On categories with arbitrary 2-cell structures

TL;DR

This paper investigates the structure of categories equipped with arbitrary 2-cell structures, emphasizing their significance in understanding the intrinsic features of the base category, with a focus on monoids as a key example.

Contribution

It explores the category of all 2-cell structures as sesquicategories, highlighting their importance beyond 2-categories and providing insights through the example of monoids.

Findings

2-cell structures form sesquicategories but not necessarily 2-categories

The study reveals intrinsic features of the base category related to 2-cell structures

Monoids as one-object categories have 2-cell structures similar to semibimodules

Abstract

When a category is equipped with a 2-cell structure it becomes a sesquicategory but not necessarily a 2-category. It is widely accepted that the latter property is equivalent to the middle interchange law. However, little attention has been given to the study of the category of all 2-cell structures (seen as sesquicategories with a fixed underlying base category) other than as a generalization for 2-categories. The purpose of this work is to highlight the significance of such a study, which can prove valuable in identifying intrinsic features pertaining to the base category. These ideas are expanded upon through the guiding example of the category of monoids. Specifically, when a monoid is viewed as a one-object category, its 2-cell structures resemble semibimodules.