2.8. Change of base [0007]

Suppose that $E$ is displayed over $B$ and $F : X\to B$ is a functor; then we may define a displayed category $\InvImg{F}E$ as over $X$ follows:

  1. An object of $(\InvImg{F}E)\Sub{x}$ is an object of $E\Sub{Fx}$.

  2. Given $\bar{x}\in (\InvImg{F}E)\Sub{x}$, $\bar{y}\in (\InvImg{F}E)\Sub{y}$ and $f : x \to y$, a morphism $\bar{x}\to\Sub{f}\bar{y}$ in $\InvImg{F}E$ is given by a morphism $\bar{x}\to\Sub{Ff}\bar{y}$ in $E$.

We visualize the change of base scenario as follows: