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1]Department of Mathematics, University of Chicago, Eckhart Hall, 5734 S
University Ave, Chicago, IL 60637, USA

2]Department of Mathematics,University of Florida, P.O. Box 118105,
Gainesville, FL 32611 USA

3]Department of Mathematics, University of Kentucky, Patterson Office Tower,
120 Patterson Dr., Lexington, KY 40508, USA

4]Department of Mathematics,Harvard University, Science Center Room 325, 1
Oxford Street, Cambridge, MA 02138 USA

[5]David N. Yetter [5]Department of Mathematics, Kansas State University, 1228
N. Martin Luther King, Jr. Drive, Manhattan, KS 66506 USA

# On Profinite Quandles

Alexander W. Byard  [<EMAIL_ADDRESS>Brian Cai  [<EMAIL_ADDRESS>Nathan P. Jones  [<EMAIL_ADDRESS>Lucy H. Vuong  [
<EMAIL_ADDRESS>*<EMAIL_ADDRESS>
###### Abstract

s We undertake the study of profinite quandles. A quandle $(Q,\triangleleft)$
is a set $Q$ with a operation $\triangleleft$ that satisfies idempotency,
right invertibility, and right self-distributivity axioms. Profinite quandles
are inverse limits of inverse systems of finite quandles and are closely
related to profinite groups. As with all profinite models of algebraic
theories, they are naturally equipped with the structure of a Stone space,
with respect to which structure their operations are continuous. 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 $\widehat{\text{Inn}(Q)}$. It is
anticipated that the results herein will find applications to the étale
homotopy theory of number fields.

##  1 Introduction

Quandles were introduced by Joyce in his 1982 seminal paper A classifying
invariant of knots, the knot quandle as an algebraic invariant on classical
knots and links. [1] However, the earliest known example of a quandle dates
back to 1940s Japan when Mituhisa Takasaki defined a kei, now known as an
involutory quandle, in an attempt to find a non-associative algebraic
structure capable of describing reflections in finite geometry. [2] Since
then, quandles have been extensively studied in relation to classical knots
and links [3, 4, 5, 6, 7, 8] as well as other mathematical structures. For
instance Zablow [9] showed that the Dehn twists of any orientable surface form
a quandle, while Crans [10] showed that quandles arise naturally in relating a
Lie group and its Lie algebra. Yetter [11] showed that many monodromy
phenomena arising in geometric topology and algebraic geometry are more
naturally described by quandle homomorphisms rather than group homomorphisms
with unalgebraic side-conditions as had been done classically. For excellent
introductory articles on quandles, see [12].

A quandle is a set $Q$ with two binary operations satisfying idempotency,
right-invertibility, and right-distributivity axioms analogous to the
Reidemeister moves in knot theory. A subquandle of $Q$ is a subset closed
under the operations of $Q$, and the collection of subquandles of $Q$ forms a
lattice. In their 2018 paper, Saki and Kiani demonstrated that the lattice of
subquandles of a finite quandle is complemented and gave an example of an
infinite quandle that does not have a complemented subquandle lattice. [13]
Amsberry et al. constructed an ind-finite quandle without a complemented
subquandle lattice, but conjectured that the subquandle lattice of a profinite
will be complemented. [14]

Our original motivation in studying profinite quandles was to resolve a
conjecture of Amsberry et al. [14] which remains unresolved. We became aware,
however, that the general theory of profinite quandles is of some importance
as the étale homotopy of number fields naturally gives rise to profinite
quandles, which have been studied by Davis and Schlank [15]. In this paper we
introduce the theory of profinite quandles. We show that products and finite
disjoint unions preserve profiniteness and give a topological characterization
for when a subquandle of a profinite quandle is profinite. Finally, we provide
a version of algebraic connectedness appropriate to profinite quandles and
characterize the structure of algebraically connected profinite quandles in
terms of the profinite completion of their inner automorphism groups.

While this paper was in its final preparation, we became aware that Singh [16]
had independently derived a number of basic results about profinite quandles,
mostly distinct from our own. We note in footnotes those of our results which
are also found in [16].

##  2 Basic Quandle Theory

######  Definition 2.1 (Quandle).

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

  1. (Q1)

$x\triangleleft x=x$,

  2. (Q2)

$(x\triangleleft y)\triangleleft^{-1}y=x=(x\triangleleft^{-1}y)\triangleleft
y$,

  3. (Q3)

$(x\triangleleft y)\triangleleft z=(x\triangleleft
z)\triangleleft(y\triangleleft z)$.

In other words $\triangleleft$ is idempotent, right-invertible, and right-
distributive.

Notice that this definition describes quandles as models of an equational
theory. Quandles can be equivalently defined using a first order theory with a
single operation $\triangleleft$ since (Q2) is equivalent to the condition
that for all $y\in Q$ the map $x\mapsto x\triangleleft y$ is a bijection. This
allows for the omission of $\triangleleft^{-1}$ when convenient.

An algebraic structure $(Q,\triangleleft)$ satisfying (Q2) and (Q3) is called
a rack.

######  Example 2.2 (Tait quandle).

Let $Q=\\{1,2,3\\}$ and define $\triangleleft:Q\times Q\to Q$ by the following
operation table:

$\triangleleft$ | 1 | 2 | 3
---|---|---|---
1 | 1 | 3 | 2
2 | 3 | 2 | 1
3 | 2 | 1 | 3

With this operation $(Q,\triangleleft)$ forms the Tait Quandle.

######  Example 2.3.

(Trivial quandle). Let $S$ be a set and define $\triangleleft:S\times S\to S$
by $(x,y)\mapsto x$ for all $x,y\in S$. Then $\triangleleft$ defines the
trivial quandle structure on a set $S$.

######  Example 2.4 (Conjugation quandle).

Let $G$ be a group. We define the operation $\triangleleft:G\times G\to G$ be
given by $(g,h)\mapsto h^{-1}gh$ for all $g,h\in G$. The group $G$ with this
operation defines a quandle $(G,\triangleleft)$, denoted by $\text{Conj}(G)$,
called the conjugation quandle on $G$.

Conj defines a functor from the category Gp of groups to the category Qndl of
quandles. Furthermore, Conj admits a left adjoint, denoted Adconj. For all
quandles $Q$ we have that $\text{Adconj}(Q):=F(Q)/\sim$, where $F(Q)$ is the
free group generated by $Q$ and $x\triangleleft y\sim y^{-1}xy$. [1] In some
literature, $\text{Adconj}(Q)$ is also referred to as the universal
augmentation group of $Q$.

######  Definition 2.5 (Involutory quandle).

An involutory quandle or kei is a quandle $(Q,\triangleleft)$ satisfying
$(x\triangleleft y)\triangleleft y=x$ for all $x,y\in Q$.

######  Example 2.6 (Takasaki quandle).

The Takasaki quandle is named after Takasaki’s early work on kei. [2] Let $A$
be an abelian group and define the operation $\triangleleft:A\times A\to A$ by
$(x,y)\mapsto 2y-x$. Together $(A,\triangleleft)$ defines a kei, denoted by
$\text{Tak}(A)$, called the Takasaki kei on $A$, or the Takasaki quandle on
$A$.

Tak defines a functor from the category Ab of abelian groups to the category
Kei of kei. Moreover, Tak admits a left adjoint, denoted AdTak. For all
involutory quandles $Q$, we have that $\text{AdTak}(Q):=\langle
Q\rangle_{\text{ab}}/\sim$, where $\langle Q\rangle_{\text{ab}}$ is the free
abelian group generated by $Q$ and $x\triangleleft y\sim 2y-x$. [14]

######  Example 2.7 (Core quandle).

The Takasaki quandle on abelian groups can be generalized to all groups. For a
group $G$, define the operation $\triangleleft:G\times G\to G$ by
$(x,y)\mapsto xy^{-1}x$. The pair $(G,\triangleleft)$ defines a kei structure
on $G$ denoted Core($G$).

######  Definition 2.8 (Subquandle).

For a quandle $(Q,\triangleleft)$, a subset $Q^{\prime}\subseteq Q$ is a
subquandle of Q provided $(Q^{\prime},\triangleleft)$ is a quandle, i.e.
$Q^{\prime}$ is closed under $\triangleleft|_{Q^{\prime}}$ and
$\triangleleft^{-1}|_{Q^{\prime}}$. We write $Q^{\prime}\preceq Q$ to denote
that $Q^{\prime}$ is a subquandle of $Q$. When $Q^{\prime}$ is a proper
subquandle of $Q$, we write $Q^{\prime}\prec Q$.

######  Proposition 2.9.

Let $Q$ be a quandle. If $Q_{1}$ and $Q_{2}$ are subquandles of $Q$, then
$Q_{1}\cap Q_{2}$ is a subquandle of $Q$.

######  Definition 2.10.

Let $(Q,\triangleleft)$ be a quandle. Given a subset $S\subseteq Q$, the
subquandle $\langle S\rangle$ generated by $S$ is given by

$\langle S\rangle:=\bigcap_{S\subseteq Q^{\prime}\preceq Q}Q^{\prime}$

######  Definition 2.11 (Quandle homomorphism).

Let $(Q_{1},\triangleleft_{1})$ and $(Q_{2},\triangleleft_{2})$ be quandles. A
quandle homomorphism $\varphi$ from $Q_{1}$ to $Q_{2}$ is a map
$\varphi:Q_{1}\to Q_{2}$ such that
$\varphi(x\triangleleft_{1}y)=\varphi(x)\triangleleft_{2}\varphi(y)$. A
quandle isomorphism is a bijective quandle homomorphism.

######  Definition 2.12 (Automorphism group of a quandle).

Let $Q$ be a quandle. A quandle isomorphism $\varphi:Q\to Q$ between $Q$ and
itself is called a quandle automorphism of $Q$. The set of quandle
automorphisms

$\text{Aut}(Q):=\\{\varphi:Q\to Q:\varphi\text{ is an automorphism of $Q$}\\}$

of $Q$ forms a group called the automorphism group of $Q$.

Let $y\in Q$. The symmetry $S_{y}:Q\to Q$ at $y$ is given by
$S_{y}(x)=x\triangleleft y$ for all $x\in Q$. It is a quandle automorphism by
axiom (Q3). The group

$\text{Inn}(Q):=\langle S_{y}\rangle_{y\in Q}$

generated by all symmetries of $Q$ is called the inner automorphism group of
$Q$. It is a theorem of Joyce [1] that $\text{Inn}(Q)$ is a normal subgroup of
$\text{Aut}(Q)$.

Finally, we recall

######  Definition 2.13.

A quandle $Q$ is said to be algebraically connected if there is exactly one
orbit under right action by $\text{Inn}(Q)$.

In the context of finite quandles, algebraically connected quandles form
“building blocks” from which all finite quandles can be iteratively assembled
using semi-disjoint unions, that is quandle structures on the disjoint union,
the operations of which restrict to the given ones on the the summands (see
[17] for a formal definition).

In [17], Ehrman et al. established a characterization of algebraically
connected finite quandles. First, a proposition following from properties of
group actions.

######  Proposition 2.14.

Let $Q$ be a connected quandle on $n$ elements. Then $n$ divides
$|\text{Inn}(Q)|$ and any choice of $q\in Q$ induces a
$\text{Inn}(Q)$-equivariant bijection between $H\backslash\text{Inn}(Q),$
where $H$ is the stabilizer of $q$ under action by $\text{Inn}(Q)$.

