# Generalized (co)homology of symmetric quandles over homogeneous Beck modules

Biswadeep Karmakar  ,  Deepanshi Saraf  and  Mahender Singh  Department of
Mathematical Sciences, Indian Institute of Science Education and Research
(IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306,
India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A quandle equipped with a good involution is referred to as symmetric. It is
known that the cohomology of symmetric quandles gives rise to strong cocycle
invariants for classical and surface links, even when they are not necessarily
oriented. In this paper, we introduce the category of symmetric quandle
modules and prove that these modules completely determine the Beck modules in
the category of symmetric quandles. Consequently, this establishes suitable
coefficient objects for constructing appropriate (co)homology theories. We
develop an extension theory of modules over symmetric quandles and propose a
generalized (co)homology theory for symmetric quandles with coefficients in a
homogeneous Beck module, which also recovers the symmetric quandle
(co)homology developed by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)].
Our constructions also apply to symmetric racks. We conclude by establishing
an explicit isomorphism between the second cohomology of a symmetric quandle
and the first cohomology of its associated group.

###### Key words and phrases:

abelian object, associated group, cohomology, quandle module, rack module,
slice category, symmetric quandle, symmetric rack

###### 2020 Mathematics Subject Classification:

Primary 57M27, 57K12; Secondary 57Q45

##  1\. Introduction

Within the realm of oriented knot theory, racks and quandles [13, 20], and
their cohomological ramifications [4, 8, 9], have undoubtedly played a
remarkable role. Their rich theories, combined with various refinements, have
given rise to a plethora of connections spanning Hopf algebras [1], mapping
class groups [6, 24], Riemannian symmetric spaces [19], the Yang–Baxter
equation [18], Yetter–Drinfeld modules [1], and contact geometry [17], to name
a few.

In knot theory, the existence of an orientation often becomes a crucial
prerequisite, particularly in scenarios where one intends to work with the
quandle cocycle invariants. The challenge of overcoming the need for an
orientation was resolved by Kamada and Oshiro [14, 16], who introduced
symmetric quandles and their homology groups. In these works, they defined a
quandle cocycle invariant $\Phi_{\theta}$ for an unoriented link with respect
to a given symmetric quandle 2-cocycle $\theta$. They further showed that
$\Phi_{\theta}$ agrees with the quandle cocycle invariant
$\Phi^{\mathrm{ori}}_{\theta}$ of an oriented link when the orientation is
arbitrarily assigned. Consequently, when the quandle 2-cocycle $\theta$ is
cohomologous to a symmetric quandle 2-cocycle, the invariant
$\Phi^{\mathrm{ori}}_{\theta}$ does not depend on the orientation of the link.
An analogous invariant has been defined for unoriented surface links via
symmetric quandle 3-cocycles, and similar results have ensued. One of the
remarkable consequences of this is enabling us to estimate the minimal triple
point numbers of non-orientable surface links.

Using certain symmetric quandles called 4-fold symmetric quandles, an
invariant of 3-manifolds has been defined in [10, 21]. Furthermore, it has
been shown that the Chern–Simons invariant of closed 3-manifolds can be
regarded as a quandle cocycle invariant. In [12], symmetric quandles have been
used to define invariants for spatial graphs and handlebody-links. The
aforementioned coupled with further modifications (see, for example [5, 22,
23]) have justified the relevance of symmetric quandles to low dimensional
topology.

Inspired by [1, 11], we explore these objects from a categorical perspective
enabling us to take a different approach than the already existing ones. As a
consequence, we are able to completely determine the Beck modules in the
category of symmetric quandles. Introduced by Beck in [2], given a category
$\mathbf{C}$ and an object $X$ in $\mathbf{C}$, a Beck module over $X$ is an
abelian group object in the slice category $\mathbf{C}|_{X}.$ Beck modules
provide a very general notion of a coefficient module for an associated
(co)homology theory. To exemplify their strength, we note that they
simultaneously generalise the notion of coefficient modules from group
cohomology, Lie algebra cohomology as well as Hochschild cohomology of
associative algebras. Once the Beck modules in a category have been
identified, it is natural to ponder over a corresponding (co)homology theory.
We effectively formulate such a (co)homology theory for symmetric quandles
with homogeneous coefficients. We also explore the related extension theory
and establish connections with low dimensional group cohomology of associated
groups. Similar constructions and results work in the category of racks as
well.

It is noteworthy to mention that this generalized (co)homology theory for
symmetric quandles with homogeneous coefficients can be conceived in a manner
reminiscent of group and Hochschild cohomology in which the coefficient
objects admit actions. The significance of this (co)homology theory lies in
its potential to yield invariants (for example, for spatial graphs and low
dimensional manifolds) that are more comprehensive and capable of capturing
additional information compared to the original (co)homology, as initially
defined by Kamada and Oshiro [16], and which is a specific case where the
action is trivial.

The paper is organised as follows. In Section 2, we review some basic
definitions and key concepts from the theory of symmetric racks and symmetric
quandles that will be used in the subsequent sections. Section 3 introduces
modules over symmetric racks and symmetric quandles as trunk maps from certain
associated trunks to the category of abelian groups, along with some examples.
In Section 4, we describe abelian group objects in the slice category over a
fixed symmetric rack. This is achieved by first defining the semi-direct
product of a symmetric rack and its module. Theorem 4.4 proves that the
category of modules over a symmetric rack $(X,\rho_{X})$ is equivalent to the
category of abelian group objects in the slice category over $(X,\rho_{X})$.
An analogous result for symmetric quandles is deduced in Theorem 4.5. In
Section 5, we define extensions of symmetric racks and symmetric quandles.
Theorem 5.8 establishes that the set of equivalence classes of extensions of a
symmetric rack $(X,\rho_{X})$ by an $(X,\rho_{X})$-module $\mathscr{F}$ are in
bijective correspondence with an abelian group
$\mathcal{H}^{2}_{\mathrm{SR}}(X,\mathscr{F})$ defined via extensions. An
analogous result for symmetric quandles is proved in Theorem 5.9. In Section
6, after defining the suitable symmetric algebra, we introduce a generalized
(co)homology theory for symmetric racks and symmetric quandles. This theory
recovers the symmetric quandle (co)homology developed by Kamada and Oshiro
[16]. Finally, in Section 7, we establish an isomorphism between the second
cohomology of a symmetric rack and the first cohomology of its associated
group when the coefficients are trivial homogeneous modules.

##  2\. Preliminaries

In this section, we review some basic definitions and key concepts required
for what follows.

###  2.1. Symmetric racks and symmetric quandles

We begin with some basic ideas introduced in [16].

######  Definition 2.1.

Let $(X,*)$ be a rack (respectively quandle). A map $\rho_{X}:X\rightarrow X$
is called a good involution if

  1. (S1)

$\rho_{X}$ is an involution, $\mathrm{i.e.}$, $\rho_{X}^{2}=\mathrm{id}_{X}$,

  2. (S2)

$\rho_{X}(x\ast y)=\rho_{X}(x)\ast y$,

  3. (S3)

$x\ast\rho_{X}(y)=x\ast^{-1}y$,

for all $x,y\in X$. The pair $(X,\rho_{X})$ is called a symmetric rack
(respectively symmetric quandle).

We refer the reader to [14, 15, 16, 23] for more details on these structures
and their applications in knot theory.

######  Example 2.2.

The following are some elementary examples:

  1. (1)

If $X$ is a trivial quandle, then every involution of $X$ is a good
involution.

  2. (2)

If $X=\operatorname{Conj}(G)$ is the conjugation quandle of a group $G$, then
$\rho_{X}:X\rightarrow X$ given by $\rho_{X}(x)=x^{-1}$ is a good involution.

  3. (3)

If $X=\operatorname{Core}(G)$ is the core quandle of a group $G$, then the
identity map $\mathrm{id}:X\rightarrow X$ is a good involution. Also, if $z\in
G$ is a central element of order two, then $\rho_{X}:X\rightarrow X$ given by
$\rho(y)=yz$ is another good involution of $X$.

######  Definition 2.3.

A symmetric rack (respectively quandle) homomorphism is a map
$f:(X,\rho_{X})\rightarrow(Y,\rho_{Y})$ such that

$f(x\ast y)=f(x)\ast f(y)\quad\textrm{and}\quad f(\rho_{X}(x))=\rho_{Y}(f(x))$

for all $x,y\in X$. A symmetric rack (respectively quandle) isomorphism is a
symmetric rack (respectively quandle) homomorphism which is also a bijection.

Analogous to the associated group of a rack (respectively quandle), Kamada and
Oshiro defined the associated group of a symmetric rack (respectively quandle)
[16, Section 4].

######  Definition 2.4.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle). Then the group

$G_{(X,\rho_{X})}=\langle e_{x},x\in X\mid e_{x\ast
y}=e_{y}^{-1}e_{x}e_{y},~{}e_{\rho_{X}(x)}=e_{x}^{-1}\text{ for }x,y\in
X\rangle$

is called the associated group of $(X,\rho_{X})$.

######  Example 2.5.

[16, Example 2.5] Let $(X,\rho_{X})$ be the fundamental symmetric quandle of
an unknot in $\mathbb{R}^{3}$ or an unknotted 2-sphere in $\mathbb{R}^{4}$. To
be precise, $X=\\{x_{1},x_{2}\\}$ and $\rho_{X}(x_{1})=x_{2}$. Then the
associated group $G_{(X,\rho_{X})}\cong\mathbb{Z}$.

Let $\mathbf{SR}$ and $\mathbf{SQ}$ denote the categories whose constituent
objects are the symmetric racks and symmetric quandles, respectively, with the
morphisms as defined above. It is clear that finite direct products exist in
both $\mathbf{SR}$ and $\mathbf{SQ}$. Further, equalisers exist in both
$\mathbf{SR}$ and $\mathbf{SQ}$. These are just the set-theoretical equalisers
with the induced $\ast$ and $\rho_{X}$. Since, both $\mathbf{SR}$ and
$\mathbf{SQ}$ have all binary products and equalisers, they admit all finite
limits. [3, Proposition 2.8.2].

