On categories of faithful quandles with surjective or injective quandle homomorphisms

TL;DR

This paper establishes category equivalences for faithful quandles with surjective or injective homomorphisms, linking them to pairs of groups and their generators, thereby advancing the algebraic understanding of quandles.

Contribution

It introduces new categorical equivalences for faithful quandles with surjective and injective homomorphisms, connecting them to group-theoretic structures.

Findings

Category of faithful quandles with surjective homomorphisms is equivalent to pairs of groups with certain homomorphisms.

Category of faithful quandles with injective homomorphisms is equivalent to a newly defined category of group pairs.

Provides a structural framework for analyzing faithful quandles via group theory.

Abstract

E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter [J. Knot theory Ramifications (2010)] pointed out that by considering inner automorphism groups of quandles, one have 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 category equivalence between the category of faithful quandles with surjective quandle homomorphisms and that of pairs of groups and their generators with suitable group homomorphisms. We are also interested in injective quandle homomorphisms. By defining suitable morphisms among pairs of groups and their generators, we obtain a category which is equivalent to the category of faithful quandles with injective quandle homomorphisms.