From the proposition, we may represent $Q$ by
$(\\{Hg_{1},...,Hg_{n}\\},\triangleleft)$ where $Hg_{i}\triangleleft
Hg_{j}=Hg_{k}$ if $q_{i}\triangleleft q_{j}=q_{k}$. Define an augmentation map
$|\cdot|:Q=H\backslash\text{Inn}(Q)\to\text{Inn}(Q)$ by $|Hg_{i}|=g$ where
$g\in\text{Inn}(Q)$ such that $x\cdot g=x\triangleleft Hg_{i}$ for all $x\in
Q$. To distinguish between the augmentation map and the order of a group, we
will denote $|H|$ as the order of the subgroup $H$ and $|Hh|$ as the
augmentation of $H$ as a right coset in $H\backslash\text{Inn}(Q)$.

######  Theorem 2.15.

[17] Let $Q$ be an algebraically connected quandle on $n$ elements, let $H$ be
the stabilizer of $q\in Q$ under action by $\text{Inn}(Q)$, and let
$\\{g_{2},...,g_{n}\\}$ be all coset representatives of $H$ not in $H$. Then
$H\subset\text{Inn}(Q)\subset\mathfrak{S}_{n}$, $\frac{|G|}{|H|}=n$, $|Hh|\in
Z(H)$, and $\text{Inn}(Q)$ is generated by $\\{|Hh|,|Hg_{2}|,...,|Hg_{n}|\\}$,
where $|Hg_{i}|=g_{i}^{-1}|Hh|g_{i}$ for $2\leq i\leq n$.

######  Theorem 2.16.

[17] Suppose that for groups $G$ and $H$, we have $H\subset
G\subset\mathfrak{S}_{n}$ and $\frac{|G|}{|H|}=n$ for $n\in\mathbb{N}$. Let
$g_{1},...,g_{n}$ be coset representatives of $H$ in $G$ and further suppose
that $G$ is generated by $\\{g_{1}^{-1}hg_{1},...,g_{n}^{-1}hg_{n}\\}$ for
some $h\in Z(H)$. Then $(H\backslash G,\triangleleft)$ where
$Hg_{i}\triangleleft Hg_{j}:=Hg_{i}g_{j}^{-1}hg_{j}$ defines an algebraically
connected quandle on $n$ elements.

Later, we will obtain analogous results for profinite quandles.

As quandles are models of an equational theory, the category of quandles
admits all limits and colimits. In particular, products:

######  Definition 2.17 (Product).

Let $(Q,\triangleleft_{q})$ and $(S,\triangleleft_{S})$ be quandles. The
product quandle $(Q\times S,\triangleright)$ is the product of the underlying
sets with the operation

$(q_{1},s_{1})\triangleright(q_{2},s_{2}):=(q_{1}\triangleright_{Q}q_{2},s_{1}\triangleright_{S}s_{2}).$

Let $(Q_{i},\triangleleft_{i})$ for $i\in{\mathcal{I}}$ be a set of quandles,
the product quandle $\prod_{i\in{\mathcal{I}}}Q_{i}$ is the product of the
underlying sets with

$\langle q_{i}\rangle_{i\in{\mathcal{I}}}\triangleleft\langle
r_{i}\rangle_{i\in{\mathcal{I}}}:=\langle
q_{i}\triangleleft_{i}r_{i}\rangle_{i\in{\mathcal{I}}}$

The coproduct of quandles in complicated in the same way that the coproduct of
groups (i.e. free product of groups) is, and as it is ill-behaved in the
profinite setting, we will not consider it.

We will, however, consider a special case of the semi-disjoint union
construction, to wit:

######  Definition 2.18 (Disjoint Union).

Let $(Q,\triangleleft_{q})$ and $(S,\triangleleft_{S})$ be quandles. The
disjoint union quandle $(Q\sqcup S,\triangleright)$ is defined as follows:

  1. (i)

If $q\in Q$ and $s\in S$, then $q\triangleleft s=q$ and $s\triangleleft q=s$.

  2. (ii)

If $q_{1},q_{2}\in Q$, then $q_{1}\triangleleft
q_{2}=q_{1}\triangleleft_{Q}q_{2}$.

  3. (iii)

If $s_{1},s_{2}\in S$, then $s_{1}\triangleleft
s_{2}=s_{1}\triangleleft_{S}s_{2}$.

It should be noted that disjoint union is not the coproduct in the category of
quandles. It can be described in terms of a univeral property, being the
quandle equipped with quandle homomorphisms from $Q$ and $S$ and universal
among all such quandles in which the images of the two quandles act trivially
on each other.

Finally, we will have cause to consider a notion from Joyce [1] which both
gives a means of constructing quandles and a relationship between quandles and
certain groups related to them.

######  Definition 2.19 (Augmented Quandle).

An augmented quandle $(Q,G)$ consists of a set $Q$, a group $G$ equipped with
a right action on $Q$, and a map $|\cdot|:Q\to G$ satisfying

  1. (AQ1)

$q|q|=q$ for all $q\in Q$

  2. (AQ2)

$|qg|=g^{-1}|q|g$ for all $q\in Q,g\in G$.

The map $|\cdot|$ is called the augmentation map.

Given an augmented quandle $(Q,G)$, we can define a quandle operation on $Q$
by $x\triangleleft y:=x|y|$. If $Q$ was already endowed with a quandle
structure, then the quandle structure prescribed by $|\cdot|$ is the same
structure, hence why we can define an augmented quandle using just a set, as
opposed to a quandle.

In [1], Joyce showed that, given a quandle $(Q,\triangleleft)$, the inclusion
of generators into $\text{Adconj}(Q)$ can be considered the universal
augmentation of $Q$ in the sense that $\triangleleft$ induces a right
$\text{Adconj}(Q)$-action on $Q$ by quandle automorphisms such that the
inclusion of generators is an augmentation. Furthermore, given a group $G$ and
augmentation $\langle\cdot\rangle:Q\to G$, there is a unique group
homomorphism $\varphi:\text{Adconj}(Q)\to G$ such that $\varphi(|q|)=\langle
q\rangle$ and $qg=q\varphi(g)$ for all $q\in Q,g\in\text{Adconj}(Q)$. This is
why $\text{Adconj}(Q)$ is occasionally referred to as the universal
augmentation group of $Q$.

##  3 Profinite Quandles

A profinite quandle is the inverse limit of a family of finite quandles
organized into an inverse system. Hence profinite quandles are, in a sense,
“approximated” by finite quandles. It is of interest to determine if
properties that hold true for finite quandles hold true for their profinite
counterparts.

###  3.1 Inverse Systems and Profinite Quandles

######  Definition 3.1.

A directed set $(\Lambda,\leq)$ is a pair consisting of a set $\Lambda$ and a
partial order $\leq$ for which every pair of elements has an upper bound.

######  Definition 3.2 (Inverse system of quandles).

Let $(\Lambda,\leq)$ be a directed set. An inverse system of quandles and
quandle homomorphisms over $\Lambda$ is a pair
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
consisting of a family of quandles $\\{Q_{\lambda}\\}_{\lambda\in\Lambda}$
indexed by $\Lambda$ and a family of quandle homomorphisms
$\varphi_{\alpha\beta}:Q_{\beta}\to Q_{\alpha}$ for all $\alpha\leq\beta$ with
$\alpha,\beta\in\Lambda$ such that the following hold:

  1. (i)

The homomorphism $\varphi_{\alpha\alpha}$ is the identity on $Q_{\alpha}$ for
all $\alpha\in\Lambda$.

  2. (ii)

We have
$\varphi_{\alpha\delta}=\varphi_{\alpha\beta}\circ\varphi_{\beta\delta}$ for
all $\alpha,\beta,\delta\in\Lambda$ with $\alpha\leq\beta\leq\delta$.

We call the maps $\varphi_{\alpha\beta}$ the transition maps of the inverse
system.

######  Definition 3.3 (Inverse limit).

Let $(\Lambda,\leq)$ be a directed poset and let
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
be an inverse system of quandles and quandle homomorphisms over $\Lambda$. The
inverse limit of this inverse system is given by

$Q=\varprojlim_{\lambda\in\Lambda}Q_{\lambda}:=\left\\{\vec{q}\in\prod_{\lambda\in\Lambda}Q_{\lambda}:q_{\alpha}=\varphi_{\alpha\beta}(q_{\beta})\text{
for all }\alpha\leq\beta\text{ in }\Lambda\right\\}.$

There are projection maps
$\pi_{\alpha}:\varprojlim_{\lambda\in\Lambda}Q_{\lambda}\to Q_{\alpha}$ from
the inverse limit to each of the factors $Q_{\alpha}$ for all
$\alpha\in\Lambda$ such that

${{\displaystyle{\varprojlim_{\lambda\in\Lambda}}\hskip
2.84526ptQ_{\lambda}}}$${{Q_{\alpha}}}$${{Q_{\beta}}}$$\scriptstyle{\varphi_{\alpha\beta}}$$\scriptstyle{\pi_{\alpha}}$$\scriptstyle{\pi_{\beta}}$

commutes for all $\alpha\leq\beta\in\Lambda$. The projection $\pi_{\alpha}$ is
given by $\pi_{\alpha}(q)=q_{\alpha}$ for all
$q=(q_{\lambda})_{\lambda\in\Lambda}\in\varprojlim_{\lambda\in\Lambda}Q_{\lambda}$.
Note that inverse limit coincides with the categorical notion of limit, or
projective limit.

######  Proposition 3.4.

Let $(\Lambda,\leq)$ be a directed poset and let
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
be an inverse system of quandles and quandle homomorphisms over $\Lambda$. The
inverse limit $Q:=\varprojlim_{\lambda\in\Lambda}Q_{\lambda}$ is a quandle
with operation given by coordinatewise operation. That is,

$q\triangleleft
s=(q_{\lambda})_{\lambda\in\Lambda}\triangleleft(s_{\lambda})_{\lambda\in\Lambda}:=(q_{\lambda}\triangleleft_{\,Q_{\lambda}}s_{\lambda})_{\lambda\in\Lambda}$

for all $q,s\in Q$, where $\triangleleft_{\,Q_{\lambda}}$ is the quandle
operation on $Q_{\lambda}$ for all $\lambda\in\Lambda$.

###### Proof.

First note that $\triangleleft$ is closed. Indeed, if $q,s\in Q$, then
$q\triangleleft
s=(q_{\lambda}\triangleleft_{\,Q_{\lambda}}s_{\lambda})_{\lambda\in\Lambda}$
and since each $\varphi_{\alpha\beta}$ is a quandle homomorphism we have that

$\varphi_{\alpha\beta}(q_{\beta}\triangleleft_{\,Q_{\beta}}s_{\beta})=\varphi_{\alpha\beta}(q_{\beta})\triangleleft_{\,Q_{\alpha}}\varphi_{\alpha\beta}(s_{\beta})=q_{\alpha}\triangleleft_{\,Q_{\alpha}}s_{\alpha}.$

The remaining quandle axioms follow from those of the factors $Q_{\lambda}$.