Let $(X,\rho_{X})$ be a fixed symmetric rack and $\mathbf{SR}|_{(X,\rho_{X})}$
denote the slice category over $(X,\rho_{X})$. More precisely, the objects in
$\mathbf{SR}|_{(X,\rho_{X})}$ are the symmetric rack homomorphisms
$f:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$, and a morphism from
$f:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ to
$g:(Z,\rho_{Z})\rightarrow(X,\rho_{X})$ is a symmetric rack homomorphism
$h:(Y,\rho_{Y})\rightarrow(Z,\rho_{Z})$ such that $g\,h=f$. The slice category
over a symmetric quandle is defined analogously.

Since $\mathbf{SR}$ and $\mathbf{SQ}$ have all finite limits, it follows that
$\mathbf{SR}|_{(X,\rho_{X})}$ and $\mathbf{SQ}|_{(X,\rho_{X})}$ also have
finite limits [3, using Proposition 2.8.2]. In particular, the product of
$f:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ and
$g:(Z,\rho_{Z})\rightarrow(X,\rho_{X})$ in $\mathbf{SR}|_{(X,\rho_{X})}$ is
the pullback

$h:(Y\times_{X}Z,\,\rho_{Y\times_{X}Z})\rightarrow(X,\rho_{X})$

of $f$ and $g$, where $Y\times_{X}Z=\\{(y,z)\in Y\times Z\mid f(y)=g(z)\\}$
has the usual product rack structure,
$\rho_{Y\times_{X}Z}(y,z)=(\rho_{Y}(y),\rho_{Z}(z))$ and $h(y,z)=f(y)=g(z)$
for all $(y,z)\in Y\times_{X}Z$. The same construction works in $\mathbf{SQ}$.
Thus, we can consider group objects in $\mathbf{SR}|_{(X,\rho_{X})}$ and
$\mathbf{SQ}|_{(X,\rho_{X})}$.

###  2.2. Trunks

Trunks were first introduced and studied by Fenn, Rourke, and Sanderson in
[8]. A trunk $\mathrm{T}$ is an object analogous to a category which consists
of a class of objects and a set $\operatorname{Hom}_{\mathrm{T}}(A,B)$ of
morphisms along with some commutative squares

${A}$${B}$${C}$${D,}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{i}$

called preferred squares. Given two trunks S and $\mathrm{T}$, a trunk map
$F:\textrm{S}\rightarrow\mathrm{T}$ is a map which assigns to each object $A$
of S, an object $F(A)$ of $\mathrm{T}$, and to every morphism $f:A\rightarrow
B$ of S, a morphism $F(f):F(A)\rightarrow F(B)$ of $\mathrm{T}$ such that a
preferred square

${A}$${B}$${C}$${D,}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{i}$

is mapped to a preferred square

${F(A)}$${F(B)}$${F(C)}$${F(D).}$$\scriptstyle{F(f)}$$\scriptstyle{F(g)}$$\scriptstyle{F(h)}$$\scriptstyle{F(i)}$

For example, for any category $\mathbf{C}$, we have a well-defined trunk
$T(\mathbf{C})$, which has the same objects and morphisms as $\mathbf{C}$, and
whose preferred squares are the commutative diagrams in $\mathbf{C}$. In
particular, we denote the trunk associated to the category of abelian groups
by $\mathbf{Ab}$.

##  3\. Modules over symmetric racks and symmetric quandles

Given a symmetric rack (respectively quandle) $(X,\rho_{X})$, we define a
trunk $\mathrm{T}(X,\rho_{X})$ as follows:

  * •

for each $x\in X$, $\mathrm{T}(X,\rho_{X})$ has precisely one object,

  * •

for each $x\in X$, there are morphisms $\gamma_{x}:x\rightarrow\rho_{X}(x)$
and $\mathrm{id}_{x}:x\rightarrow x$,

  * •

for each ordered pair $(x,y)$ of elements of $X$, there are morphisms
$\alpha_{x,y}:x\rightarrow x\ast y$ and $\beta_{x,y}:y\rightarrow x*y$ such
that the following squares are preferred squares for all $x,y,z\in X$:

${x}$${x\ast y}$${x\ast z}$${(x\ast y)\ast
z}$$\scriptstyle{\alpha_{x,y}}$$\scriptstyle{\alpha_{x,z}}$$\scriptstyle{\alpha_{x\ast
y,z}}$$\scriptstyle{\alpha_{x\ast z,y\ast z}}$${y}$${x\ast y}$${y\ast
z}$${(x\ast y)\ast
z}$$\scriptstyle{\beta_{x,y}}$$\scriptstyle{\alpha_{y,z}}$$\scriptstyle{\alpha_{x\ast
y,z}}$$\scriptstyle{\beta_{x\ast z,y\ast z}}$

${x}$${x\ast y}$${\rho_{X}(x)}$${\rho_{X}(x)\ast
y}$$\scriptstyle{\alpha_{x,y}}$$\scriptstyle{\gamma_{x}}$$\scriptstyle{\gamma_{x\ast
y}}$$\scriptstyle{\alpha_{\rho_{X}(x),y}}$${x}$${x\ast
y}$${\rho_{X}(x)}$${\rho_{X}(x)\ast
y}$$\scriptstyle{\beta_{x,y}}$$\scriptstyle{\mathrm{id}_{x}}$$\scriptstyle{\gamma_{x\ast
y}}$$\scriptstyle{\beta_{\rho_{X}(x),y}}$

${x}$${\rho_{X}(x)}$${x}$${x}$$\scriptstyle{\gamma_{x}}$$\scriptstyle{\mathrm{id}_{x}}$$\scriptstyle{\gamma_{\rho_{X}(x)}}$$\scriptstyle{\mathrm{id}_{x}}$${x}$${x\ast\rho_{X}(y)}$${x}$${x}$$\scriptstyle{\alpha_{x,\rho_{X}(y)}}$$\scriptstyle{\mathrm{id}_{x}}$$\scriptstyle{\alpha_{x\ast\rho_{X}(y),y}}$$\scriptstyle{\mathrm{id}_{x}}$

Thus, a trunk map $A:\mathrm{T}(X,\rho_{X})\rightarrow\mathbf{Ab}$ yields
abelian groups $A_{x}$ and group homomorphisms $\phi_{x,y}:A_{x}\rightarrow
A_{x\ast y}$, $\psi_{x,y}:A_{y}\rightarrow A_{x\ast y}$ and
$\eta_{x}:A_{x}\rightarrow A_{\rho_{X}(x)}$ such that

  1. (M1)

$\phi_{x\ast y,z}\phi_{x,y}=\phi_{x\ast z,y\ast z}\phi_{x,z}$,

  2. (M2)

$\phi_{x\ast y,z}\psi_{x,y}=\psi_{x\ast z,y\ast z}\phi_{y,z}$,

  3. (M3)

$\eta_{\rho_{X}(x)}\eta_{x}=\mathrm{id}$,

  4. (M4)

$\eta_{x\ast y}\phi_{x,y}=\phi_{\rho_{X}(x),y}\eta_{x}$,

  5. (M5)

$\psi_{\rho_{X}(x),y}=\eta_{x\ast y}\psi_{x,y}$,

  6. (M6)

$\phi_{x\ast^{-1}y,y}\phi_{x,\rho_{X}(y)}=\mathrm{id}$,

for all $x,y,z\in X$. We denote such a trunk map by
$\mathscr{F}=(A,\phi,\psi,\eta)$.

######  Definition 3.1.

Let $(X,\rho_{X})$ be a symmetric rack. Then an $(X,\rho_{X})$-module is a
trunk map $\mathscr{F}=(A,\phi,\psi,\eta):T(X,\rho_{X})\rightarrow\mathbf{Ab}$
such that $\phi_{x,y}:A_{x}\rightarrow A_{x\ast y}$ is an isomorphism and

  1. (M7)

$\psi_{x\ast y,z}(a)=\phi_{x\ast z,y\ast z}\psi_{x,z}(a)+\psi_{x\ast z,y\ast
z}\psi_{y,z}(a)$,

  2. (M8)

$\phi_{x\ast^{-1}y,y}\psi_{x,\rho_{X}(y)}(\eta_{y}(b))=-\psi_{x\ast\rho_{X}(y),y}(b)$,

hold for all $a\in A_{z}$, $b\in A_{y}$ and $x,y,z\in X$.

If $(X,\rho_{X})$ is a symmetric quandle, then we desire that the trunk map
$\mathscr{F}$ additionally satisfies the condition

  1. (M9)

$\phi_{x,x}(a)+\psi_{x,x}(a)=a$

for all $a\in A_{x}$ and $x\in X.$

######  Definition 3.2.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle). An
$(X,\rho_{X})$-module $\mathscr{F}=(A,\phi,\psi,\eta)$ is called homogeneous
if the constituent groups are all isomorphic, that is, $A_{x}\cong A_{y}$ for
all $x,y\in X$.

Before proceeding further, let us consider some examples of modules.

######  Example 3.3.

Let $(X,\rho_{X})$ be a symmetric rack or a symmetric quandle.

  1. (1)

Any abelian group $A$ can be considered as an $(X,\rho_{X})$-module by taking
$A_{x}=A$ $\phi_{x,y}=\mathrm{id}_{A}$, $\psi_{x,y}=0$ and
$\eta_{x}=-\mathrm{id}_{A}$ for all $x,y\in X$, which we call a trivial
homogeneous module.

  2. (2)

Consider a family of abelian groups $\\{G_{i}\\}_{i\in I}$, where $I$ is an
indexing set. Let $\alpha_{i},\beta_{i}\in\operatorname{Aut}(G_{i})$ such that
$\alpha_{i}^{2}=\mathrm{id}_{G_{i}}=\beta_{i}^{2}$ and
$\alpha_{i}\beta_{i}=\beta_{i}\alpha_{i}$ for all $i\in I$. Then,
$\mathscr{F}=(A,\phi,\psi,\eta)$ is an $(X,\rho_{X})$-module, where
$A_{x}=\prod_{i\in I}G_{i}$, $\phi_{x,y}=\prod_{i\in I}\alpha_{i}$,
$\psi_{x,y}=0$ and $\eta_{x}=\prod_{i\in I}\beta_{i}$ for all $x\in X$.

