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  "body": "\n## Summary\n\nA sheaf on a [[site|NYE0Z94MME]] $(\\mathcal{C}, J)$ is a [[presheaf|AND2WHZC8P]] on $\\mathcal{C}$ with a gluing property.\nCompatible local data on every covering [[sieve|CNCXNWKC6R]] glue to a unique global datum.\nIt generalizes a [[sheaf on a topological space|TB6TM6R4MD]], which is the case of the site of open sets.\nSheaves on a small site form a [[Grothendieck topos|S3YVE24AQK]], and every presheaf has a best approximation by a sheaf, its [[sheafification|2HYK3KBQNA]].\nSheaves on the étale site of a scheme give étale cohomology.\n\n## Explanations\n\n### L1 human\n\nA presheaf gives, for each piece of a space, a set of data, and a way to restrict data to smaller pieces.\nIt is a sheaf when data can be checked and built locally.\nIf pieces of data on a cover agree where they overlap, there is exactly one datum on the whole that restricts to them.\nFor example, continuous real functions on the open sets of a space form a sheaf, since continuity is a local property.\nConstant functions do not: a function equal to $0$ on one component and $1$ on another is locally constant but not constant.\n",
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    "name": "Sheaf on a site",
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        "url": "https://en.wikipedia.org/wiki/Sheaf_(mathematics)"
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  "markdown": "https://tamnd.github.io/math2/node/RT8A8FJ566.md",
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  "name": "Sheaf on a site",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/RT8/RT8A8FJ566.md",
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    "tag": "RT8A8FJ566",
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    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
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      },
      {
        "what": "a source with a locator",
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