The Stacks project

Lemma 7.12.6. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{C}$. Then there exists a diagram $\mathcal{I} \to \mathcal{C}$, $i \mapsto U_ i$ such that

\[ \mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} h_{U_ i}^\# \]

Moreover, if $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is a subset such that every object of $\mathcal{C}$ has a covering by elements of $E$, then we may assume $U_ i$ is an element of $E$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$.

Proof. Let $\mathcal{I}$ be the category whose objects are pairs $(U, s)$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $s \in \mathcal{F}(U)$ and whose morphisms $(U, s) \to (U', s')$ are morphisms $f : U \to U'$ in $\mathcal{C}$ with $f^*s' = s$. For each object $(U, s)$ of $\mathcal{I}$ the element $s$ defines by the Yoneda lemma a map $s : h_ U \to \mathcal{F}$ of presheaves. Hence by the universal property of sheafification a map $h_ U^\# \to \mathcal{F}$. These maps are immediately seen to be compatible with the morphisms in the category $\mathcal{I}$. Hence we obtain a map $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U \to \mathcal{F}$ of presheaves (where the colimit is taken in the category of presheaves) and a map $\mathop{\mathrm{colim}}\nolimits _{(U, s)} (h_ U)^\# \to \mathcal{F}$ of sheaves (where the colimit is taken in the category of sheaves). Since sheafification is the left adjoint to the inclusion functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{PSh}(\mathcal{C})$ (Proposition 7.10.12) we have $\mathop{\mathrm{colim}}\nolimits (h_ U)^\# = (\mathop{\mathrm{colim}}\nolimits h_ U)^\# $ by Categories, Lemma 4.24.5. Thus it suffices to show that $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U \to \mathcal{F}$ is an isomorphism of presheaves. To see this we show that for every object $X$ of $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U(X) \to \mathcal{F}(X)$ is bijective. Namely, it has an inverse sending the element $t \in \mathcal{F}(X)$ to the element of the set $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U(X)$ corresponding to $(X, t)$ and $\text{id}_ X \in h_ X(X)$.

We omit the proof of the final statement. $\square$

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