Sieve

Remarks

Equivalently, a sieve on $c$ is a subfunctor $S \subseteq \Hom_{\mathcal{C}}(-, c)$ of the representable functor, with $S(d)$ the set of members of $S$ with domain $d$. The maximal sieve on $c$ is the set of all morphisms with codomain $c$. For a sieve $S$ on $c$ and a morphism $h \colon d \to c$, the pullback sieve $h^*S = \set{g : h \circ g \in S}$ is a sieve on $d$. Example: view the poset of open subsets of a space $X$ as a category, and let $\set{U_i}$ be an open cover of $U$. The open sets $V \subseteq U$ that lie in some $U_i$ form a sieve on $U$, since $W \subseteq V \subseteq U_i$ gives $W \subseteq U_i$. Non-example: in the same category, the set $\set{U_1}$ alone is not a sieve on $U$ when $U_1$ has a smaller open subset $W \ne U_1$, since the composite $W \subseteq U_1 \subseteq U$ is missing.