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\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 7.29.5. Let $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a topos. Let $\{ \mathcal{F}_ i\} _{i \in I}$ be a set of sheaves on $\mathcal{C}$. There exists an equivalence of topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ induced by a special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$ of sites such that

  1. $\mathcal{C}'$ has a subcanonical topology,

  2. a family $\{ V_ j \to V\} $ of morphisms of $\mathcal{C}'$ is (combinatorially equivalent to) a covering of $\mathcal{C}'$ if and only if $\coprod h_{V_ j} \to h_ V$ is surjective,

  3. $\mathcal{C}'$ has fibre products and a final object (i.e., $\mathcal{C}'$ has all finite limits),

  4. every subsheaf of a representable sheaf on $\mathcal{C}'$ is representable, and

  5. each $g_*\mathcal{F}_ i$ is a representable sheaf.

Proof. Consider the full subcategory $\mathcal{C}_1 \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ consisting of all $h_ U^\# $ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, the given sheaves $\mathcal{F}_ i$ and the final sheaf $*$ (see Example 7.10.2). We are going to inductively define full subcategories

\[ \mathcal{C}_1 \subset \mathcal{C}_2 \subset \mathcal{C}_2 \subset \ldots \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]

Namely, given $\mathcal{C}_ n$ let $\mathcal{C}_{n + 1}$ be the full subcategory consisting of all fibre products and subsheaves of objects of $\mathcal{C}_ n$. (Note that $\mathcal{C}_{n + 1}$ has a set of objects.) Set $\mathcal{C}' = \bigcup _{n \geq 1} \mathcal{C}_ n$. A covering in $\mathcal{C}'$ is any family $\{ \mathcal{G}_ j \to \mathcal{G}\} _{j \in J}$ of morphisms of objects of $\mathcal{C}'$ such that $\coprod \mathcal{G}_ j \to \mathcal{G}$ is surjective as a map of sheaves on $\mathcal{C}$. The functor $v : \mathcal{C} \to \mathcal{C'}$ is given by $U \mapsto h_ U^\# $. Apply Lemma 7.29.4. $\square$


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