The Stacks project

Lemma 7.12.5. Let $\mathcal{C}$ be a site. Let $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset such that every object of $\mathcal{C}$ has a covering by elements of $E$. Let $\mathcal{F}$ be a sheaf of sets. There exists a diagram of sheaves of sets

\[ \xymatrix{ \mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_0 \ar[r] & \mathcal{F} } \]

which represents $\mathcal{F}$ as a coequalizer, such that $\mathcal{F}_ i$, $i = 0, 1$ are coproducts of sheaves of the form $h_ U^\# $ with $U \in E$.

Proof. First we show there is an epimorphism $\mathcal{F}_0 \to \mathcal{F}$ of the desired type. Namely, just take

\[ \mathcal{F}_0 = \coprod \nolimits _{U \in E, s \in \mathcal{F}(U)} (h_ U)^\# \longrightarrow \mathcal{F} \]

Here the arrow restricted to the component corresponding to $(U, s)$ maps the element $\text{id}_ U \in h_ U^\# (U)$ to the section $s \in \mathcal{F}(U)$. This is an epimorphism according to Lemma 7.11.2 and our condition on $E$. To construct $\mathcal{F}_1$ first set $\mathcal{G} = \mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0$ and then construct an epimorphism $\mathcal{F}_1 \to \mathcal{G}$ as above. See Lemma 7.11.3. $\square$

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