论文标题
架子和难题的覆盖率更高 - 第一部分
Higher coverings of racks and quandles -- Part I
论文作者
论文摘要
本文是一系列三篇文章的第二部分，我们在其中开发了较高的架子和难题理论。该项目植根于M. Eisermann在Quandle覆盖范围内的工作，V。Vev。将覆盖物描述为那些相对在琐碎的难题相对较中心的溢出的分类视角。我们通过在G. janelidze的意义上应用了更高分类的Galois理论的技术来扩展这项工作，尤其是我们确定有意义的高维中心性条件，以定义我们对机架和难民的更高覆盖率。在第一篇文章（第一部分）中，我们重新访问和阐明了覆盖感兴趣的理论的基础，我们将其扩展到了架子的更一般的环境，并在数学上描述了如何在机架和Quandles之间导航。我们解释了发挥作用的代数成分，并加强了这些成分的同位和拓扑解释。特别是，我们证明并坚持组和架子（或Quandles）之间的共轭函数conj的左伴随的关键作用。我们重命名该函数PTH，并说明它在哪种意义上将机架发送给其同型路径类别。我们使用PTH表征覆盖物和相对中心性，但也对这些条件有了更视觉范围的“几何”理解。我们使用替代性概括和视觉证明来表征架子和难题的中央扩展。我们从Galois理论的角度完成了Eisermann M. eisermann的临时结构（弱普遍覆盖和基本群体）的恢复。我们素描如何从分类Galois理论的基本定理中推断出M. Eisermann的详细分类结果。我们提出所有的思想和结果，这些思想和结果将阐明第二部分和III中发展的高维理论。
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.
