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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $c$ be an object of $\\mathcal{C}$.\nA *sieve* on $c$ is a set $S$ of morphisms with codomain $c$ with the following property.\nFor every $f \\colon d \\to c$ in $S$ and every morphism $g \\colon e \\to d$, the composite $f \\circ g$ is in $S$.\n\n## Remarks\n\nEquivalently, a sieve on $c$ is a subfunctor $S \\subseteq \\Hom_{\\mathcal{C}}(-, c)$ of the [[representable functor|KCCHH87H6E]], with $S(d)$ the set of members of $S$ with domain $d$.\nThe *maximal sieve* on $c$ is the set of all morphisms with codomain $c$.\nFor a sieve $S$ on $c$ and a morphism $h \\colon d \\to c$, the *pullback sieve* $h^*S = \\set{g : h \\circ g \\in S}$ is a sieve on $d$.\nExample: view the poset of open subsets of a space $X$ as a category, and let $\\set{U_i}$ be an open cover of $U$.\nThe open sets $V \\subseteq U$ that lie in some $U_i$ form a sieve on $U$, since $W \\subseteq V \\subseteq U_i$ gives $W \\subseteq U_i$.\nNon-example: in the same category, the set $\\set{U_1}$ alone is not a sieve on $U$ when $U_1$ has a smaller open subset $W \\ne U_1$, since the composite $W \\subseteq U_1 \\subseteq U$ is missing.\n",
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        "url": "https://en.wikipedia.org/wiki/Sieve_(category_theory)"
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