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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, J)$ be a [[site|NYE0Z94MME]] and let $P \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Set}$ be a [[presheaf|AND2WHZC8P]].\nFor $x \\in P(c)$ and $f \\colon d \\to c$, write $x|_f = P(f)(x)$.\nLet $S$ be a [[sieve|CNCXNWKC6R]] on $c$.\nA *matching family* for $S$ is a choice of an element $x_f \\in P(d)$ for each $f \\colon d \\to c$ in $S$ such that $x_{f \\circ g} = x_f|_g$ for every morphism $g \\colon e \\to d$.\nAn *amalgamation* of the family is an element $x \\in P(c)$ with $x|_f = x_f$ for every $f \\in S$.\nThe presheaf $P$ is a *sheaf* for $J$ when the following holds.\nFor every object $c$ and every covering sieve $S \\in J(c)$, each matching family for $S$ has exactly one amalgamation.\n\n## Remarks\n\nEquivalently, $P$ is a sheaf when, for every covering sieve $S \\subseteq \\Hom(-, c)$, every natural transformation $S \\to P$ extends uniquely to $\\Hom(-, c) \\to P$.\nA morphism of sheaves is a natural transformation; sheaves on $(\\mathcal{C}, J)$ form a full subcategory $\\mathrm{Sh}(\\mathcal{C}, J)$ of presheaves.\nExample: on the site of open sets of a space $X$, the presheaf $P(U) = C(U, \\R)$ of continuous functions is a sheaf.\nFunctions on the members of a covering sieve that agree on overlaps define one function on $U$, which is continuous because continuity is local.\nNon-example: on the same site, the constant presheaf $P(U) = \\set{0, 1}$ fails, because the empty cover of $U = \\emptyset$ has a unique (empty) matching family but two amalgamations.\nFor this site the definition gives back the [[sheaves on a topological space|TB6TM6R4MD]].\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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    "tag": "BAH89SW049",
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  "markdown": "https://tamnd.github.io/math2/node/BAH89SW049.md",
  "msc": [
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  "name": "Sheaf on a site",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/BAH/BAH89SW049.md",
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    "tag": "BAH89SW049",
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        "note": "the row of the inventory that the node accounts for",
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