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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA *presheaf* $\\mathcal F$ of sets on $X$ consists of the following data.\nFor each [[open set|02BH3MCBQB]] $U \\subseteq X$ there is a set $\\mathcal F(U)$.\nFor each pair of open sets $V \\subseteq U$ there is a map $\\rho_{UV} \\colon \\mathcal F(U) \\to \\mathcal F(V)$, written $s \\mapsto s|_V$.\nThese satisfy $\\rho_{UU} = \\id$ and $\\rho_{VW} \\circ \\rho_{UV} = \\rho_{UW}$ whenever $W \\subseteq V \\subseteq U$.\nThe elements of $\\mathcal F(U)$ are the *sections* of $\\mathcal F$ over $U$.\nA presheaf $\\mathcal F$ is a *sheaf* if, for every open set $U$ and every open cover $U = \\bigcup_{i \\in I} U_i$:\n\n1. (Locality) if $s, t \\in \\mathcal F(U)$ satisfy $s|_{U_i} = t|_{U_i}$ for all $i$, then $s = t$;\n2. (Gluing) if $s_i \\in \\mathcal F(U_i)$ satisfy $s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$ for all $i, j$, then there is $s \\in \\mathcal F(U)$ with $s|_{U_i} = s_i$ for all $i$.\n\nA sheaf of [[abelian groups|082GHNPJ53]] is a sheaf in which each $\\mathcal F(U)$ is an abelian group and each restriction map is a homomorphism.\nA sheaf of [[commutative rings|0310REJJY8]] is defined in the same way.\n\n## Remarks\n\nApplied to the empty cover of $U = \\emptyset$, the two conditions say that $\\mathcal F(\\emptyset)$ has exactly one element.\n\nExample: for $U \\subseteq \\R$ open, let $\\mathcal C(U)$ be the ring of [[continuous|0VGFHTM75Y]] functions $U \\to \\R$, with the usual restriction.\nLocality holds because a function is determined by its values.\nGluing holds because the glued function agrees near each point with some continuous $s_i$, so it is continuous.\nSo $\\mathcal C$ is a sheaf of commutative rings.\n\nNon-example: let $\\mathcal B(U)$ be the set of bounded continuous functions on $U$.\nGluing fails for $U = \\R$ with the cover by the intervals $U_m = (-m, m)$ and the sections $s_m(x) = x$ on $U_m$: they agree on overlaps, but their only possible gluing, $x \\mapsto x$, is not bounded.\n\nEquivalently, conditions 1 and 2 say that $\\mathcal F(U) \\to \\prod_i \\mathcal F(U_i) \\rightrightarrows \\prod_{i, j} \\mathcal F(U_i \\cap U_j)$ is an equalizer diagram.\nSheaves of abelian groups are the coefficients of [[sheaf cohomology|KB76923A1M]].\n",
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