(Q1) Let $q\in Q$. Observe that

$\displaystyle q\triangleleft q$
$\displaystyle=(q_{\lambda})_{\lambda\in\Lambda}\triangleleft(q_{\lambda})_{\lambda\in\Lambda}$
$\displaystyle=(q_{\lambda}\triangleleft_{\,Q_{\lambda}}q_{\lambda})_{\lambda\in\Lambda}$
$\displaystyle=(q_{\lambda})_{\lambda\in\Lambda},\text{ by ({Q1}) for each
$Q_{\lambda}$}$ $\displaystyle=q.$

(Q2) Let $q,s\in Q$. Observe that

$\displaystyle(q\triangleleft s)\triangleleft^{-1}s$
$\displaystyle=((q_{\lambda}\triangleleft_{\,Q_{\lambda}}s_{\lambda})\triangleleft_{\,Q_{\lambda}}^{-1}s_{\lambda})_{\lambda\in\Lambda}$
$\displaystyle=(q_{\lambda})_{\lambda\in\Lambda},\text{ by ({Q2}) for each
$Q_{\lambda}$}$ $\displaystyle=q.$

Moreover, observe that

$\displaystyle(q\triangleleft^{-1}s)\triangleleft s$
$\displaystyle=((q_{\lambda}\triangleleft_{\,Q_{\lambda}}^{-1}s_{\lambda})\triangleleft_{\,Q_{\lambda}}s_{\lambda})_{\lambda\in\Lambda}$
$\displaystyle=(q_{\lambda})_{\lambda\in\Lambda},\text{ by ({Q2}) for each
$Q_{\lambda}$}$ $\displaystyle=q.$

(Q3) Let $q,r,s\in Q$. Observe that

$\displaystyle(q\triangleleft r)\triangleleft s$
$\displaystyle=(q_{\lambda}\triangleleft_{\,Q_{\lambda}}r_{\lambda})\triangleleft_{\,Q_{\lambda}}s_{\lambda})_{\lambda\in\Lambda}$
$\displaystyle=((q_{\lambda}\triangleleft_{\,Q_{\lambda}}s_{\lambda})\triangleleft_{\,Q_{\lambda}}(r_{\lambda}\triangleleft_{\,Q_{\lambda}}s_{\lambda})$
$\displaystyle=(q\triangleleft s)\triangleleft(r\triangleleft s),$

where we have used (Q3) for each $Q_{\lambda}$. ∎

######  Definition 3.5 (Profinite quandle).

A quandle $Q$ is profinite if it is the inverse limit of an inverse system of
finite quandles and quandle homomorphisms.

######  Example 3.6.

All finite quandles are the inverse limit of the inverse system consisting of
only the finite quandle itself and its identity map.

######  Example 3.7.

A quandle $Q$ is said to be residually finite if for every pair of elements
$q\neq r\in Q$, there exists a finite quandle $Q^{\prime}$ and a quandle
homomorphism $f:Q\rightarrow Q^{\prime}$ such that $f(q)\neq f(r)$.

If $Q$ is a residually finite quandle, it maps by an injective quandle
homomorphism into a profinite quandle $\widehat{Q}$, its profinite completion,
which is the limit of the inverse system of all finite quotient quandles of
$Q$.

Residual finiteness turns out to be a property shared by many of the most
studied quandles: Bardakov, Singh and Singh [18] have shown that knot quandles
and free quandles are residually finite.

######  Remark 3.8.

An inverse system of finite quandles can without loss of generatlity be
assumed to have surjective maps. This is shown in [19] in the case of an
inverse system of finite sets, and the proof applies to an inverse system of
quandles also. We desire the quandle homomorphisms to be surjective since the
inner automorphism functor, Inn, is only a functor on the category of quandles
and surjective quandle homomorphisms, not the category of quandles and quandle
homomorphisms. Hereinafter we assume without further comment that the
homomorphisms in any inverse system of quandles are surjective.

The notions of inverse system, inverse limit, and profiniteness are defined
for groups similarly.

######  Definition 3.9 (Inverse system of groups).

An inverse system of groups and group homomorphisms over a directed poset
$(\Lambda,\leq)$ is a pair
$(\\{G_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
consisting of a family of groups $\\{G_{\lambda}\\}_{\lambda\in\Lambda}$
indexed by $\Lambda$ and a family of group homomorphisms
$\varphi_{\alpha\beta}:G_{\beta}\to G_{\alpha}$ for all $\alpha\leq\beta$ with
$\alpha,\beta\in\Lambda$ such that the following hold:

  1. (i)

The homomorphism $\varphi_{\alpha\alpha}$ is the identity on $G_{\alpha}$ for
all $\alpha\in\Lambda$.

  2. (ii)

We have
$\varphi_{\alpha\delta}=\varphi_{\alpha\beta}\circ\varphi_{\beta\delta}$ for
all $\alpha,\beta,\delta\in\Lambda$ with $\alpha\leq\beta\leq\delta$.

The inverse limit of such an inverse system is given by

$G=\varprojlim_{\lambda\in\Lambda}G_{\lambda}:=\left\\{\vec{g}\in\prod_{\lambda\in\Lambda}G_{\lambda}:g_{\alpha}=\varphi_{\alpha\beta}(g_{\beta})\text{
for all }\alpha\leq\beta\text{ in }\Lambda\right\\}.$

A group $G$ is profinite if it is the inverse limit of an inverse system of
finite groups and group homomorphisms.

Profinite groups have been studied extensively and have a well-developed
theory. [20] Because of the intimate relationships between groups and
quandles, profinite groups will occur throughout our subsequent discussion of
profinite quandle. In the first instance, they provide us with more examples:

######  Example 3.10.

The functor $\text{Conj}:\textbf{Gp}\to\textbf{Qndl}$ admits a left adjoint
$\text{Adconj}:\textbf{Qndl}\to\textbf{Gp}$. Hence Conj preserves all limits.
Therefore $\text{Conj}(G)$ is a profinite quandle whenever $G$ is a profinite
group.

######  Example 3.11.

The functor $\text{Tak}:\textbf{Ab}\to\textbf{Kei}$ admits a left adjoint
$\text{Adtak}:\textbf{Kei}\to\textbf{Ab}$. Hence Tak preserves all limits.
Therefore $\text{Tak}(A)$ is a profinite kei whenever $A$ is a profinite
abelian group.

To fill in these examples of profinite quandles we need examples of profinite
groups:

######  Example 3.12.

For any prime $p$, the additive group of the $p$-adic integers,
${\mathbb{Z}}_{p}$, is a profinite abelian group.

This is easy to see as it is the inverse limit of the totally ordered diagram

${\mathbb{Z}}/p\leftarrow{\mathbb{Z}}/p^{2}\leftarrow{\mathbb{Z}}/p^{3}\leftarrow\ldots$

.

######  Example 3.13.

The profinite completion of the integers $\widehat{\mathbb{Z}}$ is a profinite
abelian group.

######  Example 3.14.

The absolute galois group $G_{K}$ of any field $K$, that is the galois group
of $K_{sep}$, the separable closure of $K$ is a profinite (usually non-
abelian) group. (As an aside, we recall that the separable closure of a
perfect field is the same as its algebraic closure, and that imperfect fields
must be of finite characteristic and positive transcendence degree over the
prime field of the same characteristic.)

######  Example 3.15.

The automorphism group of any (infinite) rooted tree all of whose vertices are
of finite degree is a profinite group.

###  3.2 Topological Properties

Profinite quandles, while the definition as given does not a priori involve a
topology, are, in fact, examples of topological quandles.

######  Definition 3.16 (Topological Quandle).

A topological quandle $(Q,\tau,\triangleleft)$ is a topological space
$(Q,\tau)$ that is also a quandle $(Q,\triangleleft)$ such that the quandle
operations $\triangleleft:Q\times Q\to Q$ and $\triangleleft^{-1}:Q\times Q\to
Q$ are continuous.

Every profinite quandle $Q=\varprojlim_{\lambda\in\Lambda}Q_{\lambda}$ can be
endowed with a Stone topology (compact, Hausdorff, totally disconnected) as a
subspace of $\prod_{\lambda\in\Lambda}Q_{\lambda}$ equipped with the product
topology, where the finite quandles are made into Stone spaces by giving them
each the discrete topology. But any product of Hausdorff totally disconnected
spaces is again Hausdorff totally disconnected, and by Tychonoff’s Theorem,
the arbitrary product of compact spaces is again compact. Hence
$\prod_{\lambda\in\Lambda}Q_{\lambda}$ is a Stone space.

Subspaces of Hausdorff totally disconnected spaces are again Hausdorff totally
disconnected, and by [20, Lemma 1.1.2] we have $\varprojlim
Q_{\lambda}\subseteq\prod Q_{\lambda}$ is closed, hence compact. Therefore $Q$
is a Stone space.

The same argument established the well-known result that every profinite group
$G=\varprojlim_{\lambda\in\Lambda}G_{\lambda}$ can be endowed with a Stone
topology as a subspace of the product topology on
$\prod_{\lambda\in\Lambda}G_{\lambda}$. It is also known that, conversely,
every Stone topological group is a profinite group. Johnstone [21] generalized
this result to models of other algebraic theories which admit a particular
syntactical structure: if a finitary algebraic theory A has a finite complete
set of words in some variable $y$, then every Stone topological A-algebra is
profinite. Johnstone [21] gave easily verifiable criteria for this condition
in the case of theories of an associative operation, but whether the theory of
quandles admits a finite complete set of words, hence Stone topological
quandles are profinite, is an open problem.

Throughout this paper, the existence of the Stone topology on profinite
quandles will be important in a number of proofs.

######  Proposition 3.17.

Let $Q$ be a profinite quandle with inverse system
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
and projection maps $\pi_{\lambda}:Q\to Q_{\lambda}.$ The collection of
subsets

$\mathcal{B}:=\\{\pi_{\lambda}^{-1}(x):x\in Q_{\lambda},\lambda\in\Lambda\\}$

forms a basis for a Stone topology on $Q.$ We call $\mathcal{B}$ the “slim
basis” of $Q$ and its elements “slim basic open sets”.

###### Proof.

We will directly verify that $\mathcal{B}$ is a basis.

  1. 1.

Covering Property: Let $q=(q_{\lambda})_{\lambda\in\Lambda}\in Q$. We have
that $q\in\pi_{\lambda}^{-1}(q_{\lambda})$ for each $\lambda\in\Lambda$.

  2. 2.

Covering of Finite Intersections: Consider $\pi_{\alpha}^{-1}(x)$ and
$\pi_{\beta}^{-1}(y)$ for some elements $x\in Q_{\alpha}$ and $y\in
Q_{\beta}$. If the intersection is empty, there is nothing to show, so suppose
$q=(q_{\lambda})_{\lambda\in\Lambda}\in\pi_{\alpha}^{-1}(x)\cap\pi_{\beta}^{-1}(y)$.
So $\pi_{\alpha}(q)=q_{\alpha}=x$ and $\pi_{\beta}(q)=q_{\beta}=y$. But
$\Lambda$ is directed, so there exits $\gamma$ with $\alpha\leq\gamma$ and
$\beta\leq\gamma$.

