Intercategories: A framework for three-dimensional category theory

TL;DR

This paper introduces intercategories as a unifying framework for various three-dimensional categorical structures, providing new insights and tools for their analysis and applications.

Contribution

It formalizes the concept of intercategories, showing how they encompass many existing three-dimensional categories and related structures, with detailed morphism and cell analysis.

Findings

Intercategories unify multiple 3D categorical structures.

Hom functors are systematically studied within this framework.

Examples include duoidal categories, monoidal double categories, and Gray categories.

Abstract

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