Diagram chasing in Mal'cev categories

Carboni, A.; Lambek, J.; Pedicchio, Maria Cristina

doi:10.1016/0022-4049(91)90022-t

Cited by 141 publications

(133 citation statements)

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“…The aim of their definition is two-fold: first of all, to provide a categorical-algebraic characterisation of groups amongst monoids as the protomodular objects in the category Mon of monoids; and secondly, to establish an object-wise approach to certain important conditions occurring in categorial algebra such as protomodularity [2, 1] and the Mal'tsev axiom [5,6].…”

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confidence: 99%

A New Characterisation of Groups Amongst Monoids

García-Martínez

2016

Appl Categor Struct

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Abstract. We prove that a monoid M is a group if and only if, in the category of monoids, all points over M are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden [8] which characterises groups amongst monoids as the protomodular objects.In their article [8], Montoli, Rodelo and Van der Linden introduce, amongst other things, the concept of a protomodular object in a finitely complete category C as an object Y P C over which all points are stably strong. The aim of their definition is two-fold: first of all, to provide a categorical-algebraic characterisation of groups amongst monoids as the protomodular objects in the category Mon of monoids; and secondly, to establish an object-wise approach to certain important conditions occurring in categorial algebra such as protomodularity [2, 1] and the Mal'tsev axiom [5,6].We briefly recall some basic definitions; see [3,7,8] for more details. Let C be a finitely complete category, which we also take to be pointed for the sake of simplicity. In C , a pair of arrows pr : W Ñ X, s : Y Ñ Xq is jointly strongly epimorphic when if mr 1 " r, ms 1 " s for some given monomorphism m : M Ñ X and arrows r 1 : W Ñ M , s 1 : Y Ñ M , then m is an isomorphism. In the case of monoids, this means that any x P X can be written as a product rpw 1 qspy 1 q¨¨¨rpw n qspy n q for some w j P W , y j P Y . This characterisation follows easily from the fact that pr, sq is a jointly strongly epimorphic pair in Mon if and only if the induced monoid morphism W`Y Ñ X is a surjection-see, for instance, [1, Corollary A.5.4 combined with Example A.5.16]. Given an object Y in C , a point over Y is a pair of morphisms pf : X Ñ Y , s : Y Ñ Xq such that f s " 1 Y . A point pf, sq is said to be strong when the pair pkerpf q : Kerpf q Ñ X, s : Y Ñ Xq is jointly strongly epimorphic. The point pf, sq is stably strong when all of its pullbacks are strong. More precisely, if g : Z Ñ Y is any morphism, then the pullback g˚pf q together with its splitting induced by s is a strong point.Even though the concept of a protomodular object serves the intended purpose of characterising groups amongst monoids, the proof of this characterisation given in [8] is rather complicated, since it relies on another, more subtle, characterisation in terms of the so-called Mal'tsev objects. The present short note aims to improve the situation by giving a quick and direct proof of a more general result: a monoid is a group as soon as all points over it are strong. Theorem. A monoid M is a group if and only if, in Mon, all points over M are strong.

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confidence: 99%

A New Characterisation of Groups Amongst Monoids

García-Martínez

2016

Appl Categor Struct

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“…In this paper we prove the following result, which is well known in the case when n = 2 (see [6,4]): Theorem 1.1. For a regular category C, and a natural number n 1, the following conditions are equivalent:…”

Section: Introductionmentioning

confidence: 78%

“…Recall that 2-permutable categories are also called Mal'tsev categories [4] while 3-permutable categories are known as Goursat categories [4,3].…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Hagemann’s theorem for regular categories

Janelidze

Rodelo

Linden

2013

J. Homotopy Relat. Struct.

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In this paper, we extend the characterisation of n-permutable varieties of universal algebras due to J. Hagemann to regular categories. In particular, we show that a regular category has n-permutable congruences if and only if every internal reflexive relation R in it satisfies R • R n−1 , and if and only if every internal reflexive relation R in it satisfies R n R n−1 . In the case when n = 2 this result is well known.

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“…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%

“…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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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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Diagram chasing in Mal'cev categories

Cited by 141 publications

References 9 publications

A New Characterisation of Groups Amongst Monoids

A New Characterisation of Groups Amongst Monoids

Hagemann’s theorem for regular categories

Beck–Chevalley condition and Goursat categories

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