Now, by the construction of the inverse limit,
$\varphi_{\alpha\gamma}(q_{\gamma})=q_{\alpha}=x$ and
$\varphi_{\beta\gamma}(q_{\gamma})=q_{\beta}=y$. In fact, for the same reason
$\pi_{\gamma}^{-1}(q_{\gamma})\subset\pi_{\alpha}^{-1}(x)\cap\pi_{\beta}^{-1}(y)$.
But $q=(q_{\lambda})_{\lambda\in\Lambda}\in\pi_{\gamma}^{-1}(q_{\gamma})$. So,
as required, given any element of an intersection of basic opens, there is a
basic open neighborhood contained in the intersection.

Thus $\mathcal{B}:=\\{\pi_{\lambda}^{-1}(x):x\in
Q_{\lambda},\lambda\in\Lambda\\}$ is a basis for the topology on $Q$. ∎

The same argument works for profinite groups. It will generally be easiest to
verify that maps to a profinite quandle are continuous by verifying that the
inverse images of slim basic open sets are open.

###  3.3 Products and Disjoint Unions

######  Proposition 3.18.

If $Q$ and $S$ are profinite quandles, then the product $Q\times S$ is a
profinite quandle.111This result was observed without proof by Singh [16].

###### Proof.

Let
$(\\{Q_{\delta}\\}_{\delta\in\Delta},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Delta})$
and
$(\\{S_{\lambda}\\}_{\lambda\in\Lambda},\\{\psi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Delta})$
be inverse systems for $Q$ and $S$ over directed posets $\Delta$ and
$\Lambda$, respectively, and denote the projections $\pi^{Q}_{\delta}:Q\to
Q_{\delta}$ and $\pi^{S}_{\lambda}\to S\to S_{\lambda}$. Consider the
collection $\\{Q_{\delta}\times
S_{\lambda}\\}_{(\delta,\lambda)\in\Delta\times\Lambda}$. Define a collection
of quandle homomorphisms

$\left\\{\sigma_{(\alpha,\xi),(\beta,\eta)}:Q_{\beta}\times S_{\eta}\to
Q_{\alpha}\times
S_{\xi}\mid(\alpha,\xi),(\beta,\eta)\in\Delta\times\Lambda,(\alpha,\xi)\leq(\beta,\eta)\right\\}$

where
$\sigma_{(\alpha,\xi),(\beta,\eta)}:=\varphi_{\alpha\beta}\times\psi_{\xi\eta}$
and $\Delta\times\Lambda$ is endowed with the product order. It is clear that
$(\\{Q_{\delta}\times
S_{\lambda}\\}_{(\delta,\lambda)\in\Delta\times\Lambda},\\{\sigma_{(\alpha,\xi),(\beta,\eta)}\\}_{(\alpha,\xi)\leq(\beta,\eta)\in\Delta\times\Lambda})$
forms an inverse system, where $\rho_{(\alpha,\xi)}:\varprojlim
Q_{\delta}\times S_{\lambda}\to Q_{\alpha}\times S_{\xi}$ denotes the
projection maps. There are maps from $Q\times S$ to each $Q_{\delta}\times
S_{\lambda}$, namely $\pi_{\delta}^{Q}\times\pi_{\lambda}^{S}$, so by the
universal property of limits, there is a unique continuous quandle
homomorphism $\Psi:Q\times S\to\varprojlim Q_{\delta}\times S_{\lambda}$ such
that the following diagram commutes:

${{Q\times S}}$${{\varprojlim Q_{\delta}\times
S_{\lambda}}}$${{Q_{\alpha}\times S_{\xi}}}$${{Q_{\beta}\times
S_{\eta}}}$$\scriptstyle{\rho_{(\beta,\eta)}}$$\scriptstyle{\rho_{(\alpha,\xi)}}$$\scriptstyle{\sigma_{((\alpha,\xi),(\beta,\eta))}}$$\scriptstyle{\pi^{Q}_{\beta}\times\pi^{S}_{\eta}}$$\scriptstyle{\pi^{Q}_{\alpha}\times\pi^{S}_{\xi}}$$\scriptstyle{\exists!\Psi}$

We claim $\Psi$ is an isomorphism. To see that $\Psi$ is injective, let
$x,y\in Q\times S$ such that $\Psi(x)=\Psi(y).$ Then for all $\delta\in\Delta$
and $\lambda\in\Lambda,$

$\rho_{(\delta,\lambda)}(\Psi(x))=\rho_{(\delta,\lambda)}(\Psi(y))\Rightarrow\pi^{Q}_{\delta}\times\pi^{S}_{\lambda}(x)=\pi^{Q}_{\delta}\times\pi^{S}_{\lambda}(y)\Rightarrow
x=y.$

For surjectivity the quandle $Q\times S$ is a product of a compact spaces,
hence $Q\times S$ is compact. Furthermore $\Psi(Q\times S)\subseteq\varprojlim
Q_{i}\times S_{i}$ is compact and therefore closed since $\varprojlim
Q_{\delta}\times S_{\lambda}$ is Hausdorff. Thus it suffices to show that
$\Psi(Q\times S)$ is dense.

Let $x\in\varprojlim Q_{\delta}\times S_{\lambda}$ and let $y\in
Q_{\delta}\times S_{\lambda}$ such that $\rho_{(\delta,\lambda)}^{-1}(y)$ is a
neighborhood of $x$. We have $\rho_{(\delta,\lambda)}(x)=y,$ so let
$z\in(\pi^{Q}_{\delta}\times\pi^{S}_{\lambda})^{-1}(y).$ Then
$\rho_{(\delta,\lambda)}(\Psi(z))=y,$ meaning
$\Psi(z)\in\rho_{(\delta,\lambda)}^{-1}(y),$ or
$\rho_{(\delta,\lambda)}^{-1}(y)\cap(\Psi(Q\times
S)\setminus\\{x\\})\neq\varnothing$, so $x$ is a limit point of $\Psi(Q\times
S)$. Thus, $\Psi(Q\times S)\subseteq\varprojlim Q_{\delta}\times S_{\lambda}$
is dense and

$Q\times S\cong\varprojlim Q_{\delta}\times S_{\lambda}.$

∎

The same indexing by the products of the two directed sets underlying the
inverse systems as was used to prove Proposition 3.18, but using pairwise
disjoint unions of finite quandles rather than pairwise products let us show
the following.

######  Proposition 3.19.

If $Q$ and $S$ are profinite quandles, then the disjoint union $Q\sqcup S$ is
a profinite quandle.

###### Proof.

Let $Q=\varprojlim_{\delta\in\Delta}Q_{\delta}$ and
$S=\varprojlim_{\lambda\in\Lambda}S_{\lambda}$ be profinite quandles with
inverse systems
$(\\{Q_{\delta}\\}_{\delta\in\Delta},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Delta})$
and
$(\\{S_{\lambda}\\}_{\lambda\in\Lambda},\\{\psi_{\zeta\eta}\\}_{\zeta\leq\eta\in\Lambda})$
over directed posets $\Delta$ and $\Lambda$, respectively. Consider the
disjoint unions $Q_{\delta}\sqcup S_{\lambda}$ of the factors of $Q$ and $S$.
We can index the $Q_{\delta}\sqcup S_{\lambda}$ by the directed poset
$\Delta\times\Lambda$ with the product order. To construct an inverse system
out of $\\{Q_{\delta}\sqcup
S_{\lambda}\\}_{(\delta,\lambda)\in\Delta\times\Lambda}$, we define a
collection of quandle homomorphisms

$\\{\sigma_{(\alpha,\zeta),(\beta,\eta)}:Q_{\beta}\sqcup S_{\eta}\to
Q_{\alpha}\sqcup
S_{\zeta}\\}_{(\alpha,\zeta)\leq(\beta,\eta)\in\Delta\times\Lambda},$

where
$\sigma_{(\alpha,\zeta),(\beta,\eta)}:=\varphi_{\alpha\beta}\sqcup\psi_{\zeta\eta}$.
That each $\sigma_{(\alpha,\zeta),(\beta,\eta)}$ are quandle homomorphisms
follows from the fact that each $\varphi_{\alpha\beta}$ and $\psi_{\zeta\eta}$
are quandle homomorphisms, along with the identity maps
$\text{Id}_{Q_{\beta}},\text{Id}_{S_{\eta}}$. To show that
$(\\{Q_{\delta}\sqcup
S_{\lambda}\\}_{(\delta,\lambda)\in\Delta\Lambda},\\{\sigma_{(\alpha,\zeta),(\beta,\eta)}\\}_{(\alpha,\zeta)\leq(\beta,\eta)\in\Delta\times\Lambda})$
is an inverse system it remains to observe that if
$(\alpha,\zeta)\leq(\beta,\eta)\leq(\gamma,\varepsilon)$, then

$\sigma_{(\alpha,\zeta),(\gamma,\varepsilon)}=\sigma_{(\alpha,\zeta),(\beta,\eta)}\circ\sigma_{(\beta,\eta),(\gamma,\varepsilon)}.$

To see this, let $v\in Q_{\gamma}\sqcup S_{\varepsilon}$. Without loss of
generality suppose $v\in Q_{\gamma}$. Then

$\displaystyle(\sigma_{(\alpha,\zeta),(\beta,\eta)}\circ\sigma_{(\beta,\eta),(\gamma,\varepsilon)})(v)$
$\displaystyle=\sigma_{(\alpha,\zeta),(\beta,\eta)}(\sigma_{(\beta,\eta),(\gamma,\varepsilon)}(v))$
$\displaystyle=\sigma_{(\alpha,\zeta),(\beta,\eta)}((\varphi_{\beta\gamma}\sqcup\psi_{\eta\varepsilon})(v))$
$\displaystyle=\sigma_{(\alpha,\zeta),(\beta,\eta)}(\varphi_{\beta\gamma}(v))$
$\displaystyle=\varphi_{\alpha\beta}(\varphi_{\beta\gamma}(v))$
$\displaystyle=(\varphi_{\alpha\beta}\circ\varphi_{\beta\gamma})(v)$
$\displaystyle=\varphi_{\alpha\gamma}(v),\text{ since
}\varphi_{\alpha\gamma}=\varphi_{\alpha\beta}\circ\varphi_{\beta\gamma}$
$\displaystyle=(\varphi_{\alpha\gamma}\sqcup\psi_{\zeta\varepsilon})(v)$
$\displaystyle=\sigma_{(\alpha,\zeta),(\gamma,\varepsilon)}(v).$

If $v\in S_{\varepsilon}$, then
$\sigma_{(\alpha,\zeta),(\gamma,\varepsilon)}(v)=(\sigma_{(\alpha,\zeta),(\beta,\eta)}\circ\sigma_{(\beta,\eta),(\gamma,\varepsilon)})(v)$
by the same argument on the $\psi_{\zeta\eta}$. Hence $(\\{Q_{\delta}\sqcup
S_{\lambda}\\}_{(\delta,\lambda)\in\Delta\times\Lambda},\\{\sigma_{(\alpha,\zeta),(\beta,\eta)}\\}_{(\alpha,\zeta)\leq(\beta,\eta)\in\Delta\times\Lambda})$
is an inverse system, whose inverse limit is given by

$\varprojlim_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}=\left\\{\vec{v}\in\prod_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}:v_{(\alpha,\zeta)}=\sigma_{(\alpha,\zeta),(\beta,\eta)}(v_{(\beta,\eta)})\text{
for all }(\alpha,\zeta)\leq(\beta,\eta)\right\\}.$