  3. (3)

Let $A$ an abelian group whose each element has order 2 or 4. Then,
$\mathscr{F}=(A,\phi,\psi,\eta)$ is an $(X,\rho_{X})$-module, where $A_{x}=A$,
$\phi_{x,y}(a)=-a$, $\psi_{x,y}(a)=2a$ and $\eta_{x}(a)=-a$ for all $x\in X$
and $a\in A$.

  4. (4)

By [16, Section 4], we have a well-defined right action of $G_{(X,\rho_{X})}$
on $(X,\rho_{X})$, given by

$x\cdot e_{y}=x*y$

for all $x,y\in G_{(X,\rho_{X})}$. Let $[x]$ denote the orbit of $x$ under
this action. Then, we have an $(X,\rho_{X})$-module
$\mathscr{F}=(A,\phi,\psi,\eta)$ with $A_{x}=A_{\rho_{X}(x)}:=A_{[x]}$ and

$\phi_{x,y}:A_{[x]}\rightarrow A_{[x\ast y]}\text{ defined by }a\mapsto
f_{[x]}(a),$ $\psi_{x,y}:A_{[x]}\rightarrow A_{[x\ast y]}\text{ defined by
}b\mapsto 0\text{ and }$ $\eta_{x}:A_{[x]}\rightarrow A_{[\rho_{X}(x)]}\text{
defined by }c\mapsto g_{[x]}(c),$

where $f_{[x]},g_{[x]}:A_{[x]}\to A_{[x]}$ are group homomorphisms such that
$f_{[x]}^{2}=\mathrm{id}=g_{[x]}^{2}$ and $f_{[x]}g_{[x]}=g_{[x]}f_{[x]}$ for
all $x\in X.$

######  Definition 3.4.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle), and
$\mathscr{F}=(A,\phi,\psi,\eta)$ and
$\mathscr{F^{\prime}}=(A^{\prime},\phi^{\prime},\psi^{\prime},\eta^{\prime})$
be $(X,\rho_{X})$-modules. An $(X,\rho_{X})$-map is a natural transformation
$f:\mathscr{F}\rightarrow\mathscr{F}^{\prime}$ of trunk maps, that is, it is a
collection $f=\\{f_{x}:A_{x}\rightarrow A^{\prime}_{x}\mid x\in X\\}$ of group
homomorphisms such that

$\displaystyle\phi^{\prime}_{x,y}f_{x}$ $\displaystyle=$ $\displaystyle
f_{x\ast y}\phi_{x,y},$ (3.0.1) $\displaystyle\psi^{\prime}_{x,y}f_{y}$
$\displaystyle=$ $\displaystyle f_{x\ast y}\psi_{x,y},$ (3.0.2)
$\displaystyle\eta^{\prime}_{x}f_{x}$ $\displaystyle=$ $\displaystyle
f_{\rho_{X}(x)}\eta_{x},$ (3.0.3)

for all $x,y\in X$.

In addition, if each $f_{x}$ is an isomorphism of groups, then we say that
$f:\mathscr{F}\rightarrow\mathscr{F}^{\prime}$ is an isomorphism of
$(X,\rho_{X})$-modules.

######  Remark 3.5.

If $(X,\rho_{X})$ is a symmetric rack, then we can form the category
$\mathbf{SRMod}_{(X,\rho_{X})}$, whose objects are $(X,\rho_{X})$-modules and
whose morphisms are $(X,\rho_{X})$-maps. Similarly, if $(X,\rho_{X})$ is a
symmetric quandle, then we can form the category
$\mathbf{SQMod}_{(X,\rho_{X})}$ of modules over $(X,\rho_{X})$. Moreover, both
$\mathbf{SRMod}_{(X,\rho_{X})}$ and $\mathbf{SQMod}_{(X,\rho_{X})}$) are
abelian categories.

##  4\. Category of modules over symmetric racks and symmetric quandles

In this section, we describe the objects in the category
$\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$ of abelian group objects in the
slice category $\mathbf{SR}|_{(X,\rho_{X})}$ over a fixed symmetric rack
$(X,\rho_{X})$. Let $(X,\rho_{X})$ be a symmetric rack and
$\mathscr{F}=(A,\phi,\psi,\eta)$ an $(X,\rho_{X})$-module. We define the semi-
direct product $\mathscr{F}\rtimes X$ of $\mathscr{F}$ and $X$ to be the set

$\big{\\{}(a,x)\mid x\in X,a\in A_{x}\big{\\}}$

equipped with the binary operation

$(a,x)\tilde{\ast}(b,y):=\big{(}\phi_{x,y}(a)+\psi_{x,y}(b),x\ast y\big{)}$
(4.0.1)

and the map $\rho_{\mathscr{F}\rtimes X}:\mathscr{F}\rtimes
X\rightarrow\mathscr{F}\rtimes X$ defined by

$\rho_{\mathscr{F}\rtimes
X}\big{(}(a,x)\big{)}=\big{(}\eta_{x}(a),\rho_{X}(x)\big{)}$

for all $a\in A_{x}$, $b\in A_{y}$ and $x,y\in X$.

######  Proposition 4.1.

Let $(X,\rho_{X})$ be a symmetric rack and $\mathscr{F}=(A,\phi,\psi,\eta)$ an
$(X,\rho_{X})$-module. Then the semi-direct product $({\mathscr{F}\rtimes
X},\rho_{\mathscr{F}\rtimes X})$ is a symmetric rack. Furthermore, if
$(X,\rho_{X})$ is a symmetric quandle, then so is $({\mathscr{F}\rtimes
X},\rho_{\mathscr{F}\rtimes X})$.

###### Proof.

Note that ${\mathscr{F}\rtimes X}$ is a rack, as proved in [11, Proposition
2.1]. It remains to check that $\rho_{\mathscr{F}\rtimes X}$ is a good
involution.

  * •

For (S1), we see that $\rho_{\mathscr{F}\rtimes
X}^{2}(a,x)=\rho_{\mathscr{F}\rtimes
X}(\eta_{x}(a),\rho_{X}(x))=(\eta_{\rho_{X}(x)}(\eta_{x}(a)),\rho_{X}^{2}(x))=(a,x)$,
where the last equality follows from the $(X,\rho_{X})$-module axiom (M3).

  * •

For (S2), we have

$\displaystyle\rho_{\mathscr{F}\rtimes X}\big{(}(a,x)\big{)}\tilde{\ast}(b,y)$
$\displaystyle=$ $\displaystyle(\eta_{x}(a),\rho_{X}(x))\tilde{\ast}(b,y)$
$\displaystyle=$
$\displaystyle(\phi_{\rho_{X}(x),y}(\eta_{x}(a))+\psi_{\rho_{X}(x),y}(b),\rho_{X}(x)\ast
y)$ $\displaystyle=$ $\displaystyle(\eta_{x\ast
y}(\phi_{x,y}(a)+\psi_{x,y}(b)),\rho_{X}(x\ast y)),$ by $(X,\rho_{X})$-module
axioms (M4) and (M5), and $\rho_{X}$ being a good involution $\displaystyle=$
$\displaystyle\rho_{\mathscr{F}\rtimes X}(\phi_{x,y}(a)+\psi_{x,y}(b),x\ast
y)$ $\displaystyle=$ $\displaystyle\rho_{\mathscr{F}\rtimes
X}\big{(}(a,x)\tilde{\ast}(b,y)\big{)}.$

  * •

For (S3), we have

$\displaystyle(a,x)\tilde{\ast}\rho_{\mathscr{F}\rtimes X}\big{(}(b,y)\big{)}$
$\displaystyle=$ $\displaystyle(a,x)\tilde{\ast}(\eta_{y}(b),\rho_{X}(y))$
$\displaystyle=$
$\displaystyle(\phi_{x,\rho_{X}(y)}(a)+\psi_{x,\rho_{X}(y)}(\eta_{y}(b)),x\ast\rho_{X}(y)),$
by $(X,\rho_{X})$-module axioms (M6) and (M8), and $\rho_{X}$ being a good
involution $\displaystyle=$
$\displaystyle(\phi_{x\ast^{-1}y,y}^{-1}(a-\psi_{x\ast^{-1}y,y}(b)),x\ast^{-1}y)$
$\displaystyle=$ $\displaystyle(a,x)\tilde{\ast}^{-1}(b,y).$

Thus, $\rho_{\mathscr{F}\rtimes X}$ is a good involution, and hence
$({\mathscr{F}\rtimes X},\rho_{\mathscr{F}\rtimes X})$ is a symmetric rack.
Finally, suppose that $(X,\rho_{X})$ is a symmetric quandle. Then, we have the
additional condition $\phi_{x,x}(a)+\psi_{x,x}(a)=a$ for all $x\in X$ and
$a\in A_{x}$. Thus, we obtain

$(a,x)\tilde{\ast}(a,x)=\big{(}\phi_{x,x}(a)+\psi_{x,x}(a),x\big{)}=(a,x)$

for all $x\in X$ and $a\in A_{x}$, and hence $({\mathscr{F}\rtimes
X},\rho_{\mathscr{F}\rtimes X})$ is a symmetric quandle. ∎

######  Proposition 4.2.

Let $(X,\rho_{X})$ be a symmetric rack and $\mathscr{F}=(A,\phi,\psi,\eta)$ an
$(X,\rho_{X})$-module. Then there exists an abelian group object
$p:({\mathscr{F}\rtimes X},\rho_{\mathscr{F}\rtimes
X})\rightarrow(X,\rho_{X})$, which we denote by $\mathcal{T}(\mathscr{F})$, in
the slice category over $(X,\rho_{X})$. Further, let
$\mathscr{F^{\prime}}=(A^{\prime},\phi^{\prime},\psi^{\prime},\eta^{\prime})$
be another $(X,\rho_{X})$-module and
$f:\mathscr{F}\rightarrow\mathscr{F^{\prime}}$ an $(X,\rho_{X})$-map. Then the
map $\mathcal{T}(f):\mathscr{F}\rtimes X\rightarrow\mathscr{F^{\prime}}\rtimes
X$ defined by $\mathcal{T}(f)\big{(}(a,x)\big{)}=(f_{x}(a),x)$ gives a slice
morphism. Moreover,
$\mathcal{T}:\mathbf{SRMod}_{(X,\rho_{X})}\rightarrow\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$
is a functor.

