Lemma 7.47.2. Let $\mathcal{C}$ be a category. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

The collection of sieves on $U$ is a set.

Inclusion defines a partial ordering on this set.

Unions and intersections of sieves are sieves.

Given a family of morphisms $\{ U_ i \to U\} _{i\in I}$ of $\mathcal{C}$ with target $U$ there exists a unique smallest sieve $S$ on $U$ such that each $U_ i \to U$ belongs to $S(U_ i)$.

The sieve $S = h_ U$ is the maximal sieve.

The empty subpresheaf is the minimal sieve.

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