Consider the map

$\Psi:Q\sqcup
S\to\varprojlim_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}$

given by

$\Psi(v):=\begin{cases}(v_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda},&\text{if
}v=(v_{\delta})_{\delta\in\Delta}\in Q;\\\
(v_{\lambda})_{(\delta,\lambda)\in\Delta\times\Lambda},&\text{if
}v=(v_{\lambda})_{\lambda\in\Lambda}\in S.\end{cases}$

We claim that $\Psi$ is a quandle isomorphism. To see that $\Psi$ is
injective, let $u,v\in Q\sqcup S$ with $u\neq v$. If $u\in Q$ and $v\in S$,
then it is clear that $\Psi(v)\neq\Psi(v)$. Hence, we only need to consider
the cases $u,v\in Q$ and $u,v\in S$. Suppose $u,v\in Q$. Since
$Q=\varprojlim_{\delta\in\Delta}Q_{\delta}$, we may write
$u=(u_{\delta})_{\delta\in\Delta}$ and $v=(v_{\delta})_{\delta\in\Delta}$.
Since $u\neq v$, there exists $\alpha\in\Delta$ for which $u_{\alpha}\neq
v_{\alpha}$. It follows that
$\Psi(u)=(u_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$ and
$\Psi(v)=(v_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$ differ in all
coordinates $(\alpha,\lambda)\in\Delta\times\Lambda$, so that
$\Psi(u)\neq\Psi(v)$. The argument for injectivity in the case $u,v\in S$ is
the same. To see that $\Psi$ is surjective, consider an element
$\vec{v}\in\varprojlim_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}$. Since

$v_{(\alpha,\zeta)}=\sigma_{(\alpha,\zeta),(\beta,\eta)}(v_{(\beta,\eta)})=(\varphi_{\alpha\beta}\sqcup\psi_{\zeta\eta})(v_{(\beta,\eta)})$

for all $(\alpha,\zeta)\leq(\beta,\eta)$, it follows from the least upper
bound property of directed posets that if some entry $v_{(\alpha,\zeta)}$ of
$\vec{v}$ lies in $Q_{\alpha}$ (resp., in $S_{\zeta}$), then all entries of
$\vec{v}$ lie in one of the $Q_{\delta}$ (resp., one of the $S_{\lambda}$).
That is, the entries of an element
$\vec{v}\in\varprojlim_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}$ either lie entirely in the components $Q_{\delta}$ of $Q$ or the
components $S_{\lambda}$ of $S$. Moreover, if $\vec{v}$ has entries entirely
in the $Q_{\delta}$ and if $(\alpha,\zeta)\leq(\alpha,\eta)$, with
$\zeta\neq\eta$, then $v_{(\alpha,\zeta)}=v_{(\alpha,\eta)}$ since
$\sigma_{(\alpha,\zeta),(\alpha,\eta)}=\varphi_{\alpha\alpha}\sqcup\psi_{\zeta\eta}$,
and similarly if $\vec{v}$ has entries entirely in the $S_{\lambda}$. Hence,
every element
$\vec{v}=(v_{(\delta,\lambda)})_{(\delta,\lambda)\in\Delta\times\Lambda}\in\varprojlim_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}$ can be written as either
$(v_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$ or
$(v_{\lambda})_{(\delta,\lambda)\in\Delta\times\Lambda}$. This shows that
$\Psi$ is surjective.

Finally, we need to show that $\Psi$ is a quandle homomorphism. We denote the
quandle operation of $Q\sqcup S$ by $\triangleleft_{\,\sqcup}$ and the quandle
operation of
$\varprojlim_{(\delta,\lambda)\in\Delta\times\Lambda}Q_{\delta}\sqcup
S_{\lambda}$ by $\triangleleft_{\,\lim}$. The quandle operations of each of
the factors $Q_{\delta}$ and $S_{\lambda}$ are denoted
$\triangleleft_{\,Q_{\delta}}$ and $\triangleleft_{\,S_{\lambda}}$,
respectively. Let $u,v\in Q\sqcup S$. We break into cases:

Case 1: $u,v\in Q$ or $u,v\in S$.

Suppose $u,v\in Q$. Observe that

$\displaystyle\Psi(u\triangleleft_{\,\sqcup}v)$
$\displaystyle=\Psi(u\triangleleft_{\,Q}v)$
$\displaystyle=\Psi((u_{\delta}\triangleleft_{\,Q_{\delta}}v_{\delta})_{\delta\in\Delta})$
$\displaystyle=(u_{\delta}\triangleleft_{\,Q_{\delta}}v_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$
$\displaystyle=(u_{\delta}\triangleleft_{\,Q_{\delta}\sqcup
S_{\lambda}}v_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$
$\displaystyle=(u_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}\triangleleft_{\,\lim}(v_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$
$\displaystyle=\Psi(u)\triangleleft_{\,\lim}\Psi(v).$

The argument is the same when $u,v\in S$.

Case 2: $u\in Q$ and $v\in S$, or $u\in S$ and $v\in Q$.

Suppose $u\in Q$ and $v\in S$. Observe that

$\displaystyle\Psi(u\triangleleft_{\,\sqcup}v)$ $\displaystyle=\Psi(u)$
$\displaystyle=(u_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}$
$\displaystyle=(u_{\delta}\triangleleft_{\,Q_{\delta}\sqcup
S_{\lambda}}v_{\lambda})_{(\delta,\lambda)\in\Delta\times\Lambda}$
$\displaystyle=(u_{\delta})_{(\delta,\lambda)\in\Delta\times\Lambda}\triangleleft_{\,\lim}(v_{\lambda})_{(\delta,\lambda)\in\Delta\times\Lambda}$
$\displaystyle=\Psi(u)\triangleleft_{\,\lim}\Psi(v).$

The argument is the same when $u\in S$ and $v\in Q$.

Since both cases exhaust all possibilities for $u,v\in Q\sqcup S$, it follows
that $\Psi$ is a quandle homomorphism. Since $\Psi$ is also bijective, it
follows that $\Psi$ is a quandle isomorphism. Hence $Q\sqcup S$ is profinite.
∎

Proposition 3.18 in fact follows from a more general theorem, the proof of
which is more involved. We stated and proved the case of finite products
separately because of the strong parallels between its proof as given and the
proof of the preservation of profiniteness under finite disjoint unions.

######  Theorem 3.20.

Let $\\{Q_{i}\;|\;i\in\mathcal{I}\\}$ be a set of profinite quandles, then
$\prod_{i\in\mathcal{I}}Q_{i}$ with coordinatewise operations is a profinite
quandle.

###### Proof.

Let $Q_{i}=\varprojlim_{\lambda\in\Lambda_{i}}Q_{\lambda}$ for each
$i\in\mathcal{I}$. Here we suppress mention of the maps
$\varphi_{\lambda,\mu}:Q_{\lambda}\rightarrow Q_{\mu}$ for
$\mu\leq\lambda\in\Lambda_{i}$, though we will have cause to mention them
later. We proceed by showing that $\prod_{i\in\mathcal{I}}Q_{i}$ can also be
realized as the limit of an inverse system of finite quandles over a directed
set built from the $\Lambda_{i}$’s and the finite subsets of $\mathcal{I}$
ordered by reverse inclusion.

The underlying set of this directed set is

$\Lambda_{\mathcal{I}}:=\coprod_{\\{J\subseteq\mathcal{I}\;|\;0<|J|<\infty\\}}\prod_{i\in
J}\Lambda_{i}.$

We denote elements of this disjoint union of products by
$(J,\langle\lambda_{i}\rangle_{i\in J})$, where the elements of the
$J$-indexed tuple lie in the directed set corresponding to the index, that is
$\lambda_{i}\in\Lambda_{i}$.

This set is then equipped with a partial order given by
$(J,\langle\lambda_{i}\rangle_{i\in J})\leq(K,\langle\mu_{i}\rangle_{i\in K})$
precisely when $J\subseteq K$ and for all $i\in J$, $\lambda_{i}\leq\mu_{i}$
in the partial ordering which makes $\Lambda_{i}$ a directed set. Reflexivity
and anti-symmetry are trivial and transitivity is easy to check.

To see that $(\Lambda_{\mathcal{I}},\leq)$ is a directed set, consider
$(J,\langle\lambda_{i}\rangle_{i\in J})$ and $(K,\langle\mu_{i}\rangle_{i\in
K})$. Then $(J\cup K,\langle\nu_{i}\rangle_{i\in J\cup K})$ where
$\nu_{i}=\lambda_{i}$ if $i\in J\setminus K$, $\nu_{i}=\mu_{i}$ if $i\in
K\setminus J$ and $\nu_{i}\in\Lambda_{i}$ is chosen so that
$\lambda_{i}\leq\nu_{i}$ and $\mu_{i}\leq\nu_{i}$ in $\Lambda_{i}$ by the
directedness of $\Lambda_{i}$ whenever $i\in J\cap K$, plainly provides the
needed element greater than or equal to both
$(J,\langle\lambda_{i}\rangle_{i\in J})$ and $(K,\langle\mu_{i}\rangle_{i\in
K})$.

Now, consider the inverse system of finite quandles indexed by
$\Lambda_{\mathcal{I}}$, in which

$Q_{(J,\langle\lambda_{i}\rangle_{i\in J})}:=\prod_{i\in J}Q_{\lambda_{i}}$

and the projection
$\varphi_{(J,\langle\lambda_{i}\rangle),(K,\langle\mu_{i}\rangle)}$ is given
by the projection
$\pi:\prod_{J}Q_{\lambda_{i}}\rightarrow\prod_{K}Q_{\lambda_{i}}$ onto the
coordinates lying in $K$, followed by
$\prod_{K}\varphi_{\lambda_{i},\mu_{i}}$, the product of the quandle
homomorphisms in the diagrams over the $\Lambda_{i}$’s for $i\in K$ whose
limits are the original $Q_{i}$’s.

We claim that the product $\prod_{i\in\mathcal{I}}Q_{i}$ is the limit of this
inverse system. We proceed by first showing that the same data determines the
elements of both:

An element of $\prod_{i\in\mathcal{I}}Q_{i}$ is in the first instance an
$\mathcal{I}$-tuple of elements of the $Q_{i}$’s for $i\in\mathcal{I}$, but an
element of $Q_{i}$ is a $\Lambda_{i}$-tuple $\langle
q_{\lambda}\rangle_{\lambda\in\Lambda_{i}}$ with $q_{\lambda}\in Q_{\lambda}$
of elements of the finite quandles $Q_{\lambda}$ $\lambda\in\Lambda_{i}$
satisfying the compatibility condition
$\varphi_{\lambda,\mu}(q_{\mu})=q_{\lambda}$.

Now, observe that this is the same data that gives the portion of the
$\Lambda_{\mathcal{I}}$-tuple (satisfying a corresponding compatibility
condition) in entries where the set in the index is a singleton. But the
compatibility condition implies that the rest of the entries are completely
determined by those with a singleton as the indexing set.