###### Proof.

Let $\mathscr{F}=(A,\phi,\psi,\eta)$ be an $(X,\rho_{X})$-module. Then, we
have the semi-direct product $({\mathscr{F}\rtimes X},\rho_{\mathscr{F}\rtimes
X})$, which is a symmetric rack due to Proposition 4.1. Let
$p:\mathscr{F}\rtimes X\rightarrow X$ be the natural projection given as
$p(a,x)=x$ for $a\in A_{x}$ and $x\in X$. Then $p$ is a homomorphism of
symmetric racks. We claim that $p:\mathscr{F}\rtimes X\rightarrow X$ has the
canonical structure of an abelian group object. Let

$\big{(}(\mathscr{F}\rtimes X)\times_{X}(\mathscr{F}\rtimes
X),\rho_{(\mathscr{F}\rtimes X)\times_{X}(\mathscr{F}\rtimes
X)}\big{)}\rightarrow(X,\rho_{X})$

be the categorical product of $p:\mathscr{F}\rtimes X\rightarrow X$ with
itself in the slice category over $(X,\rho_{X})$. The symmetric rack structure
is defined in the same way as in the cartesian product. We define the
morphisms

  * •

$\mu:(\mathscr{F}\rtimes X)\times_{X}(\mathscr{F}\rtimes
X)\rightarrow(\mathscr{F}\rtimes X)$ given by
$((a_{1},x),(a_{2},x))\mapsto(a_{1}+a_{2},x)$,

  * •

$\sigma:X\rightarrow(\mathscr{F}\rtimes X)$ given by $x\mapsto(0,x)$,

  * •

$\nu:(\mathscr{F}\rtimes X)\rightarrow(\mathscr{F}\rtimes X)$ given by
$(a,x)\mapsto(-a,x)$,

for all $a_{1},a_{2},a\in A_{x}$ and $x\in X$. It easy to check that $\mu$,
$\sigma$ and $\nu$ are symmetric rack homomorphisms. Also, the fact that
$\mu$, $\sigma$ and $\nu$ are slice morphisms and that they turn
$p:\mathscr{F}\rtimes X\rightarrow X$ into an abelian group object follows
directly (see also [11, Theorem 2.2]).

Let us define a functor
$\mathcal{T}:\mathbf{SRMod}_{(X,\rho_{X})}\rightarrow\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$.
Given an $(X,\rho_{X})$-module $\mathscr{F}=(A,\phi,\psi,\eta)$, let
$\mathcal{T}(\mathscr{F})$ denote the object $p:\mathscr{F}\rtimes
X\rightarrow X$. Let
$\mathscr{F^{\prime}}=(A^{\prime},\phi^{\prime},\psi^{\prime},\eta^{\prime})$
be another $(X,\rho_{X})$-module and
$f:\mathscr{F}\rightarrow\mathscr{F^{\prime}}$ an $(X,\rho_{X})$-map. Define
$\mathcal{T}(f):\mathscr{F}\rtimes X\rightarrow\mathscr{F^{\prime}}\rtimes X$
by

$T(f)(a,x)=(f_{x}(a),x).$

Then, we have

$\displaystyle\mathcal{T}(f)\big{(}(a,x)\star(b,y)\big{)}$ $\displaystyle=$
$\displaystyle\mathcal{T}(f)\big{(}\phi_{x,y}(a)+\psi_{x,y}(b),x\ast y\big{)}$
$\displaystyle=$ $\displaystyle\big{(}f_{x\ast
y}(\phi_{x,y}(a)+\psi_{x,y}(b)),x\ast y\big{)}$ $\displaystyle=$
$\displaystyle\big{(}\phi^{\prime}_{x,y}f_{x}(a)+\psi^{\prime}_{x,y}f_{y}(b),x\ast
y\big{)},\quad\textrm{since $f$ is an $(X,\rho_{X})$-map}$ $\displaystyle=$
$\displaystyle\mathcal{T}(f)\big{(}(a,x)\big{)}\star^{\prime}\mathcal{T}(f)\big{(}(b,y)\big{)},$

where $\star$ is the rack operation in $\mathscr{F}\rtimes X$ and
$\star^{\prime}$ is the rack operation in $\mathscr{F^{\prime}}\rtimes X$.
Further, since $\eta^{\prime}_{x}f_{x}=f_{\rho_{X}(x)}\eta_{x}$, the diagram

${\mathscr{F}\rtimes X}$${\mathscr{F}\rtimes X}$${\mathscr{F^{\prime}}\rtimes
X}$${\mathscr{F^{\prime}}\rtimes X}$$\scriptstyle{\rho_{\mathscr{F}\rtimes
X}}$$\scriptstyle{\mathcal{T}(f)}$$\scriptstyle{\mathcal{T}(f)}$$\scriptstyle{\rho_{\mathscr{F^{\prime}}\rtimes
X}}$

commutes. Thus, $T(f)$ is a homomorphism of symmetric racks. Finally, since
the diagram

${\mathscr{F}\rtimes X}$${\mathscr{F^{\prime}}\rtimes
X}$${X}$$\scriptstyle{\mathcal{T}(f)}$$\scriptstyle{p}$$\scriptstyle{p^{\prime}}$

commutes, where $p^{\prime}$ is the projection to the second coordinate,
$\mathcal{T}(f)$ is a slice morphism. In fact, one can check that
$\mathcal{T}$ is a functor (see [11, Theorem 2.2]). ∎

######  Proposition 4.3.

Let $(X,\rho_{X})$ be a symmetric rack and
$p:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ an abelian group object with the
multiplication map $m$, the inverse map $i$, and the section $s$. Let $(X,*)$
and $(Y,\star)$ be the underlying racks. Then there exists an
$(X,\rho_{X})$-module $(R,\phi,\psi,\eta)$ given by an induced abelian group
structure on the fibre $R_{x}:=p^{-1}(x)$ for each $x\in X$, where

$\phi_{x,y}:R_{x}\rightarrow R_{x\ast y}~{}~{}\text{is given by }~{}u\mapsto
u\star s(y),$ $\psi_{x,y}:R_{y}\rightarrow R_{x\ast y}~{}~{}\text{is given by
}~{}v\mapsto s(x)\star v$

and

$\eta_{x}:R_{x}\rightarrow R_{\rho_{X}(x)}~{}~{}\text{is given by
}~{}u\mapsto\rho_{Y}(u)$

for all $u\in R_{x}$, $v\in R_{y}$ and $x,y\in X$. Furthermore, the
association gives a functor
$\mathcal{S}:\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})\to\mathbf{SRMod}_{(X,\rho_{X})}$.

###### Proof.

Let $p:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ be an abelian group object in
$\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$. Let $R_{x}=p^{-1}(x)$ for each
$x\in X$. Then an abelian group structure on $R_{x}$ is given by

$u+v:=m(u,v),$

where $s(x)$ is the identity element in $R_{x}$ and $-u:=i(u)$ for all $u,v\in
R_{x}$. To define an $(X,\rho_{X})$-module structure, let us define

$\phi_{x,y}:R_{x}\rightarrow R_{x\ast y}~{}~{}\text{given by }~{}u\mapsto
u\star s(y),$ $\psi_{x,y}:R_{y}\rightarrow R_{x\ast y}~{}~{}\text{given by
}~{}v\mapsto s(x)\star v$

and

$\eta_{x}:R_{x}\rightarrow R_{\rho_{X}(x)}~{}~{}\text{given by
}~{}u\mapsto\rho_{Y}(u)$

for all $u\in R_{x}$, $v\in R_{y}$ and $x,y\in X$. It has been proved in [11,
Theorem 2.2] that $\phi_{x,y}$ and $\psi_{x,y}$ are group homomorphisms for
all $x,y\in X$. For the case of $\eta_{x}$, if $u_{1},u_{2}\in R_{x}$, then we
see that

$\displaystyle\eta_{x}(u_{1}+u_{2})$ $\displaystyle=$
$\displaystyle\rho_{Y}(m(u_{1},u_{2}))$ $\displaystyle=$ $\displaystyle
m(\rho_{Y}(u_{1}),\rho_{Y}(u_{2})),\quad\textrm{since
$m\,(\rho_{Y}\times\rho_{Y})=\rho_{Y}\,m$}$ $\displaystyle=$
$\displaystyle\rho_{Y}(u_{1})+\rho_{Y}(u_{2})$ $\displaystyle=$
$\displaystyle\eta_{x}(u_{1})+\eta_{x}(u_{2}).$

Next, we need to check that $\phi_{x,y}$, $\psi_{x,y}$ and $\eta_{x}$ satisfy
the $(X,\rho_{X})$-module identities. Since (M1), (M2) and (M7) are proved in
[11, Theorem 2.2], we verify only the remaining identities.

  * •

Since $\rho_{Y}$ is a good involution, we have

$\eta_{\rho_{X}(x)}\eta_{x}(u)=\eta_{\rho_{X}(x)}(\rho_{Y}(u))=\rho_{Y}^{2}(u)=u,$

and hence (M3) holds.

