Definition. Sieve [FG8Y]

A sieve $S$ on an object $X$ of a category $\mathcal{C}$ is a family of morphisms with codomain $X$ such that closed under composition: If $f \in S$ then for any composable $g$ we have $f \circ g \in S$.

We denote $S \mathbin{⪧} X$ or $X \mathbin{⪦} S$.