The (Tetra) Category of Pseudocategories in an Additive 2-category with Kernels

Abstract

We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47–78, 2006), internal to an additive 2-category with kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387–410, 2004). In the context of a 2-Ab-category, we introduce the notion of a pseudo-morphism and prove the equivalence of categories: PsCat(A)~PsMor(A) between pseudo-categories and pseudo-morphisms in an additive 2-category, A, with kernels– extending thus the well known equivalence Cat(Ab)~Mor(Ab) between internal categories and morphisms of abelian groups. The leading example of an additive 2-category with kernels is Cat(Ab). In the case A=Cat(Ab) we obtain a description of the (tetra) category of internal pseudo-double categories in Ab, and particularize it to a description of the (tetra) category of internal bicategories in abelian groups. As expected, pseudo-natural transformations coincide with homotopies of 2-chain complexes (as in Bourn, J Pure Appl Algebra 66:229–249, 1990).