2.4. Displayed and fibered functors [0004]

Let $E$ be displayed over $B$ and let $F$ be displayed over $C$. If $U:B \to C$ is an ordinary functor, than a displayed functor from $E$ to $F$ over $U$ is given by the following data:

  1. for each displayed object $\bar{x}\in E\Sub{x}$, a displayed object $\bar{U}\bar{x}\in E\Sub{Ux}$,
  2. for each displayed morphism $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$, a displayed morphism $\bar{U}\bar{f} : \bar{U}\bar{x}\to\Sub{Uf}\bar{U}\bar{y}$,
  3. such that the assignment $\bar{U}f$ preserves displayed identities and displayed composition.

From this notion, we can see the variation of displayed categories over their base categories itself has a “displayed categorical” structure; up to size issues, we could speak of the displayed bicategory of displayed categories.

Note. The correct notion of morphism between cartesian fibrations is given by displayed functors that preserve cartesian maps. We will call these fibered functors.