Low-dimensional internal categorial structures in weakly Mal'cev sesquicategories

In this work, the author introduces pseudocategory as a generalization for an internal category in dimension 2. First, a pseudocategory is defined [Ch1] as a system, consisting of a precategory diagram together with special 2-cells in a 2-category, satisfying some coherence conditions: if the 2-category is of the form Cat(B), of internal categories, internal functors and internal natural transformations in some category B, then a pseudocategory in (internal to) Cat(B) simultaneously generalizes internal bicategory in B and internal double-category in B (it is a pseudo-doublecategory in B, using the terminology of M. Grandis and R. Pare); later, a pseudocategory is considered in the more general context of a sesquicategory [Ch2], with one of the main results of this thesis being the description of pseudocategories in (internal to) a weakly Mal’cev sesquicategory [Ch9]. The notions of weakly Mal’cev category and weakly Mal’cev sesquicategory are also new concepts that are introduced here. Weakly Mal’cev categories, generalize Mal’cev categories, and seem to be an appropriate setting for the study of internal categories and precategories: an internal category (here, as in a Mal’cev category) is completely determined by its underlying reflexive graph; but (here, unlike in a Mal’cev category) not every internal category is an internal groupoid [Ch3]. A weakly Mal’cev sesquicategory is specially designed to mimic, as an axiomatic abstraction, a sesquicategory of the form Cat(B), with the 2-cell structure given by internal transformations, not necessarily natural, and B a weakly Mal’cev category [Ch8]. In fact, if thinking that a weakly Mal’cev category is a kind of partially enriched category, in the sense that there are many partial ternary operations in each hom-set, then a sesquicategory is weakly Mal’cev when the 2-cells are also enriched in the previous sense. In [Ch1] and [Ch6] the name tetracategory is used for a structure with objects, morphisms, 2-cells, pseudo-cells, and tetra-cells, obtained as an abstraction for the category PsCat of pseudocategories, pseudofunctors, natural transformations, pseudo-natural transformations and modifications. An equivalence of categories PsCat(A)∼PsMor(A) is proved in [Ch6] between PsCat(A) the category of pseudocategories in A, an additive 2-category with kernels [Ch5], and PsMor(A) an ad hoc category of “pseudomorphisms” in A. This result may be seen as the 2-dimensional analogue for the well known equivalence Cat(A)∼Mor(A) if A is just an additive category with kernels. In particular, for the case A=Cat(Ab), the general result gives us a description for the tetracategory of internal bicategories in abelian groups, where it is immediate to observe that homotopies between 2-chains indeed correspond to pseudo-natural transformations. At the end, a final note is added for the case of internal bicategories in Groups, that can be derived from the more general result of pseudocategories in a weakly Mal’cev sesquicategory, taking the sesquicategory of crossed-modules with derivations as 2-cells. 1Troughout this thesis a reference such as [Chn] refers to Chapter number n.