Higher Coverings of Racks and Quandles—Part II

Renaud, François

doi:10.1007/s10485-021-09651-z

Cited by 5 publications

(11 citation statements)

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“…Our interest in the "shape" of a general element in the intersection of the kernels of two morphisms between free groups arises from research about the concept of a double covering of racks and quandles [28,29]. Understanding this research, its context and its motivations is not strictly necessary for understanding this paper's result.…”

Section: Motivations and Context Of Studymentioning

confidence: 99%

“…The links between the theory of racks and quandles on the one hand, and the theory of groups on the other, are tight. In particular, the development of a higher covering theory of racks and quandles started in [28,29] is to be viewed in parallel with corresponding breakthroughs in group theory. Based on the development and applications of the categorical Galois theory of group, [17,19,18,21], the classical and higher Hopf formulae [10,5] were rediscovered (see [20]) and generalized as in [13].…”

Section: Motivations and Context Of Studymentioning

confidence: 99%

“…Based on the development and applications of the categorical Galois theory of group, [17,19,18,21], the classical and higher Hopf formulae [10,5] were rediscovered (see [20]) and generalized as in [13]. As it is explained in the introduction of [28] and in the conclusions of [29], drawing the links between the homology of racks and quandles and the higher covering theory of these, via the study of generalized Hopf formulae [13,20] is one of the possible outcomes of such developments. Besides the links with group theory, categorical Galois theory also explains the parallels between the covering theory of racks and quandles and the covering theory of topological spaces.…”

Section: Motivations and Context Of Studymentioning

confidence: 99%

“…The first step which we took towards these outcomes, was the definition and study of double coverings of racks and quandles [28]. These are special commutative squares of surjective morphisms of racks (or quandles) which we characterized (roughly speaking) by the "trivial action" of certain "related" double extensions of groups.…”

Section: Motivations and Context Of Studymentioning

confidence: 99%

“…One dimensional symmetric paths. In [28], we introduce the following concepts and results (see also [2]). Definition 2.3.…”

Section: 2mentioning

confidence: 99%

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Rewriting the elements in the intersection of the kernels of two morphisms between free groups

Renaud¹

2021

Preprint

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Let Fg be the free group functor, left adjoint to the forgetful functor between the category of groups Grp and the category of sets Set. Let f : A → B and h : A → C be two functions in Set and let Ker(Fg(f )) and Ker(Fg(h)) be the kernels of the induced morphisms between free groups. Provided that the kernel pairs Eq(f ) and Eq(h) of f and h permute (such as it is the case when the pushout of f and h is a double extension in Set), this short article describes a method to rewrite a general element in the intersection Ker(Fg(f )) ∩ Ker(Fg(g)) as a product of generators in A which is f, h -symmetric in the sense of the higher covering theory of racks and quandles.

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Section: Motivations and Context Of Studymentioning

confidence: 99%