Definition 6.1.
Let \(C\) be a category. A sieve on an object \(X \in C\) is a subfunctor \(U \hookrightarrow y(X)\) of the representable presheaf \(y(X) = \Hom_C(-,X)\). Equivalently, it is a collection of morphisms with codomain \(X\) that is closed under precomposition.