Intercategories

TL;DR

This paper introduces intercategories, a new 3-dimensional categorical structure with unique composition laws and a non-invertible interchange, expanding the framework of weak triple categories.

Contribution

It defines intercategories, a novel weak triple category with non-invertible interchange, and develops their morphisms and coherence conditions.

Findings

Intercategories form a strict triple category.

They feature three types of arrows and cells with specific composition laws.

The interchange law is replaced by a non-invertible comparison cell.

Abstract

We introduce a 3-dimensional categorical structure which we call intercategory. This is a kind of weak triple category with three kinds of arrows, three kinds of 2-dimensional cells and one kind of 3-dimensional cells. In one dimension, the compositions are strictly associative and unitary, whereas in the other two, these laws only hold up to coherent isomorphism. The main feature is that the interchange law between the second and third compositions does not hold, but rather there is a non invertible comparison cell which satisfies some coherence conditions. We introduce appropriate morphisms of intercategory, of which there are three types, and cells relating these. We show that these fit together to produce a strict triple category of intercategories.