Triple categories

In a previous post, we described double categories. It was mentioned that double categories are categories internal to the category of categories $Cat$. This process of internalising categories in categories can be repeated $n$ times to produce $n$-fold categories. Repeating this process three times produces triple categories.

Definition:
A triple category $C$ is an internal category in double categories. A triple category has the following six components:

• A double category of objects $Ob(C)$
• A double category of arrows $Arrows(C)$
• A source double functor $source: Arrows(C) \to Ob(C)$
• A target double functor $target: Arrows(C) \to Ob(C)$
• An identity double functor $identity: Ob(C) \to Arrows(C)$
• A composition double functor $\circ: Arrows(C) \times_{Ob(C)} Arrows(C) \to Arrows(C)$

Alternatively, a triple category can be constructed from eight different types of cells with composition, source, target, and identity morphisms coming in different directions. So for example, triple categories have vertical, horizontal, and transversal types of composition. This stands in contrast to elementary categories, which only have composition defined in one direction.

Cells:
A triple category has eight different types of cells constructed from combining three different types of arrows in three different dimensions.

• Objects
• Arrows
  • Vertical arrows
  • Horizontal arrows
  • Transversal arrows
• Squares
  • Vertical-Horizontal squares
  • Vertical-Transversal squares
  • Horizontal/Transversal squares
• Cubes

A triple category $C$ has three different double categories of squares and three different categories of arrows associated to it.

References:
Triple category