Definition. Intercategory [mde-FWX6]

An intercategory is a form of triple category structure, with many examples in nature.

An intercategory has morphisms in three dimensions, usually called transverse, horizontal and vertical.

Transverse and horizontal morphisms are the boundaries of horizontal cells, which form a pseudo double category. Similarly, transverse and vertical morphisms are the boundaries of vertical cells, which also form a pseudo double category. Horizontal and vertical cells are the boundaries of basic cells. Horizontal, vertical, and basic cells are the boundaries of cubes. Diagrammatically we have the following,

Three equivalent definitions are given in grandis-paré-2015,

1. A pseudocategory in the 2-category of pseudo double categories, lax functors and loose transformations.
2. A pseudocategory in the 2-category of pseudo double categories, colax functors and loose transformations.
3. A double pseudocategory in the 2-category of categories, functors and natural transformations.

The these give rise to different notions of morphism of intercategory, leading to a strict triple category of intercategories.

An intercategory is more strict than a weak triple category in the sense that there is no dimension in which associativity and unitality hold only up to equivalence. Instead, the two weak dimensions form pseudo double categories. It is however more lax than a weak triple category in the sense that the interchanger between horizontal and vertical dimensions need not be an isomorphism.