{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThe *category of posets* $\\mathbf{Pos}$ is the [[category|A2DE66937D]] given by the following data.\n\n1. Its objects are the [[partially ordered sets|011C29Q1S1]].\n2. A morphism from $(P, \\le_P)$ to $(Q, \\le_Q)$ is a [[monotone function|0WFF6AZPBP]] $f \\colon P \\to Q$, that is, $x \\le_P y$ implies $f(x) \\le_Q f(y)$.\n3. Composition is the usual [[composition of functions|07HF9YCWPP]], and the identity morphism of $P$ is the identity map of $P$.\n\n## Remarks\n\nThe data form a category.\nThe identity map is monotone, and if $f$ and $g$ are monotone, then $x \\le y$ gives $f(x) \\le f(y)$ and then $g(f(x)) \\le g(f(y))$.\nAssociativity and the identity laws hold because they hold for functions.\nPosets with order-reversing maps do not form a category in the same way: the composite of two order-reversing maps preserves order.\nA morphism of $\\mathbf{Pos}$ is an isomorphism exactly when it is an [[order isomorphism|0Y74KN99A5]].\nA monotone bijection need not be one: the identity from a two-element antichain to a two-element chain is a monotone bijection whose inverse is not monotone.\n\n## Explanations\n\n### L1 human\n\nThe category of posets is the world of ordered sets together with the maps that respect order.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ZBKVRMWHMW",
    "kind": "definition",
    "msc": [
      "18B35",
      "06A06"
    ],
    "name": "Category of posets",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Category_of_preordered_sets"
      }
    ],
    "tag": "CNGSMAB60A",
    "uses": [
      "A2DE66937D",
      "011C29Q1S1",
      "0WFF6AZPBP",
      "07HF9YCWPP"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CNGSMAB60A/",
  "links": [
    {
      "field": "defines",
      "target": "ZBKVRMWHMW"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "011C29Q1S1"
    },
    {
      "field": "uses",
      "target": "0WFF6AZPBP"
    },
    {
      "field": "uses",
      "target": "07HF9YCWPP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CNGSMAB60A.md",
  "msc": [
    "18B35",
    "06A06"
  ],
  "name": "Category of posets",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CNG/CNGSMAB60A.md",
  "status": null,
  "tag": "CNGSMAB60A",
  "uri": "m2://CNGSMAB60A",
  "url": "https://tamnd.github.io/math2/wiki/category-of-posets-definition/",
  "writing": {
    "tag": "CNGSMAB60A",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}