  * •

Again, since $\rho_{Y}$ is a good involution, we have

$\phi_{\rho_{X}(x),y}(\eta_{x}(u))=\phi_{\rho_{X}(x),y}(\rho_{Y}(u))=\rho_{Y}(u)\star
s(y)=\rho_{Y}(u\star s(y))=\eta_{x\ast y}(u\star s(y))=\eta_{x\ast
y}(\phi_{x,y}(u)),$

and hence (M4) holds.

  * •

Since $s$ is a homomorphism of symmetric racks, we see that

$\psi_{\rho_{X}(x),y}(v)=s(\rho_{X}(x))\star v=\rho_{Y}(s(x))\star
v=\rho_{Y}(s(x)\star v)=\eta_{x\ast y}(s(x)\star(v))=\eta_{x\ast
y}(\psi_{x,y}(v)),$

and hence (M5) holds.

  * •

Since $\rho_{Y}$ is a good involution and $s$ is a homomorphism of symmetric
racks, we obtain

$\phi_{x\ast^{-1}y,y}\phi_{x,\rho_{X}(y)}(u)=\phi_{x\ast^{-1},y}(u\star
s(\rho_{X}(y)))=\phi_{x\ast^{-1},y}(u\star\rho_{Y}(s(y)))=(u\star\rho_{Y}(s(y)))\star
s(y)=u,$

and hence (M6) holds.

  * •

Note that
$\phi_{x\ast^{-1}y,y}\psi_{x,\rho_{X}(y)}(\eta_{y}(v))=\phi_{x\ast^{-1}y,y}\psi_{x,\rho_{X}(y)}(\rho_{Y}(v))=\phi_{x\ast^{-1}y,y}(s(x)\star\rho_{Y}(v))=(s(x)\star\rho_{Y}(v))\star
s(y)$ and $\psi_{x\ast\rho_{X}(y),y}(v)=s(x\ast\rho_{X}(y))\star
v=(s(x)\star\rho_{Y}(s(y)))\star v$. Then, we have

$\displaystyle\phi_{x\ast^{-1}y,y}\psi_{x,\rho_{X}(y)}(\eta_{y}(v))+\psi_{x\ast\rho_{X}(y),y}(v)$
$\displaystyle=$ $\displaystyle(s(x)\star\rho_{Y}(v))\star
s(y)+(s(x)\star\rho_{Y}(s(y)))\star v$ $\displaystyle=$ $\displaystyle
m\big{(}(s(x)\star\rho_{Y}(v))\star s(y),~{}(s(x)\star\rho_{Y}(s(y)))\star
v\big{)}$ $\displaystyle=$ $\displaystyle
m\big{(}\big{(}(s(x),s(x))\star(\rho_{Y}(v),\rho_{Y}(s(y)))\big{)}\star(s(y),v)\big{)},\quad\textrm{by
definition of rack operation in $Y\times_{X}Y$}$ $\displaystyle=$
$\displaystyle\big{(}m(s(x),s(x))\star
m\,(\rho_{Y}(v),\rho_{Y}(s(y)))\big{)}\star m(s(y),v),\quad\textrm{since $m$
is a homomorphism of racks}$ $\displaystyle=$
$\displaystyle\big{(}s(x)\star\rho_{Y}\,m(v,s(y)))\star
m(s(y),v),\quad\textrm{since $m$ is a homomorphism of symmetric racks}$
$\displaystyle=$ $\displaystyle\big{(}s(x)\star\rho_{Y}\,m(s(y),v))\star
m(s(y),v),\quad\textrm{since $\rho_{Y}$ is a good involution}$
$\displaystyle=$ $\displaystyle s(x),$

and hence (M8) holds. Hence, we have proved that $(R,\phi,\psi,\eta)$ is an
$(X,\rho_{X})$-module.

Next, we construct a functor
$\mathcal{S}:\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})\to\mathbf{SRMod}_{(X,\rho_{X})}$.
Given an abelian group object $p:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$, we set
$\mathcal{S}(p)=(R,\phi,\psi,\eta)$, which is defined above. Let $f$ be a
morphism from the abelian group object
$p_{1}:(Y_{1},\rho_{Y_{1}})\rightarrow(X,\rho_{X})$ to the abelian group
object $p_{2}:(Y_{2},\rho_{Y_{2}})\rightarrow(X,\rho_{X})$. Thus, if $m$ and
$m^{\prime}$ denote the corresponding multiplication maps, then
$f\,m=m^{\prime}\,(f,f)$. Further,
$f:(Y_{1},\rho_{Y_{1}})\rightarrow(Y_{2},\rho_{Y_{2}})$ is a homomorphism of
symmetric racks with $p_{2}\,f=p_{1}$. Let $\mathscr{F}_{1}$ and
$\mathscr{F}_{2}$ denote the $(X,\rho_{X})$-modules as described above. Define
$g:\mathscr{F}_{1}=(R_{1},\phi,\psi,\eta)\rightarrow\mathscr{F}_{2}=(R_{2},\phi^{\prime},\psi^{\prime},\eta^{\prime})$
by setting

$g_{x}:{(R_{1})}_{x}\rightarrow{(R_{2})}_{x}~{}~{}\text{as}~{}~{}g_{x}(u)=f(u)$

for all $x\in X$ and $u\in{(R_{1})}_{x}$. Since $f\,m=m^{\prime}\,(f,f)$, it
follows that $g_{x}$ is an abelian group homomorphism for each $x\in X$. The
identities $\phi^{\prime}_{x,y}g_{x}=g_{x\ast y}\phi_{x,y}$ and
$\psi^{\prime}_{x,y}g_{y}=g_{x\ast y}\psi_{x,y}$ are proved in [11, Theorem
2.2]. Further, since

$g_{\rho_{X}(x)}\eta_{x}(w)=g_{\rho_{X}(x)}(\rho_{Y_{1}}(w))=f(\rho_{Y_{1}}(w))=\rho_{Y_{2}}(f(w))=\rho_{Y_{2}}g_{x}(w)=\eta_{x}^{\prime}(g_{x}(w))$

for all $w\in(R_{1})_{x}$, we have
$\eta^{\prime}_{x}g_{x}=g_{\rho_{X}(x)}\eta_{x}$. Hence, $g$ is an
$(X,\rho_{X})$-map. A routine check shows that $\mathcal{S}$ is a functor (see
[11, Thereom 2.2]). ∎

######  Theorem 4.4.

Let $(X,\rho_{X})$ be a symmetric rack. Then the category
$\mathbf{SRMod}_{(X,\rho_{X})}$ of $(X,\rho_{X})$-modules is equivalent to the
category $\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$ of abelian group objects
in the slice category over $(X,\rho_{X})$.

###### Proof.

Let $(X,\rho_{X})$ be a symmetric rack. By Proposition 4.2 and Proposition
4.3, we have functors
$\mathcal{T}:\mathbf{SRMod}_{(X,\rho_{X})}\to\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$
and
$\mathcal{S}:\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})\to\mathbf{SRMod}_{(X,\rho_{X})}$.
To show that the categories $\mathbf{SRMod}_{(X,\rho_{X})}$ and
$\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$ are equivalent, we show that
$\mathcal{T}\mathcal{S}$ is naturally isomorphic to
$\mathrm{id}_{\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})}$ and
$\mathcal{S}\mathcal{T}$ is naturally isomorphic to
$\mathrm{id}_{\mathbf{SRMod}_{(X,\rho_{X})}}$.

Let $\mathcal{O}\in\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$ denote the object
$p:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ with the multiplication map $m$, the
inverse map $i$, and the section $s$. Then
$\mathcal{T}\mathcal{S}(\mathcal{O})$ is the object $q:(\mathscr{R}\rtimes
X,\rho_{\mathscr{R}\rtimes X})\rightarrow(X,\rho_{X})$ in the category
$\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$, where
$\mathscr{R}=(R,\phi,\psi,\eta)$ as constructed in Proposition 4.3. Let $*$ be
the rack operation on $X$, and $\tilde{*}$ the rack operation on
$\mathscr{R}\rtimes X$. Note that, the abelian group structure on
$q:(\mathscr{R}\rtimes X,\rho_{\mathscr{R}\rtimes X})\rightarrow(X,\rho_{X})$
has the multiplication, the inverse and the section given by

  * •

$\mu((a_{1},x),(a_{2},x))=(a_{1}+a_{2},x)=(m(a_{1},a_{2}),x)$,

  * •

$\sigma(x)=(0,x)=(s(x),x)$,

  * •

$\nu((a,x))=(-a,x)=(i(a),x)$,

for all $a_{1},a_{2},a\in R_{x}$ and $x\in X$. Define
$\zeta_{\mathcal{O}}:(Y,\rho_{Y})\rightarrow(\mathscr{R}\rtimes
X,\rho_{\mathscr{R}\rtimes X})$ by setting $\zeta_{\mathcal{O}}(a)=(a,x)$ for
$a\in R_{x}$. Clearly, $\zeta_{\mathcal{O}}$ is a bijection and the diagram

${(Y,\rho_{Y})}$${(\mathscr{R}\rtimes X,\rho_{\mathscr{R}\rtimes
X})}$${(X,\rho_{X})}$$\scriptstyle{\zeta_{\mathcal{O}}}$$\scriptstyle{p}$$\scriptstyle{q}$

commutes. Note that $\zeta_{\mathcal{O}}(m(a,b))=\mu(\zeta_{\mathcal{O}}(a)$,
$\zeta_{\mathcal{O}}(b)),~{}\zeta_{\mathcal{O}}(s(a))=\sigma(\zeta_{\mathcal{O}}(a))$
and $\zeta_{\mathcal{O}}(i(a))=\nu(\zeta_{\mathcal{O}}(a))$ for all $a,b\in
R_{x}$. Hence, $\zeta_{\mathcal{O}}$ respects the abelian group structure.
Now, for $a\in R_{x}$ and $b\in R_{y}$, we have

