Some remarks on connectors and groupoids in goursat categories

We prove that connectors are stable under quotients in any (regular) Goursat category. As a consequence, the category Conn(C) of connectors in C is a Goursat category whenever C is. This implies that Goursat categories can be characterised in terms of a simple property of internal groupoids.Portuguese Government through FCT/MCTES; European Regional Development Fun

Monoidal characterisation of groupoids and connectors

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids

Higher commutator conditions for extensions in Mal'tsev categories

We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed modules of Lie algebras, and compact groups.Comment: 32 page

Comprehensive factorization and I-central extensions

We show that, for a regular reflection functor I between efficiently regular categories, the reflection of an extension to an I-central extension is reduced to the comprehensive factorization of an explicit internal functor. We then analyse the Mal'tsev context where similar results are obtained under weaker conditions on I. (C) 2011 Elsevier B.V. All rights reserved.FCT/Centro de Matematica da Universidade de Coimbr

Relative Commutator Theory in Semi-Abelian Categories

Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.Comment: 22 page

A NOTE ON THE CATEGORICAL NOTIONS OF NORMAL SUBOBJECT AND OF EQUIVALENCE CLASS

In a non-pointed category E, a subobject which is normal to an equivalence relation is not necessarily an equivalence class. We elaborate this categorical distinction, with a special attention to the Mal'tsev context. Moreover, we introduce the notion of fibrant equipment, and we use it to establish some conditions ensuring the uniqueness of an equivalence relation to which a given subobject is normal, and to give a description of such a relation

Higher central extensions and cohomology

We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This allows us to obtain a duality, in a certain sense, between "internal" homology and "external" cohomology in semiabelian categories. These results depend on a geometric viewpoint of the concept of a higher central extension, as well as the algebraic one in terms of commutators. (C) 2015 Elsevier Inc. All rights reserved