A characterization of central extensions in the variety of quandles

How to centralize and normalize quandle extensions

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

Higher coverings of racks and quandles -- Part II

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 to the subject by V. Even, who characterizes coverings as those surjections which are categorically 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 co…

Coextensive varieties via Central Elements

In this paper we use the theory of central elements in order to provide a characterization for coextensive varieties. In particular, if the variety is of finite type, congruence-permutable and its class of directly indecomposable members is universal, then coextensivity is equivalent to be a variety of shells.

从共延展变种到Gaeta位面

From coextensive varieties to the Gaeta topos

本文研究了共延展变种中中心元素的函子性质，并证明在特定条件下Gaeta位面可分类无中心模型。

斜括号的同调Hopf公式

Hopf formulae for homology of skew braces

本文利用非阿贝尔同调代数方法，针对斜括号到根环、群和阿贝尔群三个子簇的反射子建立新Hopf公式，并刻画相应中心扩张，导出Stallings-Stammbach正合序列及中心列新结果。

Relative Commutator Theory in Semi-Abelian Categories

Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.