{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "FN6P3CTPTS"
    }
  ],
  "body": "\n## Summary\n\nThe free category on a [[directed graph|0KFYGSHHF5]] has the vertices of the graph as objects and the finite paths of edges as morphisms.\nPaths are composed by joining them end to end.\nIt is the most general category generated by the graph: no equations between paths hold except those forced by the axioms of a category.\nIt plays for categories the role that the [[free monoid|CV013KHVJG]] plays for monoids.\n\n## Explanations\n\n### L1 human\n\nDraw some dots and arrows between them.\nA morphism of the free category is a route you can travel by following arrows one after another, and composing two routes means doing one and then the other.\nDoing nothing at a dot is the identity route.\nIf the graph has one dot and one loop $a$, the routes are $a^0, a^1, a^2, \\dots$, and composing them adds the exponents, so the free category is the monoid $(\\N, +)$.\nIf the graph is $x \\to y \\to z$, there are exactly six routes: three identities, the two edges, and the route from $x$ to $z$ through $y$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A10",
      "18A40"
    ],
    "name": "Free category",
    "same_as": {
      "wikidata": "Q5500202",
      "wikipedia": "Free category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_category"
      }
    ],
    "specializes": [
      "A2DE66937D"
    ],
    "tag": "THRHFW3SMP"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/THRHFW3SMP/",
  "links": [
    {
      "field": "specializes",
      "target": "A2DE66937D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/THRHFW3SMP.md",
  "msc": [
    "18A10",
    "18A40"
  ],
  "name": "Free category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/THR/THRHFW3SMP.md",
  "status": null,
  "tag": "THRHFW3SMP",
  "uri": "m2://THRHFW3SMP",
  "url": "https://tamnd.github.io/math2/wiki/free-category/",
  "writing": {
    "tag": "THRHFW3SMP",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}