On categories of faithful quandles with surjective or injective quandle homomorphisms

E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter 4 pointed out that by considering inner automorphism groups of quandles, one has a functor from the category of quandles with surjective homomorphisms to that of groups with surjective homomorphisms. In this paper, we focus on faithful quandles. As main results, we give a category equivalence between the category of faithful quandles with surjective quandle homomorphisms and that of pairs of groups and their conjugation-stable generators with suitable group homomorphisms. We are also interested in injective quandle homomorphisms. By defining suitable morphisms among pairs of groups and their conjugation-stable generators, we obtain a category which is equivalent to the category of faithful quandles with injective quandle homomorphisms.