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 groupoids over $\mathcal{C}$).
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.
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:
$\mathcal{S}$ is fibred in setoids, and
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 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.
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{X}, \mathcal{Y}) \Big/ 2\text{-isomorphism} = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, Y) \]
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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