$\displaystyle\zeta_{\mathcal{O}}(a)\tilde{\ast}\zeta_{\mathcal{O}}(b)$
$\displaystyle=$ $\displaystyle(a,x)\tilde{\ast}(b,y)$ $\displaystyle=$
$\displaystyle\big{(}\phi_{x,y}(a)+\psi_{x,y}(b),x\ast y\big{)}$
$\displaystyle=$ $\displaystyle\big{(}a\star s(y)+s(x)\star b,x\ast y\big{)}$
$\displaystyle=$ $\displaystyle\big{(}m\big{(}a\star s(y),s(x)\star
b\big{)},x\ast y\big{)}$ $\displaystyle=$
$\displaystyle\big{(}m((a,s(x))\tilde{\star}(s(y),b)),x\ast y\big{)}$
$\displaystyle=$ $\displaystyle\big{(}m(a,s(x))\star m(s(y),b),x\ast
y\big{)},\quad\textrm{since $m:Y\times Y\to Y$ is a rack homomorphism}$
$\displaystyle=$ $\displaystyle(a\star b,x\ast y)$ $\displaystyle=$
$\displaystyle\zeta_{\mathcal{O}}(a\star b),$

where $\star$ is the rack operation on $Y$ and $\tilde{\star}$ is the rack
operation on $Y\times Y$. Thus, $\zeta_{\mathcal{O}}$ is a rack homomorphism.
Since $\rho_{\mathscr{R}\rtimes
X}\,\zeta_{\mathcal{O}}=\zeta_{\mathcal{O}}\,\rho_{Y}$, it follows that
$\zeta_{\mathcal{O}}$ is a symmetric rack isomorphism, and hence
$\zeta_{\mathcal{O}}:\mathcal{O}\to\mathcal{T}\mathcal{S}(\mathcal{O})$ is an
isomorphism.

Let $\mathcal{O}$ and $\mathcal{O^{\prime}}$ be the objects
$p:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ and
$p^{\prime}:(Y^{\prime},\rho_{Y^{\prime}})\rightarrow(X,\rho_{X})$ in
$\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$, respectively. Let
$f:\mathcal{O}\rightarrow\mathcal{O^{\prime}}$ be a morphism. Then, by
definition, $f:(Y,\rho_{Y})\to(Y^{\prime},\rho_{Y^{\prime}})$ is a symmetric
rack homomorphism such that $p^{\prime}\circ f=p$ and $f$ preserves the
abelian group structure. Let $x\in X$ and $a\in R_{x}$. Then
$\zeta_{\mathcal{O^{\prime}}}\,f(a)=(f_{x}(a),x)$, where $f_{x}=f|_{R_{x}}$
and

$\mathcal{T}\mathcal{S}(f)\,\zeta_{\mathcal{O}}(a)=\mathcal{T}\mathcal{S}(f)(a,x)=\mathcal{T}(f|_{R_{x}}(a))=\mathcal{T}(f_{x}(a))=(f_{x}(a),x).$

Thus, the diagram

${\mathcal{O}}$${\mathcal{T}\mathcal{S}(\mathcal{O})}$${\mathcal{O^{\prime}}}$${\mathcal{T}\mathcal{S}(\mathcal{O^{\prime}})}$$\scriptstyle{\zeta_{\mathcal{O}}}$$\scriptstyle{f}$$\scriptstyle{\mathcal{T}\mathcal{S}(f)}$$\scriptstyle{\zeta_{\mathcal{O^{\prime}}}}$

commutes, and we have shown that $\mathcal{T}\mathcal{S}$ is naturally
isomorphic to $\mathrm{id}_{\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})}$.

Next, we prove that $\mathcal{S}\mathcal{T}$ is naturally isomorphic to
$\mathrm{id}_{\mathbf{SRMod}_{(X,\rho_{X})}}$. Let $\mathcal{O}$ be the object
$\mathscr{F}=(A,\phi,\psi,\eta)$ in $\mathbf{SRMod}_{(X,\rho_{X})}$. Then
$\mathcal{T}(\mathcal{O})$ is the abelian group object $p:(\mathscr{F}\rtimes
X,\rho_{\mathscr{F}\rtimes X})\rightarrow(X,\rho_{X})$ as in Proposition 4.2.
For each $x\in X$, since $A^{\prime}_{x}:=p^{-1}(x)=(A_{x},x)$, it follows
that $\mathcal{S}\mathcal{T}(\mathcal{O})$ is the object
$\mathscr{F^{\prime}}:=(A^{\prime},\phi^{\prime},\psi^{\prime},\eta^{\prime})$,
where

$\displaystyle\phi_{x,y}^{\prime}(a,x)$
$\displaystyle=(a,x)\tilde{\ast}\sigma(y)=\
(a,x)\tilde{\ast}(0,y)=(\phi_{x,y}(a),x\ast y),$ (4.0.2)
$\displaystyle\psi_{x,y}^{\prime}(b,y)$
$\displaystyle=\sigma(x)\tilde{\ast}(b,y)=(0,x)\tilde{\ast}(b,y)=(\psi_{x,y}(b),x\ast
y),$ (4.0.3) $\displaystyle\eta_{x}^{\prime}(a,x)$
$\displaystyle=\rho_{\mathscr{F}\rtimes X}(a,x)=(\eta_{x}(a),\rho_{X}(x)),$
(4.0.4)

for all $a\in A_{x}$, $b\in A_{y}$ and $x,y\in X$.

We now construct an $(X,\rho_{X})$-module isomorphism
$\lambda_{\mathcal{O}}:\mathcal{O}\to\mathcal{S}\mathcal{T}(\mathcal{O})$. For
each $x\in X$, define $(\lambda_{\mathcal{O}})_{x}:A_{x}\to A^{\prime}_{x}$ as
$a\mapsto(a,x)$ for all $a\in A_{x}$. Clearly, $(\lambda_{O})_{x}$ is an
abelian group isomorphism for each $x\in X$. Also, using (4.0.2), (4.0.3) and
(4.0.4), we have

$\displaystyle\phi^{\prime}_{x,y}(\lambda_{O})_{x}(a)$ $\displaystyle=$
$\displaystyle\phi^{\prime}_{x,y}(a,x)=(\phi_{x,y}(a),x\ast
y)=(\lambda_{O})_{x\ast y}\phi_{x,y}(a),$
$\displaystyle\psi^{\prime}_{x,y}(\lambda_{O})_{y}(b)$ $\displaystyle=$
$\displaystyle\psi^{\prime}_{x,y}(b,y)=(\psi_{x,y}(b),x\ast
y)=(\lambda_{O})_{x\ast y}\psi_{x,y}(b),$
$\displaystyle\eta^{\prime}_{x}(\lambda_{O})_{x}(a)$ $\displaystyle=$
$\displaystyle\eta^{\prime}_{x}(a,x)=(\eta_{x}(a),\rho_{X}(x))=(\lambda_{O})_{\rho_{X}(x)}\eta_{x}(a),$

for all $a\in A_{x}$, $b\in A_{y}$ and $x,y\in X$. Thus, (3.0.1-3.0.3) hold,
and $\lambda_{O}$ is an $(X,\rho_{X})$-module isomorphism. Finally, let
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ be the objects
$\mathscr{F}_{1}=(A,\phi,\psi,\eta)$ and
$\mathscr{F}_{2}=(\tilde{A},\tilde{\phi},\tilde{\psi},\tilde{\eta})$ in
$\mathbf{SRMod}_{(X,\rho_{X})}$, respectively. Let
$g:\mathcal{O}_{1}\rightarrow\mathcal{O}_{2}$ be a $(X,\rho_{X})$-map. It is
easy to see that the diagram

${\mathcal{O}_{1}}$${\mathcal{S}\mathcal{T}(\mathcal{O}_{1})}$${\mathcal{O}_{2}}$${\mathcal{S}\mathcal{T}(\mathcal{O}_{2})}$$\scriptstyle{\lambda_{\mathcal{O}_{1}}}$$\scriptstyle{g}$$\scriptstyle{\mathcal{T}\mathcal{S}(g)}$$\scriptstyle{\lambda_{\mathcal{O}_{2}}}$

commutes. Thus, $\mathcal{S}\mathcal{T}$ is naturally isomorphic to
$\mathrm{id}_{\mathbf{SRMod}_{(X,\rho_{X})}}$, and the categories
$\mathbf{Ab}(\mathbf{SR}|_{(X,\rho_{X})})$ and $\mathbf{SRMod}_{(X,\rho_{X})}$
are equivalent. ∎

We now turn our attention to symmetric quandles. We observe that, if
$\mathscr{F}=(A,\phi,\psi,\eta)$ is a module over a symmetric quandle
$(X,\rho_{X})$, then by Proposition 4.1, the semi-direct product
$(\mathscr{F}\rtimes X,\rho_{\mathscr{F}\rtimes X})$ is a symmetric quandle.
Let $p:(Y,\rho_{Y})\rightarrow(X,\rho_{X})$ be an abelian group object with
the multiplication map $m$ and the section $s$, as in Proposition 4.3. We see
that

$\displaystyle\phi_{x,x}(u)+\psi_{x,x}(u)$ $\displaystyle=$ $\displaystyle
u\star s(x)+s(x)\star u$ $\displaystyle=$ $\displaystyle m\big{(}u\star
s(x),s(x)\star u\big{)}$ $\displaystyle=$ $\displaystyle
m\big{(}(u,s(x))\star(s(x),u)\big{)}$ $\displaystyle=$ $\displaystyle
m\big{(}(u,s(x)\big{)}\star m\big{(}(s(x),u)\big{)}$ $\displaystyle=$
$\displaystyle u\star u$ $\displaystyle=$ $\displaystyle u$

for all $x\in X$ and $u\in R_{x}$, which is condition (M9). Thus, we
immediately obtain the following result.

######  Theorem 4.5.

Let $(X,\rho_{X})$ be a symmetric quandle. Then the categories
$\mathbf{Ab}(\mathbf{SQ}|_{(X,\rho_{X})})$ and $\mathbf{SQMod}_{(X,\rho_{X})}$
are equivalent.

