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*}

4.39 Representable categories fibred in groupoids

Here is our definition of a representable category fibred in groupoids. As promised this is invariant under equivalences.

Definition 4.39.1. Let $\mathcal{C}$ be a category. A category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$ is called representable if there exists an object $X$ of $\mathcal{C}$ and an equivalence $j : \mathcal{S} \to \mathcal{C}/X$ (in the $2$-category of groupoids over $\mathcal{C}$).

The usual abuse of notation is to say that $X$ represents $\mathcal{S}$ and not mention the equivalence $j$. We spell out what this entails.

Lemma 4.39.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids.

  1. $\mathcal{S}$ is representable if and only if the following conditions are satisfied:

    1. $\mathcal{S}$ is fibred in setoids, and

    2. the presheaf $U \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)/\cong $ is representable.

  2. If $\mathcal{S}$ is representable the pair $(X, j)$, where $j$ is the equivalence $j : \mathcal{S} \to \mathcal{C}/X$, is uniquely determined up to isomorphism.

Proof. The first assertion follows immediately from Lemma 4.38.5. For the second, suppose that $j' : \mathcal{S} \to \mathcal{C}/X'$ is a second such pair. Choose a $1$-morphism $t' : \mathcal{C}/X' \to \mathcal{S}$ such that $j' \circ t' \cong \text{id}_{\mathcal{C}/X'}$ and $t' \circ j' \cong \text{id}_\mathcal {S}$. Then $j \circ t' : \mathcal{C}/X' \to \mathcal{C}/X$ is an equivalence. Hence it is an isomorphism, see Lemma 4.37.6. Hence by the Yoneda Lemma 4.3.5 (via Example 4.37.7 for example) it is given by an isomorphism $X' \to X$. $\square$

Lemma 4.39.3. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Assume that $\mathcal{X}$, $\mathcal{Y}$ are representable by objects $X$, $Y$ of $\mathcal{C}$. Then

\[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{X}, \mathcal{Y}) \Big/ 2\text{-isomorphism} = \mathop{Mor}\nolimits _\mathcal {C}(X, Y) \]

More precisely, given $\phi : X \to Y$ there exists a $1$-morphism $f : \mathcal{X} \to \mathcal{Y}$ which induces $\phi $ on isomorphism classes of objects and which is unique up to unique $2$-isomorphism.

Proof. By Example 4.37.7 we have $\mathcal{C}/X = \mathcal{S}_{h_ X}$ and $\mathcal{C}/Y = \mathcal{S}_{h_ Y}$. By Lemma 4.38.6 we have

\[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{X}, \mathcal{Y}) \Big/ 2\text{-isomorphism} = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ X, h_ Y) \]

By the Yoneda Lemma 4.3.5 we have $\mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ X, h_ Y) = \mathop{Mor}\nolimits _\mathcal {C}(X, Y)$. $\square$


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