Higher coverings of racks and quandles -- Part I

Abstract

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..

A symmetric approach to higher coverings in categorical Galois theory

“…We then say that c is split by e. In other words, c is a Γ-covering when a span of discrete fibrations x Ð t Ñ c exists in Y , where x is a primitive Γ-covering. (See, for instance, Figure 1 in [29].) We define CovpY q to be the full subcategory of ExtpY q determined by the coverings.…”

“…Our aim here is to develop the corresponding higher-dimensional Galois theory. A detailed study of the first two stages of this process (the corresponding Γ 0 and Γ 1 ) may be found in [29,30]. The remaining levels of the tower will be discussed in [31].…”