The Stacks project

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

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

  2. Inclusion defines a partial ordering on this set.

  3. Unions and intersections of sieves are sieves.

  4. 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 $S(U_ i)$.

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

  6. The empty subpresheaf is the minimal sieve.

Proof. By our definition of subpresheaf, the collection of all subpresheaves of a presheaf $\mathcal{F}$ is a subset of $\prod _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} \mathcal{P}(\mathcal{F}(U))$. And this is a set. (Here $\mathcal{P}(A)$ denotes the powerset of $A$.) Hence the collection of sieves on $U$ is a set.

The partial ordering is defined by: $S \leq S'$ if and only if $S(T) \subset S'(T)$ for all $T \to U$. Notation: $S \subset S'$.

Given a collection of sieves $S_ i$, $i \in I$ on $U$ we can define $\bigcup S_ i$ as the sieve with values $(\bigcup S_ i)(T) = \bigcup S_ i(T)$ for all $T \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We define the intersection $\bigcap S_ i$ in the same way.

Given $\{ U_ i \to U\} _{i\in I}$ as in the statement, consider the morphisms of presheaves $h_{U_ i} \to h_ U$. We simply define $S$ as the union of the images (Definition 7.3.5) of these maps of presheaves.

The last two statements of the lemma are obvious. $\square$

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