4.1.3. The definition of local smallness [000I]

Based on our explorations above, we are now prepared to write down (and understand) the proper definition of local smallness for an arbitrary cartesian fibration $E$ over $B$, which should be thought of as a (potentially large) category relative to $B$.

[001C] Definition 4.1.3·a (Hom candidates).

For any $x\in B$ and displayed objects $u,v\in E\Sub{x}$, we define a hom candidate for $u,v$ to be a span $u\leftarrow \bar{h} \rightarrow v$ in $E$ in which the left-hand leg is cartesian:

In the above, $h$ should be thought of as a candidate for the “hom object” of $u,v$, and $\epsilon\Sub{h}$ should be viewed as the structure of an “evaluation map” for $h$. This structure can be rephrased in terms of a displayed category $\CandHom{x}{u}{v}$ over $\Sl{B}{x}$:

  1. Given $h\in \Sl{B}{x}$, an object of $\CandHom{x}{u}{v}\Sub{h}$ is given by a hom candidate whose apex in the base is $h$ itself. We will write $\bar{h}$ metonymically for the entire hom candidate over $h$.

  2. Given $\alpha:l\to h\in\Sl{B}{x}$ and hom candidates $\bar{l}\in \CandHom{x}{u}{v}\Sub{l}$ and $\bar{h}\in \CandHom{x}{u}{v}\Sub{h}$, a morphism $\bar{h}\to\Sub{\alpha} \bar{l}$ is given by a cartesian morphism $\bar\alpha:\bar{l}\to\Sub{\alpha}\bar{h}$ in $E$ such that the following diagram commutes:

[001B] Definition 4.1.3·b (Locally small fibration).

A cartesian fibration $E$ over $B$ is locally small if and only if for each $x\in B$ and $u,v\in E\Sub{x}$, the total category $\TotCat{\CandHom{x}{u}{v}}$ has a terminal object.