##  5\. Abelian extensions of symmetric racks and symmetric quandles

Throughout this section, for a symmetric rack (respectively quandle)
$(X,\rho_{X})$, we denote its underlying rack (respectively quandle) by
$(X,\ast)$.

######  Definition 5.1.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle) and
$\mathscr{F}=(A,\phi,\psi,\eta)$ be an $(X,\rho_{X})$-module. An extension of
$(X,\rho_{X})$ by $\mathscr{F}$ comprises of the following data:

  1. (E1)

A symmetric rack (respectively quandle) $(E,\rho_{E})$, where $\star$ is the
rack (respectively quandle) operation in $E$.

  2. (E2)

An epimorphism $f:(E,\rho_{E})\rightarrow(X,\rho_{X})$. This epimorphism
further induces a partition $E=\cup_{x\in X}E_{x}$, where $E_{x}:=f^{-1}(x)$
for $x\in X$.

  3. (E3)

For each $x\in X$, a left $A_{x}$-action on $E_{x}$, written $(a,s)\mapsto
a\cdot s$ for $a\in A_{x}$ and $s\in E_{x}$. Further, the action satisfies the
following conditions for all $a\in A_{x}$, $b\in A_{y}$, $s\in E_{x}$ and
$t\in E_{y}$:

    1. (a)

The $A_{x}$-action on $E_{x}$ is free and transitive.

    2. (b)

$(a\cdot s)\star t=\phi_{x,y}(a)\cdot(s\star t)$.

    3. (c)

$s\star(b\cdot t)=\psi_{x,y}(b)\cdot(s\star t).$

    4. (d)

$\rho_{E}(a\cdot s)=\eta_{x}(a)\cdot\rho_{E}(s).$

For brevity, we denote an extension by
$f:(E,\rho_{E})\rightarrow(X,\rho_{X})$. Observe that $\rho_{E}(E_{x})\subset
E_{\rho_{X}(x)}$ for each $x\in X$. Next, we introduce equivalence of such
extensions.

######  Definition 5.2.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle) and
$\mathscr{F}=(A,\phi,\psi,\eta)$ be an $(X,\rho_{X})$-module. Two extensions
$f_{1}:(E_{1},\rho_{E_{1}})\rightarrow(X,\rho_{X})$ and
$f_{2}:(E_{2},\rho_{E_{2}})\rightarrow(X,\rho_{X})$ of $(X,\rho_{X})$ by
$\mathscr{F}$ are equivalent if there exists a symmetric rack (respectively
quandle) isomorphism,
$\Phi:(E_{1},\rho_{E_{1}})\rightarrow(E_{2},\rho_{E_{2}})$ which respects
projection maps and group actions, that is,

  1. (EE1)

$f_{2}\Phi(s)=f_{1}(s)$ for all $s\in E_{1}$.

  2. (EE2)

$\Phi(a\cdot s)=a\cdot\Phi(s)$ for all $s\in E_{x}$, $a\in A_{x}$ and $x\in
X$.

Let $f:(E,\rho_{E})\rightarrow(X,\rho_{X})$ be an extension of $(X,\rho_{X})$
by $\mathscr{F}$. Let $(E,\star)$ and $(X,*)$ be the underlying racks. Then, a
section $s:(X,\rho_{X})\rightarrow(E,\rho_{E})$ is a map such that
$fs=\mathrm{id}_{X}$ and $s\rho_{X}=\rho_{E}s$. Since, $f$ is a symmetric rack
homomorphism, it follows that $s(x)\star s(y)\in E_{x\ast y}$. Further, since
$A_{x\ast y}$ acts freely and transitively on $E_{x\ast y}$, there exists a
unique $\sigma_{x,y}\in A_{x\ast y}$ such that

$s(x)\star s(y)=\sigma_{x,y}\cdot s(x\ast y).$

This leads to the following definition.

######  Definition 5.3.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle) and
$\mathscr{F}=(A,\phi,\psi,\eta)$ be an $(X,\rho_{X})$-module. Given an
extension $f:(E,\rho_{E})\rightarrow(X,\rho_{X})$ of $(X,\rho)$ by
$\mathscr{F}$ and a section $s:(X,\rho_{X})\rightarrow(E,\rho_{E})$, a factor
set is a map $\sigma:X\times X\rightarrow\bigsqcup_{x\in X}A_{x}$ such that

$s(x)\star s(y)=\sigma_{x,y}\cdot s(x\ast y)$ (5.0.1)

for all $x,y\in X$, where we write $\sigma(x,y)=\sigma_{x,y}$.

Note that when $\sigma_{x,y}=0\in A_{x*y}$, then
$s:(X,\rho_{X})\rightarrow(E,\rho_{E})$ is a symmetric rack homomorphism.
Thus, $\sigma$ can be regarded as an obstruction to $s$ being a symmetric rack
homomorphism. For each $x\in X$, since the $A_{x}$-action on $E_{x}$ is free
and transitive, every element of $E_{x}$ can be written uniquely in the form
$a\cdot s(x)$ for some unique $a\in A_{x}$. Thus, for $a\in A_{x}$ and $b\in
A_{y}$, we have

$\displaystyle\big{(}a\cdot s(x)\big{)}\star\big{(}b\cdot s(y)\big{)}$
$\displaystyle=$ $\displaystyle\phi_{x,y}(a)\cdot\big{(}s(x)\star(b\cdot
s(y))\big{)}$ $\displaystyle=$
$\displaystyle\phi_{x,y}(a)\cdot(\psi_{x,y}(b)\cdot\big{(}s(x)\star
s(y))\big{)}$ $\displaystyle=$
$\displaystyle\phi_{x,y}(a)\cdot(\psi_{x,y}(b)\cdot\big{(}\sigma_{x,y}\cdot
s(x\ast y))\big{)}$ $\displaystyle=$
$\displaystyle\big{(}\phi_{x,y}(a)+\psi_{x,y}(b)+\sigma_{x,y})\cdot(s(x\ast
y)\big{)}.$

Thus, the rack structure on $E$ is completely determined by the factor set
$\sigma$.

######  Theorem 5.4.

Let $(X,\rho_{X})$ be a symmetric rack and $\mathscr{F}=(A,\phi,\psi,\eta)$ be
an $(X,\rho_{X})$-module. Let $\sigma:X\times X\rightarrow\bigsqcup_{x\in
X}A_{x}$ be a map and ${E}(\mathscr{F},\sigma):=\big{\\{}(a,x)\mid x\in
X,~{}a\in A_{x}\big{\\}}$. Define the binary operation

$(a,x)\tilde{\star}(b,y):=\big{(}\phi_{x,y}(a)+\psi_{x,y}(b)+\sigma_{x,y},x\ast
y\big{)}$

on ${E}(\mathscr{F},\sigma)$ and the map
$\rho_{E(\mathscr{F},\sigma)}:E(\mathscr{F},\sigma)\to E(\mathscr{F},\sigma)$
given by

$(a,x)\mapsto\big{(}\eta_{x}(a),\rho_{X}(x)\big{)}$

for all $a\in A_{x},b\in A_{y}$ and $x,y\in X$. Then,
$(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$ is an extension of
$(X,\rho_{X})$ by $\mathscr{F}$ with factor set $\sigma$ if and only if

  1. $\mathrm{(F1)}$

$\sigma_{x\ast y,z}+\phi_{x\ast y,z}(\sigma_{x,y})=\phi_{x\ast z,y\ast
z}(\sigma_{x,z})+\sigma_{x\ast z,y\ast z}+\psi_{x\ast z,y\ast
z}(\sigma_{y,z})$,

  2. $\mathrm{(F2)}$

$\sigma_{\rho_{X}(x),y}=\eta_{x\ast y}(\sigma_{x,y})$,

  3. $\mathrm{(F3)}$

$\phi_{x\ast\rho_{X}(y),y}(\sigma_{x,\rho_{X}(y)})=-\sigma_{x\ast\rho_{X}(y),y}$,

for all $x,y,z\in X$. Moreover, if $(E,\rho_{E})$ is an extension of
$(X,\rho_{X})$ by $\mathscr{F}$ with factor set $\sigma$, then
$\mathrm{(F1)-(F3)}$ hold and $(E,\rho_{E})$ is equivalent to
$E(\mathscr{F},\sigma).$

Further, if $(X,\rho_{X})$ is a symmetric quandle, then we have the additional
condition

  1. $\mathrm{(F4)}$

$\sigma_{x,x}=0$ for all $x\in X$.

###### Proof.

Suppose that $\mathrm{(F1)-(F3)}$ hold. That
$(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$ is a rack is proved in
[11, Proposition 3.1]. We check that $\rho_{E(\mathscr{F},\sigma)}$ is a good
involution.

