On factorization systems for surjective quandle homomorphisms

Even, Valérian; Gran, Marino

doi:10.1142/s0218216514500606

Cited by 10 publications

(10 citation statements)

References 17 publications

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“…This is an important difference with the case of topological spaces for instance, for which the connected components are connected and thus the corresponding π 0 functor is semi-left-exact. See also [28] for further insights on connectedness.…”

Section: Abmentioning

confidence: 99%

Higher coverings of racks and quandles -- Part I

Renaud

2020

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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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Section: Abmentioning

confidence: 99%

Higher coverings of racks and quandles -- Part I

Renaud

2020

Preprint

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“…When moreover N is a normal subgoup of Inn(A), ∼ N becomes a congruence on A, called an orbit congruence (see Theorem 6.1 in [4]). One can show (Lemma 2.6 in [7]) that the orbit congruences in Qnd permute with any other reflexive relation R: ∼ N •R = R• ∼ N . Since the kernel congruence Eq(η A ) of the A-component of the unit of the adjunction (1) η A : A → π 0 (A) is an orbit congruence (it is ∼ Inn(A) ), the adjunction (1) can be shown to be admissible for Galois theory.…”

Section: Trivializing An Extensionmentioning

confidence: 99%

How to centralize and normalize quandle extensions

Duckerts-Antoine

Even

Montoli

2018

J. Knot Theory Ramifications

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“…The last part of the above theorem includes Proposition 3.6 from [6] as a particular case. For more details, the reader is referred to [18].…”

Section: Formal Closure Operatorsmentioning

confidence: 99%

Epireflective subcategories and formal closure operators

Duckerts-Antoine,

Gran,

Janelidze

2016

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On a category C with a designated (well-behaved) class M of monomorphisms, a closure operator in the sense of D. Dikranjan and E. Giuli is a pointed endofunctor of M, seen as a full subcategory of the arrow-category C 2 whose objects are morphisms from the class M, which "commutes" with the codomain functor cod : M → C . In other words, a closure operator consists of a functor C : M → M and a natural transformation c : 1 M → C such that cod•C = C and cod•c = 1 cod . In this paper we adapt this notion to the domain functor dom : E → C , where E is a class of epimorphisms in C , and show that such closure operators can be used to classify E-epireflective subcategories of C , provided E is closed under composition and contains isomorphisms. Specializing to the case when E is the class of regular epimorphisms in a regular category, we obtain known characterizations of regular-epireflective subcategories of general and various special types of regular categories, appearing in the works of the second author and his coauthors. These results show the interest in investigating further the notion of a closure operator relative to a general functor. They also point out new links between epireflective subcategories arising in algebra, the theory of fibrations, and the theory of categorical closure operators.

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On factorization systems for surjective quandle homomorphisms

Cited by 10 publications

References 17 publications

Higher coverings of racks and quandles -- Part I

Higher coverings of racks and quandles -- Part I

How to centralize and normalize quandle extensions

Epireflective subcategories and formal closure operators

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