A sieve on an object $𝑥$ of $\mathrm{𝒞︀}$ is a subobject of the representable presheaf $\text{𝗒}𝑥$: a presheaf $𝑆$ with a monic morphism $𝑆\rightarrowtail \text{𝗒}𝑥$. Sieves are a generalization from the notion of ideal found in ring theory to category theory.

A morphism $𝑓:𝑦\to 𝑥$ belongs to $𝑆$, written $𝑆\ni 𝑓$, when $𝑓$ lies in the image of the inclusion at $𝑦$. Because $𝑆$ is a presheaf and the inclusion is natural, membership is closed under precomposition:

A sieve is thus a “downward closed” collection of morphisms into $𝑥$.

Sieves on $𝑥$ are ordered by refinement: $𝑆\subseteq 𝑇$ when every morphism belonging to $𝑆$ belongs to $𝑇$.