Nelson Martins-Ferreira

A Note on the "Smith is Huq" Condition

We show that two known conditions which arose naturally in commutator theory and in the theory of internal crossed modules coincide: every star-multiplicative graph is multiplicative if and only if every two effective equivalence relations commute as soon as their normalisations do. This answers a question asked by George Janelidze.

Semidirect products and Split Short Five Lemma in normal categories

In this paper we study a generalization of the notion of categorical semidirect product, as defined in [6], to a non-protomodular context of categories where internal actions are induced by points, like in any pointed variety. There we define semidirect products only for regular points, in the sense we explain below, provided the Split Short Five Lemma between such points holds, and we show that this is the case if the category is normal, as defined in [12]. Finally, we give an example of a category that is neither protomodular nor Mal’tsev where such generalized semidirect products exist.

Further Remarks on the “Smith is Huq” Condition

We compare the Smith is Huq condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators in the sense of Gran, Janelidze and Ursini are independent of the chosen weight, which is known to be false for groups but turns out to be true in any two-nilpotent semi-abelian category.

On the “Smith is Huq” Condition in S-Protomodular Categories

We study the so-called “Smith is Huq” condition in the context of S-protomodular categories: two S-equivalence relations centralise each other if and only if their associated normal subobjects commute. We prove that this condition is satisfied by every category of monoids with operations equipped with the class S of Schreier split epimorphisms. Some consequences in terms of characterisation of internal structures are explored.

New Wide Classes of Weakly Mal’tsev Categories

The following classes of categories are shown to be weakly Mal’tsev in the sense of the author: (i) a suitable class of algebras with cancellation; (ii) the dual of any quasi-adhesive category; (iii) the dual of any extensive category with pullback-stable epimorphisms; (iv) the dual of any solid quasi-topos. The examples in (i) include all the Mal’tsev varieties of algebras such as groups, rings, Lie algebras, etc., but also distributive lattices and commutative monoids with cancellation. The examples in (ii)-(iv) capture many of the familiar aspects of topological spaces.

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

Abstract The notion of pseudocategory 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 from internal transformations in internal categories, con- jugations 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 geometric aspects of the 2-cell structures from an algebraic point of view.

A Decomposition Formula for the Weighted Commutator

We decompose the weighted subobject commutator of M. Gran, G. Janelidze and A. Ursini as a join of a binary and a ternary commutator.

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

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).

Normalized Bicategories Internal to Groups and more General Mal’tsev Categories

A detailed description of a normalized internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal’tsev categories endowed with a V-Mal’tsev operation in the sense of Pedicchio. The example of bicategory of paths in a topological abelian group is presented.

Internal monoids and groups in the category of commutative cancellative medial magmas

This article considers the category of commutative medial magmas with cancellation, a structure that generalizes midpoint algebras and commutative semigroups with cancellation. In this category each object admits at most one internal monoid structure for any given unit. Conditions for the existence of internal monoids and internal groups, as well as conditions under which an internal reflexive relation is a congruence, are studied.