How to centralize and normalize quandle extensions

Duckerts-Antoine, Mathieu; Even, Valérian; Montoli, Andrea

doi:10.1142/s0218216518500207

Cited by 6 publications

(10 citation statements)

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“…Amongst other results, we derive that the centralizing relations, if they exist, should be the same in both contexts. We then prove (Section 3.4) several characterizations of these centralizing relations, and extend the results from [24] on the reflectivity of coverings in extensions. In preparation for the admissibility in dimension 2, we show that coverings are closed under quotients along double extensions (towards "Birkhoff") and we show the commutativity property of the kernel pair of the centralization unit (towards "strongly Birkhoff").…”

mentioning

confidence: 54%

“…By Corollary 3.3.6, what we deduced about the functor F 1 restricts to the domain CExtQnd, and so also describes the left adjoint to the inclusion in ExtQnd from Theorem 5.5. in [24]. In addition to Corollary 3.3.6, we further describe how centralization behaves with respect to r F q .…”

Section: 3mentioning

confidence: 93%

“…Centralizing extensions. We adapt the result from [24], showing the reflectivity of quandle coverings in the category of extensions, to the context of racks. We put the emphasis on our new characterizations of the centralizing relation which works the same for racks and for quandles.…”

Section: 3mentioning

confidence: 99%

“…Proof. Using definition (i) for the centralizing relation, the proof of Theorem 5.5 in [24] easily translates to the context of racks. Then given an extension f : A → B, the unit η 1 f = (η 1 A , id B ) is indeed a double extension since its bottom component is an isomorphism.…”

Section: 3mentioning

confidence: 99%

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Higher coverings of racks and quandles -- Part I

Renaud

2020

Preprint

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This article is the first part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann's work on quandle coverings, and the categorical perspective brought by V. Even, who characterizes coverings as those surjections which are central, relatively to trivial quandles. We extend this work by application of the techniques from higher categorical Galois theory (G. Janelidze), and in particular identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles.In this first article (Part I), we revisit and clarify the foundations of the covering theory of interest, we extend it to the more general context of racks and mathematically describe how to navigate between racks and quandles. We explain the algebraic ingredients at play, and reinforce the homotopical and topological interpretations of these ingredients. In particular we justify and insist on the crucial role of the left adjoint of the conjugation functor Conj between groups and racks (or quandles). We rename this functor Pth, and explain in which sense it sends a rack to its group of homotopy classes of paths. We characterize coverings and relative centrality using Pth, but also develop a more visual "geometrical" understanding of these conditions. We use alternative generalizable and visual proofs for the characterization of central extensions. We complete the recovery of M. Eisermann's ad hoc constructions (weakly universal cover, and fundamental groupoid) from a Galois-theoretic perspective. We sketch how to deduce M. Eisermann's detailed classification results from the fundamental theorem of categorical Galois theory. As we develop this refined understanding of the subject, we lay down all the ideas and results which will articulate the higher-dimensional theory developed in Part II and III.The author is a Ph.D. student funded by Formation à la Recherche dans l'Industrie et dans l'Agriculture (FRIA) as part of Fonds de la Recherche Scientifique -FNRS..

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confidence: 54%

Section: 3mentioning

confidence: 93%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Higher coverings of racks and quandles -- Part I

Renaud

2020

Preprint

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“…such that f (a) = f (b). A surjective morphism of racks or quandles f : A → B can be centralized, using a quotient of its domain, given by the centralization congruence C 1 A over A generated by the pairs (x, y) such that there exists (a, b) ∈ Eq(f ) such that y = x ⊳ a ⊳ −1 b (see [20] and Part I).…”

Section: Introductionmentioning

confidence: 99%

Higher coverings of racks and quandles -- Part II

Renaud¹

2021

Preprint

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This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann's work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are categorically central, relatively to trivial quandles. We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higherdimensional centrality conditions defining our higher coverings of racks and quandles.In this second article (Part II), we show that categorical Galois theory applies to the inclusion of the category of coverings into the category of surjective morphisms of racks and quandles. We characterise the induced Galois theoretic concepts of trivial covering, normal covering and covering in this two-dimensional context. The latter is described via our definition and study of double coverings, also called algebraically central double extensions of racks and quandles. We define a suitable and well-behaved commutator which captures the zero, one and two-dimensional concepts of centralization in the category of quandles. We keep track of the links with the corresponding concepts in the category of groups.

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Higher Coverings of Racks and Quandles—Part II

Renaud

2021

Appl Categor Struct

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How to centralize and normalize quandle extensions

Cited by 6 publications

References 14 publications

Higher coverings of racks and quandles -- Part I

Higher coverings of racks and quandles -- Part I

Higher coverings of racks and quandles -- Part II

Higher Coverings of Racks and Quandles—Part II

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