A characterization of central extensions in the variety of quandles (Q2808144)

A \textsl{quandle} (a rack satisfying the additional condition of idempotency) is [J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003)] a set equipped with two binary operations \(\triangleleft\) and \(\triangleleft^{-1}\) (often denoted \(\triangleright\)) satisfying three axioms, paralleling the three Reidemeister moves of link diagrams. Algebraically, a quandle encapsulates the properties of group conjugation while geometrically, the fundamental rack forms a complete invariant of irreducible framed links in a 3-manifold and is a stronger invariant than the fundamental group. A quandle is called \textsl{symmetric} if identically \(a\triangleleft{}b=b\triangleleft{}a\), and \textsl{abelian} if identically \((a\triangleleft{}b)\triangleleft(c\triangleleft{}d)=(a\triangleleft{}c)\triangleleft(b\triangleleft{}d)\). These are independent properties of a quandle. The category of quandles \textbf{Qnd} forms a variety in the sense of universal algebra. The category \textbf{SymQnd} of symmetric quandles forms a Mal'tsev variety \textit{J. D. H. Smith} [Mal'cev varieties. Lecture Notes in Mathematics. 554. Berlin-Heidelberg-New York: Springer-Verlag. (1976; Zbl 0344.08002)]. In any variety \(V\), an \textsl{internal Mal'tsev algebra} is an algebra \(A\) with a homomorphism \(p_A:A\times{}A\times{}A\rightarrow{}A\) for which \(p_A(a,a,b)=b\) and \(p_A(a,b,b)=a\); these form a category \textbf{Mal(V)}. The paper observes that \textbf{Mal(SymQnd)} is precisely the subcategory of abelian symmetric quandles \textbf{AbSymQnd} (independently observed by \textit{D. Bourn} [J. Knot Theory Ramifications 24, No. 12, Article ID 1550060, 35 p. (2015; Zbl 1331.18004)]) and goes on to show that \textbf{AbSymQnd} is an \textsl{admissible} subvariety of \textbf{Qnd} for the categorical theory of central extensions, allowing the use of Galois-type theorems on the classification of the corresponding central extensions [\textit{G. Janelidze} and \textit{G. M. Kelly}, J. Pure Appl. Algebra 97, No. 2, 135--161 (1994; Zbl 0813.18001)]. The main theorem of the paper gives an algebraic description of the quandle extensions that are central for the adjunction between \textbf{Qnd} and \textbf{AbSymQnd}. It states that, for a surjective homomorphism \(f:A\rightarrow{}B\) in \textbf{Qnd}, the following conditions are equivalent: (i) \(f\) is an algebraicaly central extension with abelian symmetric fibers; (ii) \(f\) is a normal extension; (iii) \(f\) is a central extension. The authors note that the work fits into the partial Mal'tsev context, thoroughly studied in [\textit{D. Bourn}, ``Partial Mal'tsevness and partial protomodularity'', Preprint, \url{arXiv:1507.02886}].