J-closed sieve
= J-closed sieve
{c}
{title2=$f^*S\in J(\operatorname{dom}f)\Longrightarrow f\in S$}
A sieve is J-closed when membership is local with respect to the <Grothendieck topology> $J$. Pullback preserves J-closedness. Their presheaf is the <subobject classifier> of the sheaf topos: a section of a sheaf is sent to the sieve of arrows on which it belongs to the chosen subsheaf.