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.
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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