Some remarks on Maltsev and Goursat categories

Carboni, A.; Kelly, G. M.; Pedicchio, Maria Cristina

doi:10.1007/bf00872942

Cited by 125 publications

(217 citation statements)

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“…Recall that a regular category C is called a Goursat category [10] when the composition of (effective) equivalence relations R and S on a same object in C is 3-permutable: RSR = SRS. Given a relation R = (R, r 1 , r 2 ) from an object X to an object Y , we write R o for the opposite relation (R, r 2 , r 1 ) from Y to X.…”

Section: Goursat Categoriesmentioning

confidence: 99%

“…The proof of these results also relies on the fact that the Shifting Lemma [16] holds in any regular Goursat category [6], since the lattice of equivalence relations on any object is modular [10]. A more general characterisation of Goursat categories among regular categories is also obtained without requiring the existence of pushouts along split monomorphisms and involves a stability property for regular epimorphisms (Theorem 1.3).…”

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confidence: 99%

“…The weaker 3-permutability of congruences RSR = SRS, which defines 3-permutable varieties, is also equivalent to the existence of two ternary operations r and s such that the identities r(x, y, y) = x, r(x, x, y) = s(x, y, y) and s(x, x, y) = y hold [17]. A nice feature of 3-permutable varieties is the fact that they are congruence modular, a condition that plays a crucial role in the development of commutator theory [12,16].Many interesting results have been discovered in regular Mal'tsev categories [11] and in regular Goursat categories [10], which can be seen as the categorical extensions of Mal'tsev varieties and of 3-permutable varieties, respectively. The interested reader will find many properties…”

mentioning

confidence: 99%

“…Many interesting results have been discovered in regular Mal'tsev categories [11] and in regular Goursat categories [10], which can be seen as the categorical extensions of Mal'tsev varieties and of 3-permutable varieties, respectively. The interested reader will find many properties of these categories in the references [2,3,5,9,10,11,13,14,19,20,21,24], for instance.…”

mentioning

confidence: 99%

“…The interested reader will find many properties of these categories in the references [2,3,5,9,10,11,13,14,19,20,21,24], for instance.…”

mentioning

confidence: 99%

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Beck–Chevalley condition and Goursat categories

Gran

Rodelo

2017

Journal of Pure and Applied Algebra

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Abstract. We 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.A variety of universal algebras is called a Mal'tsev variety [25] when any pair of congruences R and S on the same algebra 2-permute, meaning that RS = SR. The celebrated Mal'tsev theorem asserts that the algebraic theory of such a variety is characterised by the existence of a ternary term p(x, y, z) such that the identities p(x, y, y) = x and p(x, x, y) = y hold [22]. The weaker 3-permutability of congruences RSR = SRS, which defines 3-permutable varieties, is also equivalent to the existence of two ternary operations r and s such that the identities r(x, y, y) = x, r(x, x, y) = s(x, y, y) and s(x, x, y) = y hold [17]. A nice feature of 3-permutable varieties is the fact that they are congruence modular, a condition that plays a crucial role in the development of commutator theory [12,16].Many interesting results have been discovered in regular Mal'tsev categories [11] and in regular Goursat categories [10], which can be seen as the categorical extensions of Mal'tsev varieties and of 3-permutable varieties, respectively. The interested reader will find many properties

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Section: Goursat Categoriesmentioning

confidence: 99%

mentioning

confidence: 99%