{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *concrete category* is a pair $(\\mathcal{C}, U)$ where $\\mathcal{C}$ is a [[category|A2DE66937D]] and $U \\colon \\mathcal{C} \\to \\mathbf{Set}$ is a [[faithful|GRG8WEBSH5]] [[functor|28NQ0KRHD3]] to the category of sets and functions.\nThe functor $U$ is the *forgetful functor*, and $U(A)$ is the *underlying set* of $A$.\nA category $\\mathcal{C}$ is *concretizable* if such a faithful functor exists.\n\n## Remarks\n\nExample: the category of [[groups|G5S1M9]] with the functor sending a group to its underlying set is concrete.\nTwo [[homomorphisms|0JYTG5K9Y4]] $G \\to H$ with the same underlying function are equal, so $U$ is faithful.\nNon-example: the pair (groups, $P$), where $P$ sends every group to a one-point set, is not a concrete category.\nThe homomorphisms $x \\mapsto 0$ and $x \\mapsto x$ of $\\Z$ are different but have the same image under $P$, so $P$ is not faithful.\nEvery small category $\\mathcal{C}$ is concretizable.\nSend $A$ to the set of all morphisms with codomain $A$, and $f \\colon A \\to B$ to the map $g \\mapsto f \\circ g$.\nThis functor is faithful, since $f$ is the image of $1_A$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "FYBRP16KAN",
    "kind": "definition",
    "msc": [
      "18A05",
      "18A22"
    ],
    "name": "Concrete category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Concrete_category"
      }
    ],
    "tag": "ECDKZ8G998",
    "uses": [
      "A2DE66937D",
      "28NQ0KRHD3",
      "GRG8WEBSH5",
      "G5S1M9",
      "0JYTG5K9Y4"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ECDKZ8G998/",
  "links": [
    {
      "field": "defines",
      "target": "FYBRP16KAN"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    },
    {
      "field": "uses",
      "target": "GRG8WEBSH5"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0JYTG5K9Y4"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ECDKZ8G998.md",
  "msc": [
    "18A05",
    "18A22"
  ],
  "name": "Concrete category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ECD/ECDKZ8G998.md",
  "status": null,
  "tag": "ECDKZ8G998",
  "uri": "m2://ECDKZ8G998",
  "url": "https://tamnd.github.io/math2/wiki/concrete-category-definition/",
  "writing": {
    "tag": "ECDKZ8G998",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}