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*}

4.40 Representable 1-morphisms

Let $\mathcal{C}$ be a category. In this section we explain what it means for a $1$-morphism between categories fibred in groupoids over $\mathcal{C}$ to be representable. Note that the $2$-category of categories fibred in groupoids over $\mathcal{C}$ is a “full” sub $2$-category of the $2$-category of categories over $\mathcal{C}$ (see Definition 4.34.6). Hence if $\mathcal{S}$, $\mathcal{S}'$ are fibred in groupoids over $\mathcal{C}$ then

\[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}, \mathcal{S}') \]

denotes the category of $1$-morphisms in this $2$-category (see Definition 4.31.1). These are all groupoids, see remarks following Definition 4.34.6. Here is the $2$-category analogue of the Yoneda lemma.

Lemma 4.40.1 (2-Yoneda lemma). Let $\mathcal{S}\to \mathcal{C}$ be fibred in groupoids. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor

\[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \longrightarrow \mathcal{S}_ U \]

given by $G \mapsto G(\text{id}_ U)$ is an equivalence.

Proof. Make a choice of pullbacks for $\mathcal{S}$ (see Definition 4.32.6). We define a functor

\[ \mathcal{S}_ U \longrightarrow \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \]

as follows. Given $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the associated functor is

  1. on objects: $(f : V \to U) \mapsto f^*x$, and

  2. on morphisms: the arrow $(g : V'/U \to V/U)$ maps to the composition

    \[ (f \circ g)^*x \xrightarrow {(\alpha _{g, f})_ x} g^*f^*x \rightarrow f^*x \]

    where $\alpha _{g, f}$ is as in Lemma 4.34.2.

We omit the verification that this is an inverse to the functor of the lemma. $\square$

Remark 4.40.2. We can use the $2$-Yoneda lemma to give an alternative proof of Lemma 4.36.3. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. We define a contravariant functor $F$ from $\mathcal{C}$ to the category of groupoids as follows: for $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ let

\[ F(U) = \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}). \]

If $f : U \to V$ the induced functor $\mathcal{C}/U \to \mathcal{C}/V$ induces the morphism $F(f) : F(V) \to F(U)$. Clearly $F$ is a functor. Let $\mathcal{S}'$ be the associated category fibred in groupoids from Example 4.36.1. There is an obvious functor $G : \mathcal{S}' \to \mathcal{S}$ over $\mathcal{C}$ given by taking the pair $(U, x)$, where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in F(U)$, to $x(\text{id}_ U) \in \mathcal{S}$. Now Lemma 4.40.1 implies that for each $U$,

\[ G_ U : \mathcal{S}'_ U = F(U)= \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \to \mathcal{S}_ U \]

is an equivalence, and thus $G$ is an equivalence between $\mathcal{S}$ and $\mathcal{S}'$ by Lemma 4.34.8.

Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $F : \mathcal{X} \to \mathcal{Y}$ and $G : \mathcal{C}/U \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $\mathcal{C}$. We want to describe the $2$-fibre product

\[ \xymatrix{ (\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d]^ F \\ \mathcal{C}/U \ar[r]^ G & \mathcal{Y} } \]

Let $y = G(\text{id}_ U) \in \mathcal{Y}_ U$. Make a choice of pullbacks for $\mathcal{Y}$ (see Definition 4.32.6). Then $G$ is isomorphic to the functor $(f : V \to U) \mapsto f^*y$, see Lemma 4.40.1 and its proof. We may think of an object of $(\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X}$ as a quadruple $(V, f : V \to U, x, \phi )$, see Lemma 4.31.3. Using the description of $G$ above we may think of $\phi $ as an isomorphism $\phi : f^*y \to F(x)$ in $\mathcal{Y}_ V$.

Lemma 4.40.3. In the situation above the fibre category of $(\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X}$ over an object $f : V \to U$ of $\mathcal{C}/U$ is the category described as follows:

  1. objects are pairs $(x, \phi )$, where $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V)$, and $\phi : f^*y \to F(x)$ is a morphism in $\mathcal{Y}_ V$,

  2. the set of morphisms between $(x, \phi )$ and $(x', \phi ')$ is the set of morphisms $\psi : x \to x'$ in $\mathcal{X}_ V$ such that $F(\psi ) = \phi ' \circ \phi ^{-1}$.

Proof. See discussion above. $\square$

Lemma 4.40.4. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. Let $G : \mathcal{C}/U \to \mathcal{Y}$ be a $1$-morphism. Then

\[ (\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{C}/U \]

is a category fibred in groupoids.

Proof. We have already seen in Lemma 4.34.7 that the composition

\[ (\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{C}/U \longrightarrow \mathcal{C} \]

is a category fibred in groupoids. Then the lemma follows from Lemma 4.34.12. $\square$

Definition 4.40.5. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. We say $F$ is representable, or that $\mathcal{X}$ is relatively representable over $\mathcal{Y}$, if for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $G : \mathcal{C}/U \to \mathcal{Y}$ the category fibred in groupoids

\[ (\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{C}/U \]

is representable.

