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 to the subject 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 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 study 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 suitable constructions of weakly universal covers, and fundamental groupoids 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 complementary 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.
Keywords: 2-category, Kan injectivity, KZ-pseudomonad, small object argument
2020 MSC: 18E50; 57K12; 08C05; 55Q05; 18A20; 18B40; 20L05
Theory and Applications of Categories, Vol. 40, 2024, No. 17, pp 479-536.
Published 2024-05-31.
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