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

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.

💡 Research Summary

The paper expands the notion of a pseudocategory from the traditional setting of a 2‑category to the more flexible framework of a sesquicategory, i.e., a category equipped with a 2‑cell structure that does not necessarily satisfy the full interchange law. After recalling the definition of a sesquicategory—objects, 1‑cells (arrows), and 2‑cells (transformations) together with horizontal and vertical compositions—the author points out that the usual definition of a pseudocategory relies on the strict coherence provided by a 2‑category. In a sesquicategory this coherence can break down, so the paper introduces an explicit “naturality” condition for the 2‑cell structure: for any 1‑cell h, the equations ε · h = h · ε (and similarly for other 2‑cells) must hold. This condition guarantees that the associator and unitors, now realized as 2‑cells, behave consistently under composition.

With this groundwork, the author defines a pseudocategory internal to a sesquicategory. The data consist of objects, source and target 1‑cells, a composition 1‑cell, and unit 1‑cells, together with two distinguished 2‑cells (often denoted ε and η) that encode the associativity and unit constraints. The naturality requirement ensures that these constraints satisfy the usual pentagon and triangle identities up to the prescribed 2‑cell structure, thereby preserving the essential coherence of a pseudocategory even without full interchange.

The paper then presents four families of examples that illustrate the breadth of the new definition. First, internal transformations in an internal category give rise to 2‑cells that are precisely the natural transformations between internal functors; their naturality is built‑in. Second, conjugation in a group provides a 2‑cell structure where each element acts by inner automorphism; the interchange law fails in general, but the conjugation 2‑cells satisfy the required naturality. Third, crossed modules admit derivations (or differentials) that can be viewed as 2‑cells; the compatibility of the derivation with the module action yields the naturality condition. Fourth, homotopies between chain maps in an abelian chain complex serve as 2‑cells; the standard homotopy relations guarantee the naturality of vertical and horizontal compositions. In each case the author checks that the sesquicategory axioms hold and that the pseudocategory axioms are satisfied, thereby validating the general theory.

A substantial portion of the work is devoted to analyzing the naturality condition itself. The author shows that when naturality fails, the resulting structure cannot support a coherent pseudocategory; conversely, imposing naturality can promote a sesquicategory to a “weak” 2‑category. The paper also discusses possible augmentations of the 2‑cell structure to enforce naturality, suggesting a systematic way to repair non‑natural sesquicategories.

In the concluding section the author summarizes the contributions: (1) a robust definition of pseudocategory internal to any category equipped with a 2‑cell structure, (2) a clear set of coherence (naturality) requirements that replace the strict interchange law, and (3) a suite of concrete examples spanning group theory, homological algebra, and higher category theory. The work opens several avenues for future research, including higher‑dimensional generalizations, connections with homotopical and homological invariants, and applications to non‑commutative algebraic structures such as quantum groups. Overall, the paper provides a solid algebraic foundation for studying geometric aspects of 2‑cell structures beyond the confines of strict 2‑categories.