It remains only to show that the topologies induced by the two constructions
agree. Elements of the product of the profinite quandles are those elements of
$\prod_{i\in\mathcal{I}}\prod_{\lambda\in\Lambda_{i}}Q_{i}$, with the entries
in the $\Lambda_{i}$-tuples satisfying the compatibility condition for each
$i\in\mathcal{I}$. We may identify them with the coordinates of elements in
$\varprojlim_{\lambda\in\Lambda_{i}}Q_{(J,\langle\lambda_{i}\rangle_{\in}J)}$
in which $J$ is a singleton. The argument above showed that the restriction of
the projection from $\prod_{\mathcal{I}}\prod_{\Lambda_{i}}Q_{\lambda_{i}}$ to
those coordinates gives a bijection between the limit we claim witnesses the
profiniteness of the product and the product itself. But projections are
continuous open maps, and therefore the restriction is a homeomorphism. ∎

###  3.4 Subquandles of Profinite Quandles

Given a profinite quandle, it is natural to inquire as to whether or not its
subquandles are also profinite. We provide a necessary and equivalent
condition for when subquandles of a profinite quandle are profinite.

It is easy to see that in general subquandles of profinite quandles need not
themselves be profinite: any infinite residually finite quandle (with the
discrete toplogy) may be identified with a proper non-profinite subquandle or
its profinite completion, as for example
$\text{Tak}({\mathbb{Z}})\prec\text{Tak}(\hat{\mathbb{Z}}).$

In giving a characterization of those subquandles of a profinite quandle which
are themselves profinite, it will be useful to have a characterization of
dense subquandles.

######  Proposition 3.21.

If $Q$ is a profinite quandle with inverse system
$\\{\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda}\\}$
and $S\preceq Q$, then $S$ is dense if and only if the restrictions of the
projection homomorphisms $\pi_{\lambda}:Q\rightarrow Q_{\lambda}$ to $S$ are
surjective.

###### Proof.

Recall that a subspace being dense is equivalent to having a non-empty
intersection with every non-empty open set, which is in turn equivalent to
having a non-empty intersection with every open set in a basis. Applying this
observation to the slim basis, we see that $S$ is dense if and only if it
intersects every $\pi_{\lambda}^{-1}(x)$. This is equivalent to every $x\in
Q_{\lambda}$ having a nonempty preimage under $\pi_{\lambda}|_{S}$ for all
$\lambda\in\Lambda$, thus establishing the proposition. ∎

######  Proposition 3.22.

Let $Q$ be a profinite quandle. A subquandle $S\preceq Q$ is profinite if and
only if $S$ is closed in $Q$. 222This result was independently obtained by
Singh [16] Proposition 4.8.

###### Proof.

Let
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
be an inverse system for $Q$ over a directed poset $\Lambda$ and denote the
projections $\pi_{\lambda}:Q\to Q_{\lambda}$.

$(\Rightarrow)$ Suppose $S\preceq Q$ is profinite. This implies that $S$ is
compact. Since $Q$ is Hausdorff, this implies that $S$ is closed in $Q$.

$(\Leftarrow)$ Suppose $S$ is closed in $Q$. We can restrict
$\varphi_{\alpha\beta}$ to $\pi_{\beta}(S)$, denoted
$\hat{\varphi}_{\alpha\beta}:=\varphi_{\alpha\beta}|_{\pi_{\beta}(S)},$ which
forms an inverse system
$(\\{\pi_{\lambda}(S)\\}_{\lambda\in\Lambda},\\{\hat{\varphi}_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
with projections $\rho_{\alpha}:\varprojlim\pi_{\alpha}(S)\to\pi_{\alpha}(S)$.
There are maps from $S$ to each $\pi_{\alpha}(S),$ so by the universal
property of limits, there is a unique continuous quandle homomorphism
$\Psi:S\to\varprojlim\pi_{\alpha}(S).$

${{S}}$${{\varprojlim\pi_{\alpha}(S)}}$${{\pi_{\alpha}(S)}}$${{\pi_{\beta}(S)}}$$\scriptstyle{\rho_{\beta}}$$\scriptstyle{\rho_{\alpha}}$$\scriptstyle{\hat{\varphi}_{\alpha\beta}}$$\scriptstyle{\pi_{\beta}}$$\scriptstyle{\pi_{\alpha}}$$\scriptstyle{\exists!\Psi}$

We claim $\Psi$ is an isomorphism. To see that $\Psi$ is injective, let
$q_{1},q_{2}\in S$ such that $\Psi(q_{1})=\Psi(q_{2}).$ Then for all
$\alpha\in\Lambda,$

$\rho_{\alpha}(\Psi(q_{1}))=\rho_{\alpha}(\Psi(q_{2}))\Rightarrow\pi_{\alpha}(q_{1})=\pi_{\alpha}(q_{2})\Rightarrow
q_{1}=q_{2}.$

For surjectivity, $S$ is a closed subspace of a compact space, hence $S$ is
compact. Furthermore, $\Psi(S)\subseteq\varprojlim\pi_{\alpha}(S)$ is compact
and therefore closed since $\varprojlim\pi_{\alpha}(S)$ is Hausdorff. But by
construction all of the maps $\pi_{\alpha}:S\rightarrow\pi_{\alpha}(S)$ are
surjective, thus by Proposition 3.21 $S$ is dense in the profinite
$\varprojlim\pi_{\alpha}(S)$ and we are done.

∎

An identical proof to that of Proposition 3.23, excepting that the word
“quandle” replaced with ”group” throughout, gives us

######  Proposition 3.23.

Let $G$ be a profinite group. A subgroup $S<G$ is profinite if and only if $S$
is closed in $G$.

We will have occasion to use this in one of our examples below.

Topological closure gives a way of producing profinite subquandles from
arbitrary subquandles due the the following result about general topological
quandles.

######  Proposition 3.24.

Let $Q$ be a topological quandle and $Q^{\prime}\preceq Q$ be a subquandle.
Then $\overline{Q^{\prime}},$ the topological closure of $Q^{\prime},$ is a
subquandle of $Q$.

###### Proof.

Let $x,y\in\overline{Q^{\prime}}$ and suppose $x\triangleleft
y\not\in\overline{Q^{\prime}}.$ We have $\overline{Q^{\prime}}^{c}$ is open,
so there is a neighborhood $U\subseteq Q\times Q$ of $(x,y)$ such that
$u\triangleleft v\not\in\overline{Q^{\prime}}$ for every $(u,v)\in U.$
Furthermore, there exists neighborhoods $V_{x},V_{y}\subseteq Q$ of $x$ and
$y,$ respectively, such that $V_{x}\times V_{y}\subseteq U.$ Let $u\in
V_{x}\cap Q^{\prime}$ and $v\in V_{y}\cap Q^{\prime}$. Then $u\triangleleft
v\in Q^{\prime}\subseteq\overline{Q^{\prime}}$, but $(u,v)\in U$ by
construction, a contradiction. ∎

##  4 Profinite Analogs of Inn(Q)

It is fairly easy to show that the inner automorphism group of a profinite
group is always profinite, being the quotient of the profinite group by its
center which is necessarily a closed normal subgroup. However, as
Counterexample 4.3 shows, the inner automorphism group of a profinite quandle
need not be a profinite group. Therefore, we introduce profinite analogs of
the inner automorphism group (i.e. profinite groups that act continously on
the profinite quandle in a way analogous to the action of the inner
automorphism group).

######  Definition 4.1 (Profinite inner automorphism group).

Let $Q$ be a profinite quandle with inverse system
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$.
The functor Inn provides an inverse system over
$\\{\text{Inn}(Q_{\lambda})\\}_{\lambda\in\Lambda}$, so define the profinite
inner automorphism group over D to be
$\widehat{\text{Inn}}_{D}(Q):=\varprojlim\text{Inn}(Q_{\alpha})$, where $D$
refers to the diagram of the prescribed inverse system.

This construction explicitly depends on the diagram used to witness
profiniteness; given two filtered diagrams $D$ and $D^{\prime}$ under a
profinite quandle $Q$, $\widehat{\text{Inn}}_{D}(Q)$ and
$\widehat{\text{Inn}}_{D^{\prime}}(Q)$ are not necessarily isomorphic. While
this presents an obvious disadvantage, $\widehat{\text{Inn}}_{D}(Q)$ still
provides a description of a $Q$ in terms of the inner automorphism groups of
the component quandles while also being profinite. Furthermore, there are
properties of $\widehat{\text{Inn}}_{D}(Q)$ that arise independent of the
choice of diagram.

######  Proposition 4.2.

Let $Q$ be a profinite quandle with inverse system
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$.
Then $\text{Inn}(Q)$ is isomorphic to a dense subgroup of
$\widehat{\text{Inn}}_{D}(Q).$

###### Proof.

Denote projections
$\pi_{\lambda}:\widehat{\text{Inn}}_{D}(Q)\to\text{Inn}(Q_{\lambda}).$ For an
inner automorphism $S_{x}\in\text{Inn}(Q)$ where $q\mapsto q\triangleleft x,$
there are induced inner automorphisms
$S_{x_{\lambda}}\in\text{Inn}(Q_{\lambda})$ by $q\mapsto q\triangleleft
x_{\lambda}$ where $\pi_{\lambda}(x)=x_{\lambda}$ for each
$\lambda\in\Lambda,$ so there are induced maps
$\rho_{\lambda}:\text{Inn}(Q)\to\text{Inn}(Q_{\lambda})$ where $S_{x}\mapsto
S_{x_{\lambda}}.$ Then there is a unique homomorphism
$\Psi:\text{Inn}(Q)\to\widehat{\text{Inn}}_{D}(Q)$ by the universal property
of limits. Moreover, it is easily seen that $\Psi$ is injective. We conclude
by showing $\Psi(\text{Inn}(Q))\subseteq\widehat{\text{Inn}}_{D}(Q)$ is dense.

Let $x\in\Psi(\text{Inn}(Q))^{c}$ and let $y\in\text{Inn}(Q_{i})$ such that
$\pi_{i}^{-1}(y)$ is a neighborhood of $x.$ We have $\pi_{i}(x)=y,$ so let
$z\in\rho_{i}^{-1}(y).$ Then $\pi_{i}(\Psi(z))=y,$ meaning
$\Psi(Z)\in\pi_{i}^{-1}(y)$, or
$\pi_{i}^{-1}(y)\cap\Psi(\text{Inn}(Q))\neq\varnothing,$ so $x$ is a limit
point of $\Psi(\text{Inn}(Q)).$ Thus,
$\Psi(\text{Inn}(Q))\subseteq\widehat{\text{Inn}}_{D}(Q)$ is dense. ∎

######  Counterexample 4.3.

Proposition 3.23 and Proposition 4.2 now allow us to construct a profinite
quandle whose algebraic inner automorphism groups is not profinite.

Let $M_{n}$ be the subquandle of $\text{Conj}(\mathfrak{S}_{n})$ consisting of
all $2$-cycles, for $n\geq 2$. (As an aside, $M_{3}=T$). Observe
$\text{Inn}(M_{n})\cong\mathfrak{S}_{n}$ and that symmetries are the
transpositions. Now consider $Q=\prod_{n=2}^{\infty}M_{n}$. $Q$ is profinite
by Theorem 3.20, but for simplicity we consider its profiniteness being
witnessed by it being the limit of

$M_{2}\leftarrow M_{2}\times M_{3}\leftarrow M_{2}\times M_{3}\times
M_{4}\leftarrow\ldots.$

(Here the map from the $n+1$-fold product to the $n$-fold product is
projection onto the first $n$ coordinates.)

