Galois theory and a general notion of central extension

Janelidze, George; Kelly, G. M.

doi:10.1016/0022-4049(94)90057-4

Cited by 118 publications

(218 citation statements)

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“…We follow [19] in calling a (regular epi)-reflective subcategory X of an exact category C a Birkhoff subcategory if, moreover, X is closed in C under regular quotients. where U : X → C is a full inclusion.…”

Section: Reflections For Goursat Categoriesmentioning

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: Reflections For Goursat Categoriesmentioning

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.…”

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confidence: 99%

Beck–Chevalley condition and Goursat categories

Gran

Rodelo

2017

Journal of Pure and Applied Algebra

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“…Birkhoff subcategories. The notion of central extension introduced in [19] is defined with respect to a chosen subcategory B of the base category A: a Birkhoff subcategory B of a semi-abelian category A is a full and reflective subcategory, closed under subobjects and regular quotients. We denote the left adjoint by I : A Ñ B and write the components of its unit η A : A Ñ IpAq.…”

Section: Semi-abelian Categoriesmentioning

confidence: 99%

“…In this article we explain how, using Janelidze and Kelly's general notion of central extension [19], the classical theory of universal central extensions may be considered in the context of semi-abelian categories [20]. Our main point is that, while most of the results valid for groups and Lie algebras (see, for instance, [23,24]) generalise without any difficulty to extensions, central with respect to a chosen Birkhoff subcategory B of a semi-abelian category A, this setting turns out to be too weak for some of the most basic results, valid in the classical examples, to hold-even when B " AbpAq is determined by the abelian objects in A.…”

Section: Introductionmentioning

confidence: 99%