5.3. The full internal subcategory associated to a displayed object [0011]

The full internal subfibration (Construction 2.9·a [0010]) associated to a displayed object $\bar{u}$ of a locally small cartesian fibration $E$ over $B$ can be seen to be equivalent to the externalization of an internal category $\gl{\bar{u}}$ in $B$. In particular, we let the object of objects $\gl{\bar{u}}\Sub{0}$ be $u$ itself; defining the object of arrows $\gl{\bar{u}}\Sub{1}$ is more complex, making critical use of the local smalness of $E$ over $B$.

We will think of the fiber $E\Sub{u\times u}$ as the category of objects indexed in the boundary (source and target) of a morphism. Restricting $\bar{u}$ along the source and target projections, we obtain the displayed objects of “points of the source” and “points of the target” respectively:

Because $E$ is locally small, there is an object $\brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}\in \Sl{B}{u\times u}$ that behaves as the “generic hom set”. We define $\gl{\bar{u}}\Sub{1}\in B$ and its source and target projections to be this very object.

[001S] Theorem 5.3·a.

The externalization $\brk{\gl{\bar{u}}}$ of the internal category $\gl{\bar{u}}$ is equivalent to $\FullSubfib{\bar{u}}$.

Proof. We will define a fibred equivalence $F : \brk{\gl{\bar{u}}}\to \FullSubfib{\bar{u}}$ over $B$.

1. Fix $x\in B$ and $\chi\Sub{x} \in \brk{\gl{\bar{u}}}\Sub{x}$, i.e. $\chi\Sub{x} : x\to u$; we define $F\prn{\chi\Sub{x}}$ to be an arbitrary cartesian map $\phi\Sub{x} : \bar{x}\to\Sub{\chi\Sub{x}} \bar{u}$. (Here we have used the axiom of choice for collections [000R].)

2. Fix $f : x\to y\in B$ and $\chi\Sub{x} :x\to u$ and $\chi\Sub{y}:y\to u$ and a diagram representing a displayed morphism $h$ from $\chi\Sub{x}$ to $\chi\Sub{y}$ over $f$ as below:

We must define $F\prn{h}:\bar{x}\to\Sub{f} \bar{y}$, fixing arbitrary cartesian maps $\bar\chi\Sub{x}:\bar{x}\to\Sub{\chi\Sub{x}}\bar{u}$ and $\bar\chi\Sub{y}:\bar{y}\to\Sub{\chi\Sub{y}}\bar{u}$. First we lift $h:x\to \brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}$ into $E$ using the universal property of the cartesian lift:

By composition with the “evaluation map” for our hom object, we have a map $\bar{x}\to\Sub{f;\chi\Sub{y}}\bar{u}$:

Next we define $F\prn{h}:\bar{x}\to\Sub{f}\bar{y}$ using the universal property of (another) cartesian lift: