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

Variations of the Shifting Lemma and Goursat categories

We prove that Mal'tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences R, S and T, and characterizes congruence modular varieties. We first show that a regular category C is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in C. Moreover, we prove that a regular category C is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation S and reflexive and positive relations R and T in C. In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras.European Regional Development FundEuropean Union (EU)Fonds de la Recherche Scientifique-FNRS Credit Bref Sejour a l'etrangerFonds de la Recherche Scientifique - FNRS [2018/V 3/5/033-IB/JN-11440

Selected topics in Goursat categories

Over the last thirty years, many interesting results have been discovered in the categorical extension of 2-permutable varieties, called Mal’tsev categories. Many of these results still hold in regular categories satisfying the strictly weaker property of 3-permutability, called Goursat categories. A nice feature of regular Mal’tsev and Goursat categories is that Gumm’s Shifting Lemma holds in these categories, a property which allowed, for example, to develop commutator theory in universal algebras. The aim of this thesis is twofold: on the one hand, we extend to Goursat categories the main results obtained for the theory of projective covers and internal structures in Mal’tsev categories. In particular, we give some characterizations of the categories which are the projective covers of Goursat categories. Then, we show that the structure of internal connector is stable under quotients in any Goursat category. As a consequence, the category of internal connectors in a Goursat category is again a Goursat category. This implies that Goursat categories can be characterized in terms of a simple property of internal groupoids and internal categories. On the other hand, we study the Shifting Lemma in regular Mal’tsev and Goursat categories. We prove that regular Mal’tsev and Goursat categories can be characterized through suitable variations of the Shifting Lemma. We also investigate two properties related to the Shifting Lemma and called the Triangular Lemma and the Trapezoid Lemma in universal algebras. We establish some characterizations of regular Mal’tsev and Goursat categories with distributive lattice of equivalence relations through variations of the Triangular Lemma and Trapezoid Lemma.(SC - Sciences) -- UCL, 202

Goursat completions

We characterize categories with weak finite limits whose regular completions give rise to Goursat categories

Facets of congruence distributivity in Goursat categories

We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations. (C) 2020 Elsevier B.V. All rights reserved.Centre for Mathematics of the University of Coimbra - Portuguese Government through FCT/MCTES [UIDB/00324/2020]Fonds de la Recherche Scientifique -FNRS Credit Sejour a l'etranger [2020/V 3/5/036 - 35714631 - JG/MF - 6977]info:eu-repo/semantics/publishedVersio