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- A criterion for reflectiveness of normal extensionsPublication . Montoli, Andrea; Rodelo, Diana; Van der Linden, TimWe give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when X is the (protomodular) reflective subcategory of S-special objects of a Barr-exact S-protomodular category C, where S is the class of split epimorphic trivial extensions in C. Next to some concrete examples where the criterion may be applied, we also study the adjunction between a Barr-exact unital category and its abelian core, which we prove to be admissible.
- Some remarks on connectors and groupoids in goursat categoriesPublication . Gran, Marino; Nguefeu, Idriss Tchoffo; Rodelo, DianaWe 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.
- Internal structures in n-permutable varietiesPublication . Rodelo, DianaWe analyze the notions of reflexive multiplicative graph, internal category and internal groupoid for n-permutable varieties. (C) 2012 Elsevier B.V. All rights reserved.
- Beck-Chevalley condition and Goursat categoriesPublication . Gran, Marino; Rodelo, DianaWe characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising 3-permutable varieties of universal algebras.
- Stability properties characterising n- permutable categoriesPublication . Jacqmin, Pierre-Alain; Rodelo, DianaThe purpose of this paper is two-fold. A first and more concrete aim is to characterise n-permutable categories through certain stability properties of regular epimorphisms. These characterisations allow us to recover the ternary terms and the (n + 1)-ary terms describing n-permutable varieties of universal algebras. A second and more abstract aim is to explain two proof techniques, by using the above characterisation as an opportunity to provide explicit new examples of their use: an embedding theorem for n-permutable categories which allows us to follow the varietal proof to show that an n-permutable category has certain properties; the theory of unconditional exactness properties which allows us to remove the assumption of the existence of colimits, in particular when we use the approximate co-operations approach to show that a regular category is n-permutable.
- Variations of the Shifting Lemma and Goursat categoriesPublication . Gran, Marino; Rodelo, Diana; Nguefeu, Idriss TchoffoWe 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.
- 3 x 3 lemma for star-exact sequencesPublication . Gran, Marino; Janelidze, Zurab; Rodelo, DianaA regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 x 3 lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with "exact forks" and then, the corresponding 3 x 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such 3 x 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.
- Comprehensive factorization and I-central extensionsPublication . Bourn, Dominique; Rodelo, DianaWe 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.
- Higher central extensions and cohomologyPublication . Rodelo, Diana; Van der Linden, TimWe 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.