The Stacks project

\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*}

8.9 Stackification of categories fibred in groupoids

Here is the result.

Lemma 8.9.1. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. There exists a stack in groupoids $p' : \mathcal{S}' \to \mathcal{C}$ and a $1$-morphism $G : \mathcal{S} \to \mathcal{S}'$ of categories fibred in groupoids over $\mathcal{C}$ (see Categories, Definition 4.34.6) such that

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the map

    \[ \mathit{Mor}(x, y) \longrightarrow \mathit{Mor}(G(x), G(y)) \]

    induced by $G$ identifies the right hand side with the sheafification of the left hand side, and

  2. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any $x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ such that for every $i \in I$ the object $x'|_{U_ i}$ is in the essential image of the functor $G : \mathcal{S}_{U_ i} \to \mathcal{S}'_{U_ i}$.

Moreover the stack in groupoids $\mathcal{S}'$ is determined up to unique $2$-isomorphism by these conditions.

Proof. Apply Lemma 8.8.1. The result will be a stack in groupoids by applying Lemma 8.5.2. $\square$

Lemma 8.9.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. Let $p' : \mathcal{S}' \to \mathcal{C}$ and $G : \mathcal{S} \to \mathcal{S}'$ the stack in groupoids and $1$-morphism constructed in Lemma 8.9.1. This construction has the following universal property: Given a stack in groupoids $q : \mathcal{X} \to \mathcal{C}$ and a $1$-morphism $F : \mathcal{S} \to \mathcal{X}$ of categories over $\mathcal{C}$ there exists a $1$-morphism $H : \mathcal{S}' \to \mathcal{X}$ such that the diagram

\[ \xymatrix{ \mathcal{S} \ar[rr]_ F \ar[rd]_ G & & \mathcal{X} \\ & \mathcal{S}' \ar[ru]_ H } \]

is $2$-commutative.

Proof. This is a special case of Lemma 8.8.2. $\square$

Lemma 8.9.3. Let $\mathcal{C}$ be a site. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Z} \to \mathcal{Y}$ be morphisms of categories fibred in groupoids over $\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications.

Proof. This is a special case of Lemma 8.8.4. $\square$


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