Mal’cev, Protomodular, Homological and Semi-Abelian Categories

Borceux, Francis; Bourn, Dominique

doi:10.1007/978-1-4020-1962-3

Cited by 228 publications

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“…As recalled before, in this case, the kernel functor takes values in the base category C. Moreover, when C is protomodular, regular (or Barrexact), then also Pt A (C) is protomodular, regular (or Barr-exact) respectively (see [4], for instance). These circumstances suggest to investigate the exactness properties of kernel functors, in the sense of homological algebra.…”

Section: Exactness Properties Of Kernel Functorsmentioning

confidence: 92%

“…If they are in fact pointed, they are called homological, and they are termed semiabelian when they are also Barr exact and with finite coproducts (see [4]). …”

Section: Action Of Quotientsmentioning

confidence: 99%

“…A relevant application of this definition is related to the fibration of points. When all the change of base functors are normal, the category is called strongly-protomodular (see [6,4]). In the presence of initial object, it suffices to consider the kernel functors K A , for every object A.…”

Section: Strong Protomodularitymentioning

confidence: 99%

“…Clearly Pt A (C) is pointed, moreover it is protomodular and regular (see [4]) as C is. Actually, as showed in the proof of point 1, γ = coker(ϕ · k c ) in C underlies a regular epimorphism.…”

Section: Exactness Properties Of Kernel Functorsmentioning

confidence: 99%

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A note on strong protomodularity, actions and quotients

Metere

2017

Journal of Pure and Applied Algebra

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In order to study the problems of extending an action along a quotient of the acted object and along a quotient of the acting object, we investigate some properties of the fibration of points. In fact, we obtain a characterization of protomodular categories among quasi-pointed regular ones, and, in the semi-abelian case, a characterization of strong protomodular categories. Eventually, we return to the initial questions by stating the results in terms of internal actions. IntroductionThe present work originates from the investigation of the categorical properties related to two well-known features of group actions. Actions on quotientsSuppose we are given a pair (ξ, g):where ξ is a left-action of groups, and g is a surjective homomorphism. We discuss the following problem: under what conditions does the action ξ induces an action on the quotient Z? Indeed, it is not difficult to see that ξ is well-defined on the cosets of Y mod X = Ker(g), precisely when it is well-defined on the 0-coset X, i.e. when it restricts to X. We shall state this property as follows:(KC) An action passes to the quotient if, and only if, it restricts to the kernel. Action of quotientsSuppose now that we are given a group action ξ as before, and a surjective group homomorphism q : A → Q. A natural question arises: when does the given A-action induce a Q-action? In this case, the restriction of the action ξ to the kernel K of q always exists, and the condition under which the action of the quotient is well defined, amounts to the fact that the kernel of q acts trivially. 1These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category (see [10]). Indeed, we will show that the property (KC) characterizes strongly protomodular categories among semi-abelian categories, and that, in such contexts, actions of quotients behave substantially in the same way as in the case of groups.On the other hand these issues can be dealt with also in more general contexts. Indeed, when an object A acts on object X, just like in the case of group, one can consider the split epimorphism X ⋊ A → A given by the semidirect product projection together with its canonical section. Vice-versa, any split epimorphism with codomain A gives rise to the conjugation A-action on the kernel of the split epimorphism.This allows to formulate our issues in terms of split epimorphisms, or points, even in contexts where the machinery of internal actions is not at all available. This line of investigation will lead us to the study of some new classifying aspects of the fibration of points. In particular, with Proposition 3.3, we will give a characterization of protomodular categories among quasi-pointed regular ones as those with kernel functors that reflect short exact sequences. Then, we will show that the problem of extending actions along quotients translates (in term of points) in a property closely connected with strong protomodularity, i.e. the fact that kernel functors reflect kernels In fact, th...

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Section: Exactness Properties Of Kernel Functorsmentioning

confidence: 92%

“…If they are in fact pointed, they are called homological, and they are termed semiabelian when they are also Barr exact and with finite coproducts (see [4]). …”

Section: Action Of Quotientsmentioning

confidence: 99%

Section: Strong Protomodularitymentioning

confidence: 99%

Section: Exactness Properties Of Kernel Functorsmentioning

confidence: 99%

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A note on strong protomodularity, actions and quotients

Metere

2017

Journal of Pure and Applied Algebra

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“…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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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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An Invitation to Topological Semi-abelian Algebras

Clementino

2021

New Perspectives in Algebra, Topology and Categories

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Mal’cev, Protomodular, Homological and Semi-Abelian Categories

Cited by 228 publications

References 0 publications

A note on strong protomodularity, actions and quotients

A note on strong protomodularity, actions and quotients

Beck–Chevalley condition and Goursat categories

An Invitation to Topological Semi-abelian Algebras

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