On Profinite Quandles

TL;DR

This paper explores the structure and properties of profinite quandles, providing constructions, characterizations, and potential applications to étale homotopy theory of number fields.

Contribution

It introduces new constructions and characterizations of profinite quandles, including criteria for subquandles and a link to their automorphism groups, advancing the understanding of their algebraic structure.

Findings

Characterization of subquandles of profinite quandles as profinite.

Connection between algebraically connected profinite quandles and their automorphism groups.

Counterexample showing not all Stone topological quandles are profinite.

Abstract

We undertake the study of profinite quandles. We provide several constructions of profinite quandles from profinite groups, and from other profinite quandle. We characterize which subquandles of profinite quandles are again profinite. Finally, we provide a characterization of algebraically connected profinite quandles in terms of the profinite completion of their inner automorphism groups $\widehat{\Inn(Q)}$. It is anticipated that the results herein will find applications to the \'{e}tale homotopy theory of number fields. v.2 has been updated to include an example due to Ariel Davis settling in the negative the question of whether all Stone topological quandles are profinite.