Section 4.40 (04SE): Representable categories fibred in groupoids

4.40 Representable categories fibred in groupoids

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

Definition 4.40.1. Let $\mathcal{C}$ be a category. A category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$ is called representable if there exist an object $X$ of $\mathcal{C}$ and an equivalence $j : \mathcal{S} \to \mathcal{C}/X$ (in the $2$-category of categories fibred in 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 4.40.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids.

$\mathcal{S}$ 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.
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 4.39.5. 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 4.38.6. Hence by the Yoneda Lemma 4.3.5 (via Example 4.38.7 for example) it is given by an isomorphism $X' \to X$. $\square$

Lemma 4.40.3. 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

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

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 4.38.7 we have $\mathcal{C}/X = \mathcal{S}_{h_ X}$ and $\mathcal{C}/Y = \mathcal{S}_{h_ Y}$. By Lemma 4.39.6 we have

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

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

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Comment #11424 by Elías Guisado on May 13, 2026 at 06:53

Here's a Lemma that could fit in this section. Before stating it, we note representability can be understood in more generality.

Definition. Let $\mathcal{C}$ be a category. Let $\mathcal{S}$ be another category and let $\mathcal{S}\to\mathcal{C}$ be a functor. We say that $\mathcal{S}$ is a representable category over $\mathcal{C}$ if there is an object $U$ in $\mathcal{C}$ and an equivalence $\mathcal{S}\simeq\mathcal{C}/U$ in the $2$ -category of categories over $\mathcal{C}$ (Definition 4.32.1).

By Lemma 4.39.5, a representable category over $\mathcal{C}$ is fibred in setoids over $\mathcal{C}$ . The terminology of the Definition just introduced is compatible with that of Definition 4.40.1, for the $2$ -category of Definition 4.35.6 is a full $2$ -subcategory of that of Definition 4.32.1.

Lemma. Let $\mathcal{C}$ be a category. Let $U$ be an object in $\mathcal{C}$ . Let $\mathcal{S}$ be a category over $\mathcal{C}/U$ (i.e., it comes with a functor $\mathcal{S}\to\mathcal{C}/U$ ), so $\mathcal{S}$ is also a category over $\mathcal{C}$ . Then $\mathcal{S}$ is a representable category over $\mathcal{C}/U$ if and only if it is a representable category over $\mathcal{C}$ .

This result seems to be implicitly used in Algebraic Stacks, Sect. 94.6, first paragraph (I suspect it might be used in more places).

Proof. For the proof, note that if $f:V\to U$ is a morphism in $\mathcal{C}$ , then there is a canonical isomorphism $(\mathcal{C}/U)/f\cong\mathcal{C}/V$ of categories over $\mathcal{C}/U$ .

( $\Rightarrow$ ). Suppose there is a morphism $f:V\to U$ such that $(\mathcal{C}/U)/f\simeq\mathcal{S}$ as categories over $\mathcal{C}/U$ . Composition with the isomorphism mentioned before gives an equivalence $\mathcal{C}/V\simeq\mathcal{S}$ as categories over $\mathcal{C}/U$ ; hence an equivalence as categories over $\mathcal{C}$ .

( $\Leftarrow$ ). Suppose there is an object $V$ in $\mathcal{C}$ and an equivalence $\mathcal{C}/V\simeq\mathcal{S}$ of categories over $\mathcal{C}$ . Postcomposition with $\mathcal{S}\to\mathcal{C}/U$ gives a $1$ -morphism $G:\mathcal{C}/V\to\mathcal{C}/U$ over $\mathcal{C}$ . By #11423, we have $G=f_*$ , where $f=G(\operatorname{id}_V):V\to U$ . The composite $(\mathcal{C}/U)/f\cong\mathcal{C}/V\simeq\mathcal{S}\to\mathcal{C}/U$ is the canonical forgetful functor. That is, we have a $1$ -morphism $(\mathcal{C}/U)/f\to\mathcal{S}$ over $\mathcal{C}/U$ . It is also a $1$ -morphism over $\mathcal{C}$ which is an equivalence over $\mathcal{C}$ . By Lemma 4.35.9 it is also an equivalence over $\mathcal{C}/U$ ( $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}/U$ by #11410). $\square$

Comment #11535 by Stacks Project on May 29, 2026 at 15:02

Leaving as is. Instead, I have edited the discussion in the first paragraph of Section 94.6; see this commit.

The Stacks project

4.40 Representable categories fibred in groupoids

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