Higher coverings of racks and quandles -- Part II

TL;DR

This paper develops a higher-dimensional covering theory for racks and quandles using categorical Galois theory, extending previous work by characterizing double coverings and defining a new commutator to understand centrality.

Contribution

It introduces a higher categorical framework for coverings of racks and quandles, including the concept of double coverings and a new commutator capturing multi-dimensional centrality.

Findings

Characterization of trivial, normal, and covering in a 2D categorical context.

Definition of a well-behaved commutator for centralization in quandles.

Establishment of links between quandle coverings and group-theoretic concepts.

Abstract

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 higher-dimensional 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.