$\text{Inn}(Q)$ is generated by the symmetries $S_{\langle
q_{i}\rangle_{i=2}^{\infty}}=\langle S_{q_{i}}\rangle_{i=2}^{\infty}$, each of
which is of order 2, as each $S_{q_{i}}$ is of order 2. And the same applies
to the finite quandles $\prod_{n=2}^{N}M_{n}$ in the directed system.

By Proposition 4.2 it is a dense subgroup of $\widehat{\text{Inn}}_{D}(Q)$,
and thus by Proposition 3.23 would be all of $\widehat{\text{Inn}}(Q)$ were it
profinite.

However, it is not all of $\widehat{\text{Inn}}_{D}(Q)$. To see this we need
to find both $\text{Inn}(Q)$ and $\widehat{\text{Inn}}_{D}(Q)$.

Now, $\text{Inn}(\prod_{n=2}^{N}M_{n})$ is the subgroup of
$\prod_{n=2}^{N}\text{Inn}(M_{n})=\prod_{n=2}^{N}\mathfrak{S}_{n}$ generated
by elements which are $N$-tuples of transpositions. And $\text{Inn}(Q)$ is the
subgroup of
$\prod_{n=2}^{\infty}\text{Inn}(M_{n})=\prod_{n=2}^{\infty}\mathfrak{S}_{n}$
generated by sequences of transpositions, one in each symmetric group. Thus,
as elements of the infinite product of the symmetric groups, the coordinates
of an element of $\text{Inn}(Q)$ can all be expressed as products of $k$
transpositions, for some fixed $k$.

However, $\widehat{\text{Inn}}_{D}(Q)$ is the limit of the sequence of
$\text{Inn}(\prod_{n=2}^{N}M_{n})$’s. Because an element of a symmetric group
expressible as a product of $k$ transpositions can also be expressed as a
product of $k+2j$ transpositions for $j=0,1,2,...$ this limit contains
elements like

$\ell=\langle\ell_{n}\rangle_{n=2}^{\infty},$

where

$\ell_{n}=(12\ldots 2\lfloor\frac{n}{2}\rfloor).$

The projection of this element onto $\text{Inn}(\prod_{n=2}^{N}M_{n})$ has
coordinates all of which which can be expressed as a product of
$2\lfloor\frac{n}{2}\rfloor$ transpositions.

But $\ell$ is not an element of $\text{Inn}(Q)$, as the number of
transpositions need to express its coordinates is unbounded.

While $\widehat{\text{Inn}}_{D}(Q)$ is dependent on the diagram used witness
profiniteness of $Q,$ there are certain properties that are true independent
of diagram. This is especially useful because we are afforded advantages from
a profinite perspective that we would otherwise lack with the standard inner
automorphism group

######  Proposition 4.4.

Let $Q$ be a profinite quandle with inverse system
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$.
Then the profinite inner automorphism group $\widehat{\text{Inn}}_{D}(Q)$ acts
continuously on $Q$.

###### Proof.

Define $\rho_{\lambda}:\text{Inn}(Q_{\lambda})\times Q_{\lambda}\to
Q_{\lambda}$ by $(\varphi,q)\mapsto\varphi(q).$ Each $\rho_{\lambda}$ is
continuous as both spaces are discrete. Now define
$\rho:\prod\text{Inn}(Q_{\lambda})\times\prod Q_{\lambda}\to\prod Q_{\lambda}$
by $\rho=\prod\rho_{\lambda}.$ Then $\rho$ is continuous by construction, and
$\widehat{\text{Inn}}_{D}(Q)\times
Q\subseteq\prod\text{Inn}(Q_{\lambda})\times\prod Q_{\lambda}$ is a subspace,
meaning the restriction $\rho|_{\widehat{\text{Inn}}_{D}(Q)\times Q}$ is also
continuous. Furthermore, for
$(\varphi,q)=((\prod_{\lambda\in\Lambda}\varphi_{q^{\prime}_{\lambda}}),(\prod_{\lambda\in\Lambda}q_{\lambda}))$,
we have

$\rho((\varphi,q))=\prod_{\lambda\in\Lambda}\rho_{\lambda}(\varphi_{q^{\prime}_{\lambda}},q_{\lambda})=\prod_{\lambda\in\Lambda}\varphi_{q^{\prime}_{\lambda}}(q_{\lambda})\in
Q$

by commutativity of the diagrams $D$ and $\text{Inn}(D)$. Thus
$\rho|_{\widehat{\text{Inn}}_{D}(Q)\times Q}$ is precisely the action by
$\widehat{\text{Inn}}_{D}(Q)$ on $Q$. ∎

While $\widehat{\text{Inn}}_{D}(Q)$ has several nice properties, it is still a
construction dependent on choice of diagram. The profinite completion
$\widehat{\text{Inn}(Q)}$ of $\text{Inn}(Q)$ admits many of the same
properties while arising independent of choice of diagram.

######  Proposition 4.5.

If $Q$ is a profinite quandle, then $\widehat{\text{Inn}(Q)}$ acts
continuously on $Q.$

###### Proof.

Let
$(\\{Q_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
be an inverse system for $Q$ over a directed poset $\Lambda$ and denote the
projections $\pi_{\lambda}:Q\to Q_{\lambda}$. For an inner automorphism
$S_{x}\in\text{Inn}(Q)$ where $q\mapsto q\triangleleft x,$ there are induced
inner automorphisms $S_{x_{\lambda}}\in\text{Inn}(Q_{\lambda})$ by $q\mapsto
q\triangleleft x_{\lambda}$ where $\pi_{\lambda}(x)=x_{\lambda}$ for each
$\lambda\in\Lambda,$ so there are induced maps
$\rho_{\lambda}:\text{Inn}(Q)\to\text{Inn}(Q_{\lambda})$ where $S_{x}\mapsto
S_{x_{\lambda}}.$

Therefore, by the universal property of profinite completions, there is a map
$\text{Inn}(Q)\to\widehat{\text{Inn}(Q)}$, which induces maps
$\widehat{\text{Inn}(Q)}\to\text{Inn}(Q_{\lambda})$ for each
$\lambda\in\Lambda.$ So by the universal property of limits, there exists a
unique continuous homomorphism
$\Psi:\widehat{\text{Inn}(Q)}\to\widehat{\text{Inn}}_{D}(Q)$ \- where $D$ is
the diagram of the above inverse system - making the diagram below commute:

${{\widehat{\text{Inn}(Q)}}}$${{\widehat{\text{Inn}}_{D}(Q)}}$${{\text{Inn}(Q_{\alpha})}}$${{\text{Inn}(Q_{\beta})}}$$\scriptstyle{\rho_{\beta}}$$\scriptstyle{\rho_{\alpha}}$$\scriptstyle{\hat{\varphi}_{ij}}$$\scriptstyle{\exists!\Psi}$

But then the action $\widehat{\text{Inn}(Q)}\times Q\to Q$ factors through
continuous maps

${{\widehat{\text{Inn}(Q)}\times Q}}$${{\hskip 14.22636ptQ\hskip
14.22636pt}}$${{\widehat{\text{Inn}}_{D}(Q)\times
Q}}$$\scriptstyle{\Psi\times\text{Id}}$

Thus, $\widehat{\text{Inn}(Q)}$ acts continuously on $Q$. ∎

As in the finite setting where unions of orbits of the action of
$\text{Inn}(Q)$ give rise of subquandles, in the profinite setting we have

######  Proposition 4.6.

If $Q$ is a profinite quandle, and $C$ is a compact subset of $Q$, then
$Q_{C}$, the union of the orbits of the elements of $C$ under the action of
$\widehat{\text{Inn}(Q)}$, is a profinite subquandle of $Q$.

###### Proof.

The set $Q_{C}$ is a subquandle since it is closed under the action of
$\widehat{\text{Inn}(Q)}$, and thus a fortiori under the actions of the
symmetries of its own elements and their inverses. It is also the image of the
action map restricted to $C\times\widehat{\text{Inn}(Q)}$. The continuous
image of a compact space is compact and $Q$ is Hausdorff, hence $Q_{C}$ is
closed. Thus by Proposition 3.22 it is a profinite subquandle. ∎

Observe that if the hypothesis of compactness were omitted or
$\widehat{\text{Inn}(Q)}$ were replaced with $\text{Inn}(Q)$, the union of
orbits would still be a subquandle of $Q$ but it would not in general be
profinite.

##  5 (Profinitely) Algebraically Connected Profinite Quandles

We establish a characterization of profinite quandles with a single orbit
under action by the profinite completion of their inner automorphism groups
analogous to that of Ehrman et al. [17] in the finite case.

######  Definition 5.1.

A profinite quandle $Q$ is said to be (profinitely) algebraically connected if
there is a single orbit under the action by $\widehat{\text{Inn}(Q)}.$

As we saw in Counterexample 4.3 above, the inner automorphism group of a
profinite quandle is not in general a profinite group. In the constructions
which follow, we will see that actions of profinite analogues of the inner
automorphism group give results in the profinite context closely analogous to
those involving actions of the inner automorphism group in the finite setting.
Therefore in the profinite context, despite the conflict with the usual usage,
we will drop the adverb “profinitely” and simple call quandles fitting
Definition 5.1 “algebraically connected profinite quandles.”

######  Definition 5.2.

If $G$ is a topological group and $S\subseteq G$, we say that $S$
topologically generates $G$ if the topological closure of the subgroup
generated by $S$ is equal to $G$.

The previous definition will be used throughout our discussion of
$\widehat{\text{Inn}(Q)}$ as a topological group

For an algebraically connected profinite quandle $Q,$ any choice of $q\in Q$
induces a $\widehat{\text{Inn}(Q)}$-equivariant bijection between $Q$ and
$H\backslash\widehat{\text{Inn}(Q)}$ where $H$ is the stabilizer of $q.$
Therefore, we may represent $Q$ by
$(\\{Hg_{\delta}\\}_{\delta\in\Delta},\triangleleft)$ where
$Hg_{1}\triangleleft Hg_{2}=Hg_{3}$ if $q_{1}\triangleleft q_{2}=q_{3}.$ We
define an augmentation map
$|\cdot|:Q=H\backslash\widehat{\text{Inn}(Q)}\to\widehat{\text{Inn}(Q)}$ by
$|Hg_{\delta}|=g$ where $g\in\widehat{\text{Inn}(Q)}$ such that $x\cdot
g=x\triangleleft Hg_{\delta}$ for all $x\in Q$. To distinguish between the
augmentation map and the order of a group, we will denote $|H|$ as the order
of the subgroup $H$ and $|Hh|$ as the augmentation of $H$ as a right coset in
$H\backslash\widehat{\text{Inn}(Q)}.$

######  Theorem 5.3.

Let $Q$ be an algebraically connected profinite quandle, let $H$ be the
stabilizer of $q\in Q$ under action by $\widehat{\text{Inn}(Q)}$, and let
$\\{g_{\delta}\\}_{\delta\in\Delta}$ be all coset representatives of $H$ not
in $H$. Then $|Hh|\in Z(H)$ and $\widehat{\text{Inn}(Q)}$ is topologically
generated by $\\{|Hh|,\\{|Hg_{\delta}|\\}_{\delta\in\Delta}\\}$, where
$|Hg_{\delta}|=g_{\delta}^{-1}|Hh|g_{\delta}.$

###### Proof.

First, since $\\{|Hh|,\\{|Hg_{\delta}|\\}_{\delta\in\Delta}\\}$ attains a
representative for each coset of $H,$ the representation from the discussion
above implies that $\\{|Hh|,\\{|Hg_{\delta}|\\}_{\delta\in\Delta}\\}$
algebraically generates $\text{Inn}(Q)$. Since $\text{Inn}(Q)$ is dense in
$\widehat{\text{Inn}(Q)}$, we have
$\\{|Hh|,\\{|Hg_{\delta}|\\}_{\delta\in\Delta}\\}$ topologically generates
$\widehat{\text{Inn}(Q)}$.

