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We show how the notion of intercategory encompasses a wide variety ofthree-dimensional structures from the literature, notably duoidal categories,monoidal double categories, cubical bicategories, double bicategories and Graycategories. Variations on the notion of span provide further examples ofinterest, an important one being the intercategory of sets. We consider thethree kinds of morphism of intercategory as well as the cells binding them withapplications to the above structures. In particular hom functors are studied.

Intercategories: A framework for three-dimensional category theory