A Galois-Theoretic Approach to the Covering Theory of Quandles

On factorization systems for surjective quandle homomorphisms

“…The restriction of the notion of factorisation system to the class F of surjective homomorphisms is related to the fact that the functor π 0 has a nice exactness property only with respect to the class F of surjective homomorphisms in Qnd. This fact is explained in the following result from [11], which is now based on Corollary (1.8):…”

“…It is not possible to weaken the assumption on φ : X → π 0 (B), which has to be required to be a surjective homomorphism. Indeed, as explained in [11], the functor π 0 does not preserve pullbacks of the form (3.1) when φ : X → π 0 (B) is not required to be surjective. In other words the functor π 0 is not semi-left-exact [7].…”

“…Nevertheless, this functor still has some nice left exactness properties [11], in the sense that it preserves a suitable class of pullbacks (see Theorem 2.2). This fact implies the existence of a canonical factorisation system (E, M) for surjective homomorphisms of quandles associated with the adjunction (A).…”

“…We describe this factorisation system in Section 2, by using an important property of permutability of a class of congruences in Qnd (Lemma 1.3), explicitly described in Section 1, that is of independent interest. This factorisation system (E, M) satisfies a characteristic property of the so-called reflective ones [7]: E is the class of surjective homomorphisms which are inverted (= sent to an isomorphism) by the reflector π 0 : Qnd → Qnd * , and for two composable surjective homomorphisms f and g, then g ∈ E whenever f • g ∈ E and f ∈ E. The class M is the class of trivial extensions (also called trivial coverings) in the sense of categorical Galois theory [1,14] (see also [9,10,11]). …”

How to centralize and normalize quandle extensions

“…A quandle homomorphism f : A → B is a quandle covering if it is surjective and f (a) = f (a ′ ) implies c ⊳ a = c ⊳ a ′ for all c ∈ A. The coincidence of the two notions was proved in [6].…”