Double coverings of racks and quandles

In this Ph.D. thesis we lay down the foundations of 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 quandle coverings as those surjections which are central, relatively to trivial quandles. We revisit and extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze. In particular we extend the study of quandle coverings to the more general context of racks, we consolidate the understanding of their relationship with central extensions of groups on the one hand and topological coverings on the other. We further identify and study a meaningful two-dimensional centrality condition defining our double coverings of racks and quandles. We also introduce the definition of a suitable commutator which describes the zero, one and two-dimensional concepts of centralization in this context.(SC - Sciences) -- UCL, 202