[000L] Theorem 4.2.1·b (Globally small categories).

An ordinary category $C$ is equivalent to a globally small category if and only if the fibration $\FAM{C}$ has a generic object.

Proof. To see that this is the case, suppose that $C$ has a set of objects. Then $C\in\SET$ and we define $\lfloor{C}\rfloor$ to be the displayed object $\brc{x}\Sub{x\in C}\in \FAM{C}[C]$. Fixing $I\in \SET$ and $z\in C^I$, we consider the cartesian map displayed over $z : I \to C$:

Conversely assume that $\FAM{C}$ has a generic object $\bar{u}\in\FAM{C}[U]$ for some $U\in \SET$; then we may equip $U$ with the structure of a globally small category such that $U$ is equivalent to $C$, using a construction that is similar to our implementation of the opposite fibration (Section 2.6 [000Q]). In particular we define a morphism $x\to y\in U$ to be given by the following data:

  1. a cartesian map $a\to\Sub{x} \bar{u}$ over $x : 1\to U$,
  2. a cartesian map $b\to\Sub{y} \bar{u}$ over $y : 1\to U$,
  3. and a vertical map $h:a\to b$ in $\FAM{C}[1]\simeq C$,

such that $(a\Sub{1},b\Sub{1},h\Sub{1})$ is identified with $(a\Sub{2},b\Sub{2},h\Sub{2})$ when $h_1$ and $h_2$ are equal modulo the (unique) vertical isomorphisms between the cartesian lifts in the sense depicted below:

Remember that a cartesian map $a\to\Sub{x}\bar{u}$ is standing for a choice of an object of $C$ encoded by $x\in U$. Because such choices are unique only up to isomorphism, we must include them explicitly in the data.