Sieve (category theory, $S\hookrightarrow yU$)

A sieve on $U$ is a subfunctor of the representable functor $yU$, equivalently a family of arrows into $U$ closed under precomposition. For $f:V\to U$, the pullback sieve consists of arrows $g$ into $V$ for which $fg$ belongs to the original sieve.

A sieve $R$ on $A$ is dense when every arrow $f:B\to A$ has a further composite $fh$ in $R$. Equivalently, every pullback sieve $f^{\ast }R$ is nonempty. This condition is stable under pullback and transitive refinement and defines the double-negation topology.

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.