{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "290G1NGAMR"
    },
    {
      "field": "uses",
      "target": "8P9PJG95H3"
    },
    {
      "field": "uses",
      "target": "9VY3JT31RW"
    },
    {
      "field": "uses",
      "target": "BAH89SW049"
    },
    {
      "field": "uses",
      "target": "D6DF1ARPVD"
    },
    {
      "field": "uses",
      "target": "EPAZPBPFF9"
    },
    {
      "field": "uses",
      "target": "F9RQZ6EPS7"
    },
    {
      "field": "uses",
      "target": "GGE4KYWCNT"
    },
    {
      "field": "uses",
      "target": "Z4JZPEBM68"
    },
    {
      "field": "about",
      "target": "8P9PJG95H3"
    },
    {
      "field": "about",
      "target": "D6DF1ARPVD"
    },
    {
      "field": "defines",
      "target": "DV3PWY8NVW"
    },
    {
      "field": "specializes",
      "target": "RT8A8FJ566"
    },
    {
      "field": "specializes",
      "target": "WW81ARB8K9"
    }
  ],
  "body": "\n## Summary\n\nA presheaf on a [[category|A2DE66937D]] $\\mathcal{C}$ is a contravariant [[functor|28NQ0KRHD3]] from $\\mathcal{C}$ to the category of sets.\nPresheaves on $\\mathcal{C}$ and the natural transformations between them form a category that contains $\\mathcal{C}$ and has all limits and colimits.\nThe name comes from topology, where a presheaf on a space assigns data to open sets with restriction maps to smaller open sets.\n\n## Explanations\n\n### L1 human\n\nA presheaf attaches a set to each object.\nFor each arrow $X \\to Y$ it gives a way to pull elements back from the set at $Y$ to the set at $X$.\nThe model case is a [[topological space|0CQHFPGJ45]]: to each open set $U$ attach the set of continuous real functions on $U$.\nIf $V \\subseteq U$, a function on $U$ restricts to a function on $V$, so the maps go from big sets to small sets, against the inclusion.\nRestricting twice is the same as restricting once, which is the functor rule.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18F20",
      "18A25"
    ],
    "name": "Presheaf",
    "same_as": {
      "wikidata": "Q7241077",
      "wikipedia": "Presheaf (category theory)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Presheaf_(category_theory)"
      }
    ],
    "specializes": [
      "28NQ0KRHD3"
    ],
    "tag": "AND2WHZC8P"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AND2WHZC8P/",
  "links": [
    {
      "field": "specializes",
      "target": "28NQ0KRHD3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AND2WHZC8P.md",
  "msc": [
    "18F20",
    "18A25"
  ],
  "name": "Presheaf",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AND/AND2WHZC8P.md",
  "status": null,
  "tag": "AND2WHZC8P",
  "uri": "m2://AND2WHZC8P",
  "url": "https://tamnd.github.io/math2/wiki/presheaf/",
  "writing": {
    "tag": "AND2WHZC8P",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}