Lemma 4.40.6. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. If $F$ is representable then every one of the functors

\[ F_ U : \mathcal{X}_ U \longrightarrow \mathcal{Y}_ U \]

between fibre categories is faithful.

Proof. Clear from the description of fibre categories in Lemma 4.40.3 and the characterization of representable fibred categories in Lemma 4.39.2. $\square$

Lemma 4.40.7. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. Make a choice of pullbacks for $\mathcal{Y}$. Assume

  1. each functor $F_ U : \mathcal{X}_ U \longrightarrow \mathcal{Y}_ U$ between fibre categories is faithful, and

  2. for each $U$ and each $y \in \mathcal{Y}_ U$ the presheaf

    \[ (f : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathcal{X}_ V, \phi : f^*y \to F(x)\} /\cong \]

    is a representable presheaf on $\mathcal{C}/U$.

Then $F$ is representable.

Proof. Clear from the description of fibre categories in Lemma 4.40.3 and the characterization of representable fibred categories in Lemma 4.39.2. $\square$

Before we state the next lemma we point out that the $2$-category of categories fibred in groupoids is a $(2, 1)$-category, and hence we know what it means to say that it has a final object (see Definition 4.30.1). And it has a final object namely $\text{id} : \mathcal{C} \to \mathcal{C}$. Thus we define $2$-products of categories fibred in groupoids over $\mathcal{C}$ as the $2$-fibred products

\[ \mathcal{X} \times \mathcal{Y} := \mathcal{X} \times _\mathcal {C} \mathcal{Y}. \]

With this definition in place the following lemma makes sense.

Lemma 4.40.8. Let $\mathcal{C}$ be a category. Let $\mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. Assume $\mathcal{C}$ has products of pairs of objects and fibre products. The following are equivalent:

  1. The diagonal $\mathcal{S} \to \mathcal{S} \times \mathcal{S}$ is representable.

  2. For every $U$ in $\mathcal{C}$, any $G : \mathcal{C}/U \to \mathcal{S}$ is representable.

Proof. Suppose the diagonal is representable, and let $U, G$ be given. Consider any $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $G' : \mathcal{C}/V \to \mathcal{S}$. Note that $\mathcal{C}/U \times \mathcal{C}/V = \mathcal{C}/U \times V$ is representable. Hence the fibre product

\[ \xymatrix{ (\mathcal{C}/U \times V) \times _{(\mathcal{S} \times \mathcal{S})} \mathcal{S} \ar[r] \ar[d] & \mathcal{S} \ar[d] \\ \mathcal{C}/U \times V \ar[r]^{(G, G')} & \mathcal{S} \times \mathcal{S} } \]

is representable by assumption. This means there exists $W \to U \times V$ in $\mathcal{C}$, such that

\[ \xymatrix{ \mathcal{C}/W \ar[d] \ar[r] & \mathcal{S} \ar[d] \\ \mathcal{C}/U \times \mathcal{C}/V \ar[r] & \mathcal{S} \times \mathcal{S} } \]

is cartesian. This implies that $\mathcal{C}/W \cong \mathcal{C}/U \times _\mathcal {S} \mathcal{C}/V$ (see Lemma 4.30.11) as desired.

Assume (2) holds. Consider any $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $(G, G') : \mathcal{C}/V \to \mathcal{S} \times \mathcal{S}$. We have to show that $\mathcal{C}/V \times _{\mathcal{S} \times \mathcal{S}} \mathcal{S}$ is representable. What we know is that $\mathcal{C}/V \times _{G, \mathcal{S}, G'} \mathcal{C}/V$ is representable, say by $a : W \to V$ in $\mathcal{C}/V$. The equivalence

\[ \mathcal{C}/W \to \mathcal{C}/V \times _{G, \mathcal{S}, G'} \mathcal{C}/V \]

followed by the second projection to $\mathcal{C}/V$ gives a second morphism $a' : W \to V$. Consider $W' = W \times _{(a, a'), V \times V} V$. There exists an equivalence

\[ \mathcal{C}/W' \cong \mathcal{C}/V \times _{\mathcal{S} \times \mathcal{S}} \mathcal{S} \]

namely

\begin{eqnarray*} \mathcal{C}/W' & \cong & \mathcal{C}/W \times _{(\mathcal{C}/V \times \mathcal{C}/V)} \mathcal{C}/V \\ & \cong & \left(\mathcal{C}/V \times _{(G, \mathcal{S}, G')} \mathcal{C}/V\right) \times _{(\mathcal{C}/V \times \mathcal{C}/V)} \mathcal{C}/V \\ & \cong & \mathcal{C}/V \times _{(\mathcal{S} \times \mathcal{S})} \mathcal{S} \end{eqnarray*}

(for the last isomorphism see Lemma 4.30.12) which proves the lemma. $\square$

Bibliographic notes: Parts of this have been taken from Vistoli's notes [Vis2].


Comments (1)

Comment #1779 by Kiran Kedlaya on

Is "Biographical notes" perhaps a typo for "Bibliographic notes"? (It being a reference to Vistoli's book rather than his life story.)


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