= Heyting operations on subsheaves
{c}
{title2=$\operatorname{Sub}_{\operatorname{Sh}(\mathcal C,J)}(B)$}
Meets of subsheaves are pointwise intersections. Joins are J-closures of pointwise unions: a section lies in the join when its restrictions locally belong to some member, with the member allowed to vary. Implication consists of sections every restriction of which belongs to $D$ whenever it belongs to $A$; this gives $C\leq(A\Rightarrow D)$ exactly when $C\cap A\leq D$. Negation is implication into the initial <subobject>. Empty covers mean that the initial sheaf need not be pointwise empty.
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