Now, by Claim 4.4 of [17], for all $g\in\widehat{\text{Inn}(Q)}$ we have
$|Hg_{\delta}g|=g^{-1}|Hg_{\delta}|g$. Then
$|Hg_{\delta}|=|Hhg_{\delta}|=g_{\delta}^{-1}|Hh|g_{\delta}.$ For $g\in H$, we
have $|Hh|=|Hhg|=g^{-1}|Hh|g$, meaning $|Hh|\in Z(H)$. ∎

######  Proposition 5.4.

For a group $G$, a proper subgroup $H<G$, and some $h\in Z(H),$ define
$\triangleleft:H\backslash G\times H\backslash G\to H\backslash G$ by
$(Hg,Hk)\mapsto Hgk^{-1}hk$ and $\triangleleft^{-1}:H\backslash G\times
H\backslash G\to H\backslash G$ by $(Hg,Hk)\mapsto Hgk^{-1}h^{-1}k$. Then
$Q_{G,H,h}:=(H\backslash G,\triangleleft,\triangleleft^{-1})$ is a quandle,
called the right coset quandle of G with respect to H.

###### Proof.

Let $Hg,Hk,Hm\in Q_{G,H,h}$ be distinct.

  1. (Q1)

We have

$Hg\triangleleft Hg=Hgg^{-1}hg=Hhg=Hg.$

  2. (Q2)

We have

$\displaystyle(Hg\triangleleft Hm)\triangleleft(Hk\triangleleft Hm)$
$\displaystyle=Hgm^{-1}hm\triangleleft Hkm^{-1}hm$
$\displaystyle=Hgm^{-1}hmm^{-1}h^{-1}mk^{-1}hkm^{-1}hm$
$\displaystyle=Hgk^{-1}hkm^{-1}hm$ $\displaystyle=Hgk^{-1}hk\triangleleft Hm$
$\displaystyle=(Hg\triangleleft Hk)\triangleleft Hm.$

  3. (Q3)

We have

$\displaystyle(Hg\triangleleft Hk)\triangleleft^{-1}Hk$
$\displaystyle=Hgk^{-1}hk\triangleleft^{-1}Hk=Hgk^{-1}hkk^{-1}h^{-1}k=Hg,$
$\displaystyle(Hg\triangleleft^{-1}Hk)\triangleleft Hk$
$\displaystyle=Hgk^{-1}h^{-1}k\triangleleft Hk=Hgk^{-1}h^{-1}kk^{-1}hk=Hg.$

Thus, $Q_{G,H,h}$ is a quandle. ∎

Given group homomorphisms, there is an induced quandle homomorphism on their
respective right coset quandles.

######  Lemma 5.5.

For a group $G$, let $H<G$ and $h\in Z(H)$. Define a surjective group
homomorphism $\varphi:G\to\Gamma.$ Then there is an induced surjective quandle
homomorphism $\hat{\varphi}:Q_{G,H,h}\to Q_{\Gamma,\varphi(H),\varphi(h)}$ by
$Hg\mapsto\varphi(H)\varphi(g)$.

###### Proof.

First, note $\varphi(h)\in Z(\varphi(H)).$ Let $Hg,Hk\in Q_{G,H,h}.$ Then

$\displaystyle\hat{\varphi}(Hg\triangleleft_{G}Hk)$
$\displaystyle=\hat{\varphi}(Hgk^{-1}hk)$
$\displaystyle=\varphi(H)\varphi(g)\varphi(k^{-1})\varphi(h)\varphi(k)$
$\displaystyle=\varphi(H)\varphi(g)\varphi(k)^{-1}\varphi(h)\varphi(k)$
$\displaystyle=\varphi(H)\varphi(g)\triangleleft_{\Gamma}\varphi(H)\varphi(k),$

hence $\hat{\varphi}$ is a quandle homomorphism.

To see that $\hat{\varphi}$ is surjective, let $\varphi(H)\gamma\in
Q_{\Gamma,\varphi(H),\varphi(h)}.$ Since $\varphi$ is surjective, there exists
some $g\in G$ such that $\varphi(g)=\gamma.$ Then
$\hat{\varphi}(Hg)=\varphi(H)\varphi(g)=\varphi(H)\gamma.$

∎

######  Theorem 5.6.

Let $G$ be a profinite group and $H<G$. Further, for coset representatives
$\\{g_{\delta}\\}_{\delta\in\Delta}$ of $H$, suppose that $G$ is topologically
generated by $\\{g_{\delta}^{-1}hg_{\delta}\\}_{\delta\in\Delta}$ for $h\in
Z(H).$ Then $Q_{G,H,h}$ is an algebraically connected profinite quandle.

###### Proof.

Note coset representative notation is omitted when useful.

Let
$(\\{G_{\lambda}\\}_{\lambda\in\Lambda},\\{\varphi_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
be an inverse system for $G$ over a directed poset $\Lambda$ and denote the
projections $\pi_{\lambda}:G\to G_{\lambda}$. We begin by showing that
$Q_{G,H,h}$ is profinite. From the above lemma, there are induced surjective
maps $\hat{\pi}_{\lambda}:Q_{G,H,h}\to
Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}$ and
$\hat{\varphi}_{\alpha\beta}:Q_{G_{\beta},\pi_{\beta}(H),\pi_{\beta}(h)}\to
Q_{G_{\alpha},\pi_{\alpha}(H),\pi_{\alpha}(h)},$ meaning
$(\\{Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}\\}_{\lambda\in\Lambda},\\{\hat{\varphi}_{\alpha\beta}\\}_{\alpha\leq\beta\in\Lambda})$
forms an inverse system. By the universal property of limits, we obtain a
unique continuous quandle homomorphism $\Psi:Q_{G,H,h}\to\varprojlim
Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}$ such that the following
diagram commutes.

${{Q_{G,H,h}}}$${{\varprojlim
Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}}}$${{Q_{G_{\alpha},\pi_{\alpha}(H),\pi_{\alpha}(h)}}}$${{Q_{G_{\beta},\pi_{\beta}(H),\pi_{\beta}(h)}}}$$\scriptstyle{\rho_{\beta}}$$\scriptstyle{\rho_{\alpha}}$$\scriptstyle{\hat{\varphi}_{\alpha\beta}}$$\scriptstyle{\hat{\pi}_{\beta}}$$\scriptstyle{\hat{\pi}_{\alpha}}$$\scriptstyle{\exists!\Psi}$

We claim $\Psi$ is an isomorphism. To see that $\Psi$ is injective, let
$q_{1},q_{2}\in Q_{G,H,h}$ such that $\Psi(q_{1})=\Psi(q_{2}).$ Then for all
$\lambda\in\Lambda,$

$\rho_{\lambda}(\Psi(q_{1}))=\rho_{\lambda}(\Psi(q_{2}))\Rightarrow\hat{\pi}_{\lambda}(q_{1})=\hat{\pi}_{\lambda}(q_{2})\Rightarrow
q_{1}=q_{2}.$

For surjectivity, we have $Q_{G,H,h}$ is a (topological) quotient of $G$ which
is a Stone space, hence $Q_{G,H,h}$ is compact. Then
$\Psi(Q_{G,H,h})\subseteq\varprojlim
Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}$ is also compact. We have by
the diagram that
$\rho_{\alpha}(\Psi(Q_{G,H,h}))=\hat{\pi}_{\alpha}(Q_{G,H,h})=Q_{G_{\alpha},\pi_{\alpha}(H),\pi_{\alpha}(h)}$
by the surjectivity of $\hat{\pi}_{\alpha}$. Hence by 3.21 the subquandle
$\Psi(Q_{G,H,h})$ is dense in $\varprojlim
Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}$ and

$Q_{G,H,h}\cong\varprojlim Q_{G_{\lambda},\pi_{\lambda}(H),\pi_{\lambda}(h)}.$

Finally, we show that $Q_{G,H,h}$ is algebraically connected. Since $G$ is
topologically generated by
$\\{g_{\delta}^{-1}hg_{\delta}\\}_{\delta\in\Delta}$, the action on
$Q_{G,H,h}$ by $\widehat{\text{Inn}(Q_{G,H,h})}$ is transitive (the action is
just the multiplication action on cosets of a subgroup). Thus there is a
single orbit under this action and $Q_{G,H,h}$ is algebraically connected. ∎

The previous two theorems provide a program for constructing all algebraically
connected profinite quandles.

##  6 Directions for future research

The results herein leave our knowledge of profinite quandles in a state
slightly less satisfactory that that attained in [17] Ehrman et al. in the
finite case. There the notion of semi-disjoint union gave a way of
constructing all finite quandles from algebraically connected finite quandle
by iterated semi-disjoint union. We lack an analogous constuction. The example
of the “Cantor quandle”, that is the trivial quandle structure on the Cantor
set with the usual topology inherited from ${\mathbb{R}}$ shows that
topological considerations must be taken into account: algebraically the
Cantor quandle is an uncountable disjoint union of singleton trivial quandles
(and thus a semi-disjoint union with trivial actions), but the topology which
makes it profinite cannot be accounted for by the semi-disjoint union
construction of [17] Ehrman et al. One line of future research would be to
investigate the sufficiency of a construction akin to the semi-disjoint union
construction, performed not on a set of profinite quandles, but a family of
profinite quandles indexed by a Stone space, together with transfinite
induction as a means of constructing all profinite quandles from algebraically
connected profinite quandles.

A second obvious line of research is the resolution the conjecture of [14]
Amesbury, et al. which originally motivated the project: is the lattice of
subquandles of a profinite quandle always complemented? We will remark here
that the lattice of closed subquandles of a profinite quandle is not in
general complemented. The Cantor quandle mentioned above provides the needed
counterexample. Observe that any subquandle in the Cantor quandle consisting
of a single element is closed but does not have a closed complement. The
singleton subquandles do have a complement in the full subquandle lattice –
their set-theoretic complement – since the subquandle lattice of a trivial
quandle is simply the Boolean algebra of its subsets. A subsidiary question to
the conjecture of Amesbury et al. is whether close subquandles of a profinite
quandle have complements in the full subquandle lattice.

Another open question is whether all Stone topological quandles are profinite,
and as a subsidiary question, whether the theory of quandles admits a finite
complete set of words in some variable $y$ (cf. [21]).

Finally, the authors hope that the results herein together the interpretation
of quandles as a means of encoding monodromy [11] will prove useful in the
applications of profinite quandles to the étale homotopy theory of number
fields found in forthcoming work of Davis and Schlank [15].

##  7 Acknowledgements

This paper is the result of a program of research begun under the mentorship
of the fifth author while the first four authors were participants at Kansas
State University’s mathematics REU site, SUMaR 2023. The first four authors
wish to thank Kansas State for hospitality, and all of the authors wish to
thank the National Science Foundation for its financial support of this
project under award #DMS-2243854.

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