Some remarks on pullbacks in Gumm categories

We extend some properties of pullbacks which are known to hold in a Mal'tsev context to the more general context of Gumm categories. The varieties of universal algebras which are Gumm categories are precisely the congruence modular ones. These properties lead to a simple alternative proof of the known property that central extensions and normal extensions coincide for any Galois structure associated with a Birkhoff subcategory of an exact Goursat category.Comment: 12 page

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

On closure operators and reflections in Goursat categories

By defining a closure operator on effective equivalence relations in a regular category $C$, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories $L$ of $C$. When $C$ is an exact Goursat category this correspondence restricts to a bijection between the Birkhoff closure operators on effective equivalence relations and the Birkhoff subcategories of $C$. In this case it is possible to provide an explicit description of the closure, and to characterise the congruence distributive Goursat categories.Comment: 14 pages. Accepted for publication in "Rendiconti dell'Istituto Matematico di Trieste

Effective descent morphisms of regular epimorphisms

Let $A$ be a regular category with pushouts of regular epimorphisms by regular epimorphism and $Reg(A)$ the category of regular epimorphisms in $A$. We prove that every regular epimorphism in $Reg(A)$ is an effective descent morphism if, and only if, $Reg(A)$ is a regular category. Then, moreover, every regular epimorphism in $A$ is an effective descent morphism. This is the case, for instance, when $A$ is either exact Goursat, or ideal determined, or is a category of topological Mal'tsev algebras, or is the category of $n$-fold regular epimorphisms in any of the three previous cases, for any $n\geq 1$

An observation on n-permutability

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is n-permutable for some n.Comment: 6 page

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

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

Internal structures in n-permutable varieties

We analyze the notions of reflexive multiplicative graph, internal category and internal groupoid for n-permutable varieties. (C) 2012 Elsevier B.V. All rights reserved.CMUC; FCT (Portugal) through European Program COMPETE/FEDERinfo:eu-repo/semantics/publishedVersio