
## 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 $\mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ X, h_ Y) = \mathop{Mor}\nolimits _\mathcal {C}(X, Y)$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).