Central reflections and nilpotency in exact Mal’tsev categories

Berger, Clemens; Bourn, Dominique

doi:10.1007/s40062-016-0165-8

Cited by 12 publications

(17 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The two are known to be different in general; for instance, in the context of loops, the definition of Bruck [6] is of the former kind, while the commutator-associator filtration of Mostovoy [27,26] and his co-authors is of the latter type. Another example, in the context of Moufang loops, is given in [1].In this article, we show that the two streams of development agree in any algebraically coherent semi-abelian category. Such are, for instance, all Orzech categories of interest [31].…”

mentioning

confidence: 66%

“…

…”

mentioning

confidence: 99%

See 1 more Smart Citation

On the "three subobjects lemma" and its higher-order generalisations

Simeu,

Van der Linden

2019

Preprint

View full text Add to dashboard Cite

We solve a problem mentioned in the article [1] of Berger and Bourn: we prove that in the context of an algebraically coherent semi-abelian category, two natural definitions of the lower central series coincide.In a first, "standard" approach, nilpotency is defined as in group theory via nested binary commutators of the form rrX, Xs, Xs. In a second approach, higher Higgins commutators of the form rX, X, Xs are used to define nilpotent objects [18,16,20]. The two are known to be different in general; for instance, in the context of loops, the definition of Bruck [6] is of the former kind, while the commutator-associator filtration of Mostovoy [27,26] and his co-authors is of the latter type. Another example, in the context of Moufang loops, is given in [1].In this article, we show that the two streams of development agree in any algebraically coherent semi-abelian category. Such are, for instance, all Orzech categories of interest [31]. Our proof of this result is based on a higher-order version of the Three Subobjects Lemma of [8], which extends the classical Three Subgroups Lemma from group theory to categorical algebra. It says that any n-fold Higgins commutator rK 1 , . . . , Kns of normal subobjects K i X may be decomposed into a join of nested binary commutators.

show abstract

mentioning

confidence: 66%

“…

…”

mentioning

confidence: 99%

On the "three subobjects lemma" and its higher-order generalisations

Simeu,

Van der Linden

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The homotopy nilpotency and co-nilpotency 3 Before describing our purpose, it is worthwhile to mention a few recent contributions along these lines. Berger and Bourn [4] study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus.…”

Section: Introductionmentioning

confidence: 99%

A survey of homotopy nilpotency and co-nilpotency

Golasiński

2020

PIGC

View full text Add to dashboard Cite

We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces.Next, we take up the systematic study of homotopy nilpotency of homogenous spaces $G/K$ for a Lie group $G$ and its closed subgroup $K<G$.The homotopy nilpotency of the loop spaces $\Omega (G_{n,m}(\mathbb{K}))$ and $\Omega (V_{n,m}(\mathbb{K}))$ of Grassmann $G_{n,m}(\mathbb{K})$and Stiefel $V_{n,m}(\mathbb{K})$ manifolds for $\mathbb{K}=\mathbb{R},\,\mathbb{C}$, the field of reals or complex numbers and $\mathbb{H}$, the skew$\mathbb{R}$-algebra of quaternions is shown.

show abstract

“…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 present paper shows that a suitable Beck-Chevalley condition characterises Goursat categories. Independently, Clemens Berger and Dominique Bourn have discovered a stronger condition that holds in the special case of exact Mal'tsev categories (Proposition 1.24 in [2]). We would like to thank Francis Borceux for some useful remarks concerning the proof of Theorem 1.3.…”

mentioning

confidence: 99%

Beck–Chevalley condition and Goursat categories

Gran

Rodelo

2017

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

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

show abstract

Central reflections and nilpotency in exact Mal’tsev categories

Cited by 12 publications

References 51 publications

On the "three subobjects lemma" and its higher-order generalisations

On the "three subobjects lemma" and its higher-order generalisations

A survey of homotopy nilpotency and co-nilpotency

Beck–Chevalley condition and Goursat categories

Contact Info

Product

Resources

About