An intercategory is a particular sort of triple category that is pseudo in some ways and lax in others. Many different kinds of higher-categorical structures can be viewed as, or give rise to, intercategories. In particular, an intercategory can be viewed as a double category with “extra strictness/weakness information” given by the third direction.


The data of an intercategory is like that of a triple category: it has

All three kinds of arrows can be composed in the obvious way, all three kinds of 2-cell can be composed like in a double category, and cubes can be composed like in a triple category. Composition of transversal arrows, and transversal composition of horizontal and vertical squares and cubes, is strictly associative and unital. Horizontal and vertical composition is associative and unital up to coherent transversal isomorphism, as in a bicategory. In particular, the vertical cells and the horizontal cells each form the cells in a pseudo double category of the usual sort. Finally, composition of basic cells in the vertical and horizontal directions only satisfies the “interchange law” laxly (up to a not-necessarily-invertible comparison cell).

For a precise definition, see the references.