No Access

How to centralize and normalize quandle extensions

Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal

Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

E-mail Address: valerian.even@gmail.com

Search for more papers by this author

, and

Andrea Montoli

Dipartimento di Matematica “Federigo Enriques”, Università Degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy

E-mail Address: andrea.montoli@unimi.it

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218216518500207Cited by:2 (Source: Crossref)

Abstract

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

Keywords:

AMSC: 57M27, 18A32, 08B05

References

1. M. Barr , Exact categories, in Exact Categories and Categories of Sheaves, Lecture Notes in Mathematics, Vol. 236 (Springer, 1971), pp. 1–120. Crossref, Google Scholar
2. M. Barr and C. Wells , Toposes, Triples and Theories (Springer-Verlag, 1985). Crossref, Google Scholar
3. G. Birkhoff , On the structure of abstract algebras, Proc. Camb. Philos. Soc. 31 (1935) 433–454. Crossref, Google Scholar
4. E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter , On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156. Link, Web of Science, Google Scholar
5. M. Eisermann , Quandle coverings and their Galois correspondence, Fund. Math. 225(1) (2014) 103–168. Crossref, Web of Science, Google Scholar
6. V. Even , A Galois-theoretic approach to the covering theory of quandles, Appl. Categ. Structures 22(5–6) (2014) 817–831. Crossref, Web of Science, Google Scholar
7. V. Even and M. Gran , On factorization systems for surjective quandle homomorphisms, J. Knot Theory Ramifications 23(11) (2014) 1450060. Link, Web of Science, Google Scholar
8. T. Everaert , Higher central extensions in Mal’tsev categories, Appl. Categ. Struct. 22 (2014) 961–979. Crossref, Web of Science, Google Scholar
9. G. Janelidze , Pure Galois theory in categories, J. Algebra 132 (1990) 270–286. Crossref, Web of Science, Google Scholar
10. G. Janelidze and G. M. Kelly , Galois theory and a general notion of central extension, J. Pure Appl. Algebra 97 (1994) 135–161. Crossref, Web of Science, Google Scholar
11. D. Joyce , A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37–65. Crossref, Web of Science, Google Scholar
12. A. I. Mal’cev , On the general theory of the algebraic systems, Mat. Sbornik (N.S.) 35 (1954) 3–20. Google Scholar
13. M. Takasaki , Abstractions of symmetric transformations, Tôhoku Math. J. 49 (1943) 143–207. Google Scholar