4.3. Separators for cartesian fibrations [002E]

Let $E$ be an ordinary category. In general, to compare two morphisms $f,g:x\to y$ in $E$, it is not enough to see if they agree on global points $u:1\to x$, because the behavior of $u,v$ may differ only on generalized elements. In some cases, however, there is a family of objects $\prn{s\Sub{i}}\Sub{i\in I}\in E$ are together adequate for comparing morphisms of $E$ in the sense of Definition 4.3·a [002G] below.

[002G] Definition 4.3·a (Separating family for a category).

Given an ordinary category $E$, a set-indexed family $\prn{s\Sub{i}}\Sub{i\in I}$ of $E$-objects is called a small separating family for $E$ when, assuming that for all $i\in I$ and all $u:s_i\to x$ we have $u;f=u;g$, we then have $f=g$.

The intuition of Definition 4.3·a [002G] is that to compare two morphisms $f,g:x\to y\in E$, it suffices to check that they behave the same on all $s\Sub{i}$-shaped figures when $\prn{s\Sub{i}}\Sub{i\in I}$ is a separating family for $E$.

[002F] Example 4.3·b (Well-pointedness of the category of sets).

In the category of sets, to compare two morphisms it is enough to check that they agree on global points. This means that the unary family $\brc{\ObjTerm{\SET}}$ is a separator for $\SET$, a property referred to more generally as well-pointedness.

We will now generalize Definition 4.3·a [002G] to the case of a cartesian fibration.

[002I] Definition 4.3·c (Agreement on a class of figure shapes).

Let $\bar{s}$ be a displayed object in a cartesian fibration $E$ over $B$. A pair of displayed morphisms $f,g:\bar{x}\to \bar{y}\in E$ are said to agree on $\bar{s}$-figures when for any $\bar{s}$-figure $h : \bar{z}\to \bar{x}$ in the sense of Definition 2.9·b [002K], we have $h;f = h;g : \bar{z}\to \bar{y}$.

[002H] Definition 4.3·d (Small separator for a fibration).

Let $E$ be a cartesian fibration over $B$ such that $B$ has binary products. A displayed object $\bar{s}\in E\Sub{s}$ is said to be a small separator for $E$ when any two vertical maps $f,g:\bar{u}\to\Sub{\Idn{x}}\bar{v}\in E\Sub{x}$ are equal when they are agree on $\bar{s}$-figures in the sense of Definition 4.3·c [002I].