Higher coverings of racks and quandles -- Part I

TL;DR

This paper develops a higher covering theory for racks and quandles, extending previous work with categorical Galois theory, and clarifies foundational aspects while connecting algebraic, homotopical, and topological perspectives.

Contribution

It introduces a higher-dimensional centrality framework for racks and quandles, generalizing existing covering theories with new algebraic and geometric insights.

Findings

Characterization of coverings via the Pth functor

Recovery of Eisermann's constructions from Galois theory

Foundational clarification and extension of rack and quandle covering theory

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 by V. Even, who characterizes coverings as those surjections which are 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 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 of racks and quandles. 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. We lay down all the ideas and results which will articulate the higher-dimensional theory developed in Part II and III.