A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:

(Q1) Idempotency: $x\triangleleft x=x$.

(Q2) Right-invertibility: $(x\triangleleft y)\triangleleft^{-1}y=x=(x\triangleleft^{-1}y)\triangleleft y$.

(Q3) Right-distributivity: $(x\triangleleft y)\triangleleft z=(x\triangleleft z)\triangleleft(y\triangleleft z)$.

These axioms may seem somewhat arbitrary, but I claim they aren't: They correspond to the Reidemeister moves in knot theory. Moreover, if $G$ is a group, then defining the operation $x\triangleleft y=y^{-1}xy$ on $G$ gives a quandle structure, and as shown by Joyce in 1982, these three axioms are sufficient to capture the equational content of conjugation.

A profinite quandle is an inverse limit $\varprojlim_{\lambda\in\Lambda}Q_\lambda$ of an inverse system of finite quandles. Those familiar with the theory of profinite groups will recognize this construction.

It is known for profinite groups the following result: Every profinite group can be endowed with a Stone topology, i.e., compact, Hausdorff, and totally disconnected. On the other hand, every Stone-topological group is profinite. This result can be found in Johnstone's book on Stone spaces, though it is presented in much more generality such that any finitary algebraic category with a "complete word set" in some variable admits this Stone-topological/profinite equivalence. Johnstone also gives sufficient requirements for having a complete word set. However, one of these requirements (distributivity) is not met by quandles.

Is there reason to believe that quandles admit such an equivalence? Perhaps there is some reason to believe this in relation to quandles characterizing group conjugation, but I haven't been able to work it out. I know there is a direct proof for Stone-topological groups being profinite, see my previous question. However, this relies on having an identity element, which quandles don't have. Any insight would be appreciated.

3more comments