4.1. Locally small fibrations [000F]

There are a number of (equivalent) variations on the definition of a locally small fibration. We attempt to provide some intuition for these definitions.

4.1.1. Warmup: locally small family fibrations [000G]

An ordinary category $E$ is called locally small when for any $x,y\in E$ the collection of morphisms $x\to y$ is a set. This property of $E$ can be rephrased in terms of its family fibration (Section 2.7 [0006]) $\FAM{E}$ over $\SET$ as follows.

Consider an index set $I\in \SET$ and two families $u,v\in C^I$. We may define an $I$-indexed collection $[u,v]\Sub{i\in I}$ consisting (pointwise) of all the morphisms $u\Sub{i}\to v\Sub{i}$ in $C$:

If $C$ is locally small, $[u,v]\Sub{i\in I}$ is in fact a family of sets for any $I\in\SET$ as each $[u,v]\Sub{i}$ is a set. Conversely, if $[u,v]\Sub{i\in I}$ is a family of sets for any $I\in \SET$, then $C$ is locally small as we may consider in particular the case that $I=\mathbf{1}$.

4.1.2. A more abstract formulation [000H]

We will reformulate the above in a way that uses only the language that makes sense for an arbitrary cartesian fibration, though for now we stick with $\FAM{C}$. Given $u,v\in \FAM{C}[I]$, we have a “relative hom family” $[u,v]\in\Sl{\SET}{I}$, defined in Section 4.1.1 [000G]. The fact that each $[u,v]\Sub{i}$ is the set of all morphisms $u\Sub{i}\to v\Sub{i}$ can be rephrased more abstractly.

First we consider the restriction of $u\in \FAM{C}[I]$ to $\FAM{C}[[u,v]]$ as follows: Explicitly the family $\InvImg{[u,v]}u$ is indexed in a pair of an element $i\in I$ and a morphism $u\Sub{i}\to v\Sub{i}$. We can think of $\InvImg{[u,v]}u$ as the object of elements of $u\Sub{i}$ indexed in pairs $(i,u\Sub{i}\to v\Sub{i})$.

There is a canonical map $\epsilon\Sub{[u,v]}:\InvImg{[u,v]}u\to\Sub{p\Sub{[u,v]}} v$ that “evaluates” each indexing morphism $u\Sub{i}\to v\Sub{i}$.

That each $[u,v]\Sub{i}$ is the set of all morphisms $u\Sub{i}\to v\Sub{i}$ can be rephrased as a universal property: for any family $h\in\Sl{\SET}{I}$ and morphism $\epsilon\Sub{h} : \InvImg{h}u\to\Sub{h} v$ in $\FAM{C}$, there is a unique cartesian map $\InvImg{h}u\to \InvImg{[u,v]}u$ factoring $\epsilon\Sub{h}$ through $\epsilon\Sub{[u,v]}$ in the sense depicted below:

To convince ourselves of this, we note that the family $H \coloneqq \brc{u\Sub{i}\to v\Sub{i}}\Sub{i\in I}$ itself satisfies the universal property above. Indeed, fix a candidate $h \in \Sl{\SET}{I}$ equipped with a map $\epsilon\Sub{h} : \InvImg{h}u \to_h v$. Unfolding the meaning of this map in set theoretical notation, we see that it amounts to a family of maps $\epsilon\Sub{h}[i] : \prod\Sub{x\in h_i} \brc{u_i\to v_i}$ for each $i\in I$; such a family immediately induces the desired map $h\to H$.

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.