J-closed sieve ($f^{\ast }S\in J(\mathrm{dom}f)⟹f\in S$)

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.