On the notion of pseudocategory internal to a category with a 2-cell structure

TL;DR

This paper extends the concept of pseudocategories from 2-categories to sesquicategories, exploring their algebraic and geometric properties through various examples of 2-cell structures.

Contribution

It generalizes the notion of pseudocategories to sesquicategories and investigates the naturality and behavior of 2-cells in this broader context.

Findings

Examples of 2-cells from internal transformations, conjugations, derivations, and homotopies.

Analysis of naturality conditions for 2-cell structures.

Foundations for geometric and algebraic study of 2-cell structures.

Abstract

The notion of pseudocategory, as considered in [11], is extended from the context of a 2-category to the more general one of a sesquicategory, which is considered as a category equipped with a 2-cell structure. Some particular examples of 2-cells arising form internal transformations in internal categories, conjugations in groups, derivations in crossed-modules or homotopies in abelian chain complexes are studied in this context, namely their behaviour as abstract 2-cells in a 2-cell structure. Issues such as naturality of a 2-cell structure are investigated. This article is intended as a preliminary starting work towards the study of the geometrical aspects of the 2-cell structures from an algebraic point of view.