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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nFor a [[sieve|CNCXNWKC6R]] $S$ on $c$ and a morphism $h \\colon d \\to c$, let $h^*S = \\set{g : h \\circ g \\in S}$ be the pullback sieve on $d$.\nA *Grothendieck topology* $J$ on $\\mathcal{C}$ assigns to each object $c$ a set $J(c)$ of sieves on $c$, the *covering sieves*, such that:\n\n1. (maximal sieve) the sieve of all morphisms with codomain $c$ is in $J(c)$;\n2. (stability) if $S \\in J(c)$ and $h \\colon d \\to c$ is a morphism, then $h^*S \\in J(d)$;\n3. (transitivity) if $S \\in J(c)$ and $R$ is a sieve on $c$ such that $h^*R \\in J(d)$ for every $h \\colon d \\to c$ in $S$, then $R \\in J(c)$.\n\n## Remarks\n\nExample: let $X$ be a topological space and $\\mathcal{O}(X)$ the poset of open sets as a category.\nLet $J(U)$ be the set of sieves $S$ on $U$ whose members have union $U$.\nClause 1 holds because $U \\subseteq U$ is in the maximal sieve.\nClause 2 holds because the sets $V \\cap W$ with $V \\in S$ cover $W$ when $S$ covers $U \\supseteq W$.\nClause 3 holds because covers of the members of a cover together give a cover.\nNon-example: on $\\mathcal{O}(X)$ with $X$ nonempty, the rule $J(U) = \\emptyset$ for all $U$ fails clause 1.\nOn every category, the rule in which only maximal sieves cover is a Grothendieck topology, and so is the rule in which every sieve covers.\nMany authors start from a *pretopology*, a choice of covering families of morphisms; it generates a Grothendieck topology by taking the sieves that contain a covering family.\n",
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