different.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A05",
      "18A22"
    ],
    "name": "Functor",
    "same_as": {
      "wikidata": "Q864475",
      "wikipedia": "Functor"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Functor"
      }
    ],
    "tag": "28NQ0KRHD3"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/28NQ0KRHD3/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/28NQ0KRHD3.md",
  "msc": [
    "18A05",
    "18A22"
  ],
  "name": "Functor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/28N/28NQ0KRHD3.md",
  "status": null,
  "tag": "28NQ0KRHD3",
  "uri": "m2://28NQ0KRHD3",
  "url": "https://tamnd.github.io/math2/wiki/functor/",
  "writing": {
    "tag": "28NQ0KRHD3",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}