[0000] Definition 2·a (Displayed category).

Let $B$ be a category. A displayed category $E$ over $B$ is defined by the following data (Ahrens & Lumsdaine, 2019):

  1. for each object $x\in B$, a collection of displayed objects $E\Sub{x}$,
  2. for each morphism $f : x \to y\in B$ and displayed objects $\bar{x}\in E\Sub{x}$ and $\bar{y}\in E\Sub{y}$, a family of collections of displayed morphisms $E\Sub{f}(\bar{x},\bar{y})$,
  3. for each $x\in B$ and $\bar{x}\in E\Sub{x}$, a morphism $\Idn{\bar{x}} \in E\Sub{\Idn{x}}(\bar{x},\bar{x})$ which we may also write $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$,
  4. for each $f : x \to y$ and $g:y \to z$ in $B$ and objects $\bar{x}\in E\Sub{x}, \bar{y}\in E\Sub{y}, \bar{z}\in E\Sub{z}$, a function \[ E\Sub{f}(\bar{x},\bar{y}) \times E\Sub{g}(\bar{y},\bar{z}) \to E\Sub{f;g}(\bar{x},\bar{z}) \] that we will denote like ordinary (diagrammatic) function composition,
  5. such that the following equations hold: \[ \Idn{\bar{x}};\bar{h} = \bar{h}\qquad \bar{h};\Idn{\bar{y}} = \bar{h}\qquad \bar{f};(\bar{g};\bar{h}) = (\bar{f};\bar{g});\bar{h} \] Note that these are well-defined because of the corresponding laws for the base category $B$.

Notation. When we have too many subscripts, we will write $E[x]$ instead of $E\Sub{x}$.