On pseudocategories in a category with a 2-cell structure

For a given (fixed) category, we consider the category of all 2-cell structures (over it) and study some naturality properties. A category with a 2-cell structure is a sesquicategory; we use additive notation for the vertical composition of 2-cells; instead of a law for horizontal composition we consider a relation saying which pairs of 2-cells can be horizontally composed; for a 2-cell structure with every 2-cell invertible, we also consider a notion of commutator, measuring the obstruction for horizontal composition. We compare the concept of naturality in an abstract 2-cell structure with the example of internal natural transformations in a category of the form Cat(B), of internal categories in some category B, and show that they coincide. We provide a general construction of 2-cell structures over an arbitrary category, under some mild assumptions. In particular, the canonical 2-cell structures over groups and crossed-modules, respectively “conjugations” and “derivations”, are instances of these general constructions. We define cartesian 2-cell structure and extend the notion of pseudocategory from the context of a 2-category (as in [6]) to the more general context of a sesquicategory. As an example of application we consider pseudocategories in the sesquicategory of abelian chain complexes.