[0010] Construction 2.9·a (The full subfibration associated to a displayed object).

Let $E$ be a cartesian fibration over $B$; then any displayed object $\bar{x} \in E\Sub{x}$ induces a full subfibration $\FullSubfib{\bar{u}}\subseteq E$ spanned by displayed objects that are classified by $\bar{u}$, i.e. arise from $\bar{u}$ by cartesian lift.

  1. An object of $\FullSubfib{\bar{u}}\Sub{x}$ is specified by an object $\bar{x}\in E\Sub{x}$ together with a cartesian morphism $\bar{x}\to \bar{u}$.

  2. Given $f:x\to y\in B$, a morphism from $\bar{x}\to \bar{u}$ to $\bar{y}\to\bar{u}\in$ over $f$ is given by any displayed morphism $\bar{x}\to\Sub{f}\bar{y}$.