  * •

For (S1), we have

$\rho_{E(\mathscr{F},\sigma)}^{2}((a,x))=\rho_{E(\mathscr{F},\sigma)}(\eta_{x}(a),\rho_{X}(x))=(\eta_{\rho_{X}(x)}(\eta_{x}(a)),\rho_{X}^{2}(x))=(a,x).$

  * •

For (S2), we see that

$\displaystyle\big{(}\rho_{E(\mathscr{F},\sigma)}(a,x)\big{)}\tilde{\star}(b,y)$
$\displaystyle=$
$\displaystyle\big{(}\eta_{x}(a),\,\rho_{X}(x)\big{)}\tilde{\star}(b,y)$
$\displaystyle=$
$\displaystyle\big{(}\phi_{\rho_{X}(x),y}(\eta_{x}(a))+\psi_{\rho_{X}(x),y}(b)+\sigma_{\rho_{X}(x),y},\,\rho_{X}(x)\ast
y\big{)}$ $\displaystyle=$ $\displaystyle\big{(}\eta_{x\ast
y}(\phi_{x,y}(a)+\psi_{x,y}(b)+\sigma_{x,y}),\,\rho_{X}(x\ast
y)\big{)},\quad\textrm{by (M4), (M5) and (F2)}$ $\displaystyle=$
$\displaystyle\rho_{E(\mathscr{F},\sigma)}\big{(}\phi_{x,y}(a)+\psi_{x,y}(b)+\sigma_{x,y},\,x\ast
y\big{)}$ $\displaystyle=$
$\displaystyle\rho_{E(\mathscr{F},\sigma)}\big{(}(a,x)\tilde{\star}(b,y)\big{)}.$

  * •

For (S3), we have

$\displaystyle(a,x)\tilde{\star}\rho_{E(\mathscr{F},\sigma)}(b,y)$
$\displaystyle=$
$\displaystyle(a,x)\tilde{\star}\big{(}\eta_{y}(b),\,\rho_{X}(y)\big{)}$
$\displaystyle=$
$\displaystyle\big{(}\phi_{x,\rho_{X}(y)}(a)+\psi_{x,\rho_{X}(y)}(\eta_{y}(b))+\sigma_{x,\rho_{X}(y)},\,x\ast\rho_{X}(y)\big{)}$
$\displaystyle=$
$\displaystyle\big{(}\phi_{x\ast^{-1}y,y}^{-1}(a-\sigma_{x\ast^{-1}y,y}-\psi_{x\ast^{-1}y,y}(b)),\,x\ast^{-1}y\big{)},\quad\textrm{by
(M6), (M8) and (F3)}$ $\displaystyle=$
$\displaystyle(a,x)\tilde{\star}^{-1}(b,y).$

Hence, $(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$ is a symmetric
rack. Define
$f_{1}:(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})\rightarrow(X,\rho_{X})$
to be the projection onto the second coordinate. Let $A_{x}$ act on
$E(\mathscr{F},\sigma)_{x}=f_{1}^{-1}(x)$ by $a\cdot(b,x)=(a+b,x)$ for each
$x\in X$ and $a,b\in A_{x}$. Note that these actions are free and transitive.
Also, as proved in [11, Proposition 3.1], the actions satisfy

$\big{(}\alpha\cdot(a,x)\big{)}\tilde{\star}(b,y)=\phi_{x,y}(\alpha)\cdot\big{(}(a,x)\tilde{\star}(b,y)\big{)}\quad\textrm{and}\quad(a,x)\tilde{\star}\,\big{(}\beta\cdot(b,y)\big{)}=\psi_{x,y}(\beta)\cdot\big{(}(a,x)\tilde{\star}(b,y)\big{)}$

for all $\alpha\in A_{x}$, $\beta\in A_{y}$, $(a,x)\in
E(\mathscr{F},\sigma)_{x}$ and $(b,y)\in E(\mathscr{F},\sigma)_{x}$. Further,
we have

$\rho_{E(\mathscr{F},\sigma)}\big{(}\alpha\cdot(a,x)\big{)}=\rho_{E(\mathscr{F},\sigma)}\big{(}(\alpha+a,x)\big{)}=\big{(}\eta_{x}(\alpha+a),\rho_{X}(x)\big{)}=\eta_{x}(\alpha)\cdot(\rho_{E(\mathscr{F},\sigma)}(a,x)).$

Thus, $(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$ is an extension
of $(X,\rho_{X})$ by $\mathscr{F}$. Define $s:X\rightarrow
E(\mathscr{F},\sigma)$ as $s(x)=(0,x)$. This is clearly a section satisfying

$s(x)\tilde{\star}s(y)=(0,x)\ast(0,y)=(\sigma_{x,y},x*y)=\sigma_{x,y}\cdot
s(x\ast y)$

and

$\rho_{E(\mathscr{F},\sigma)}(s(x))=\rho_{E(\mathscr{F},\sigma)}(0,x)=(0,\rho_{X}(x))=s(\rho_{X}(x)).$

Thus, $\sigma$ is the factor set corresponding to the section s. The converse
follows by reversing the steps.

For the second assertion, let $f_{2}:(E,\rho_{E})\rightarrow(X,\rho_{X})$ be
an extension of $(X,\rho_{X})$ by $\mathscr{F}$ with factor set $\sigma$
corresponding to a section $s:(X,\rho)\rightarrow(E,\rho_{E})$. Define

$\Phi:(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})\rightarrow(E,\rho_{E})$

as $\Phi\big{(}(a,x)\big{)}=a\cdot s(x)$. Since, $A_{x}$ acts freely and
transitively, it follows that $\Phi$ is a bijection. Also, using (E3), we have

$\displaystyle\Phi\big{(}(a,x)\tilde{\star}(b,y)\big{)}$ $\displaystyle=$
$\displaystyle\Phi\big{(}\phi_{x,y}(a)+\psi_{x,y}(b)+\sigma_{x,y},\,x\ast
y\big{)}$ $\displaystyle=$
$\displaystyle(\phi_{x,y}(a)+\psi_{x,y}(b)+\sigma_{x,y})\cdot s(x\ast y)$
$\displaystyle=$
$\displaystyle(\phi_{x,y}(a)+\psi_{x,y}(b))\cdot(s(x)\tilde{\star}s(y))$
$\displaystyle=$
$\displaystyle\phi_{x,y}(a)\cdot\big{(}s(x)\tilde{\star}(b\cdot s(y))\big{)}$
$\displaystyle=$ $\displaystyle(a\cdot s(x))\tilde{\star}(b\cdot s(y))$
$\displaystyle=$ $\displaystyle\Phi(a,x)\tilde{\star}\Phi(b,y).$

Hence, $\Phi$ is a rack isomorphism. Further, we see that

$\displaystyle\Phi\rho_{E(\mathscr{F},\sigma)}(a,x)$ $\displaystyle=$
$\displaystyle\Phi\big{(}\eta_{x}(a),\rho_{X}(x)\big{)}$ $\displaystyle=$
$\displaystyle\eta_{x}(a)\cdot s(\rho_{X}(x))$ $\displaystyle=$
$\displaystyle\eta_{x}(a)\cdot\rho_{E}(s(x)),\quad\textrm{since
$\rho_{E}\,s=s\,\rho_{X}$}$ $\displaystyle=$ $\displaystyle\rho_{E}(a\cdot
s(x)),\quad\textrm{by (E3)}$ $\displaystyle=$ $\displaystyle\rho_{E}\Phi(a,x)$

for all $x\in X$ and $a\in A_{x}$. Hence, $\Phi$ is an isomorphism of
symmetric racks. Clearly, $f_{2}\Phi(a,x)=f_{2}(a\cdot s(x))=x=f_{1}(a,x)$ for
all $(a,x)\in E(\mathscr{F},\sigma)_{x}$ and
$\Phi(\alpha\cdot(a,x))=\Phi((\alpha+a,x))=(\alpha+a)\cdot
s(x)=\alpha\cdot\Phi(a,x)$ for all $\alpha,a\in A_{x}$ and $x\in X$. Hence,
$\Phi$ is an equivalence of extensions.

Finally, if $(X,\rho_{X})$ is a symmetric quandle, then the condition (F4)
follows. ∎

######  Proposition 5.5.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle) and
$\mathscr{F}$ an $(X,\rho_{X})$-module. Let $\sigma$ and $\tau$ be factor sets
corresponding to sections $s$ and $t$, respectively, for extensions of
$(X,\rho_{X})$ by $\mathscr{F}$. Then the following assertions are equivalent:

  1. (1)

The extensions $(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$ and
$(E(\mathscr{F},\tau),\rho_{E(\mathscr{F},\tau)})$ are equivalent.

  2. (2)

There exists a set of elements $\\{v_{x}\in A_{x}\mid x\in X\\}$ such that
$\tau_{x,y}=\sigma_{x,y}+\phi_{x,y}(v_{x})+\psi_{x,y}(v_{y})-v_{x\ast y}$ and
$\eta_{x}(v_{x})=v_{\rho_{X}(x)}$ for $x,y\in X$.

  3. (3)

$\sigma$ and $\tau$ are factor sets of the same extension of $(X,\rho_{X})$ by
$\mathscr{F}$ corresponding to different sections $s$ and $t$, respectively.

###### Proof.

First, we prove $(1)\Leftrightarrow(2)$. Let
$\Phi:(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})\rightarrow(E(\mathscr{F},\tau),\rho_{E(\mathscr{F},\tau)})$
be an equivalence of extensions. Since $\Phi$ respects the projections, we
obtain $\Phi(0,x)=(v_{x},x)$ for some $v_{x}\in A_{x}$. Further, since $\Phi$
respects the group actions, this gives
$\Phi(a,x)=\Phi(a\cdot(0,x))=a\cdot\Phi(0,x)=a\cdot(v_{x},x)=(a+v_{x},x)$ for
all $a\in A_{x}$. Then,

$\Phi\big{(}(a,x)\tilde{\star}(b,y)\big{)}=\big{(}\phi_{x,y}(a)+\psi_{x,y}(b)+\tau_{x,y}+v_{x\ast
y},\,x\ast y\big{)}$

and

$\Phi(a,x)\bar{\star}\Phi(b,y)=(a+v_{x},x)\bar{\star}(b+v_{y},y)=\big{(}\phi_{x,y}(a+v_{x})+\psi_{x,y}(b+v_{y})+\sigma_{x,y},\,x\ast
y\big{)},$

where $\tilde{\star}$ and $\bar{\star}$ are the rack operations in
$E(\mathscr{F},\sigma)$ and $E(\mathscr{F},\tau)$, respectively. Thus,
$\Phi\big{(}(a,x)\bar{\star}(b,y)\big{)}=\Phi(a,x)\bar{\star}\Phi(b,y)$ if and
only if $\tau_{x,y}=\sigma_{x,y}+\phi_{x,y}(v_{x})+\psi_{x,y}(v_{y})-v_{x\ast
y}$ for $x,y\in X$. We have

$\Phi\rho_{E(\mathscr{F},\sigma)}(a,x)=\Phi(\eta_{x}(a),\rho_{X}(x))=(\eta_{x}(a)+v_{\rho_{X}(x)},\rho_{X}(x))$

and

$\rho_{E(\mathscr{F},\tau)}\Phi(a,x)=\rho_{E(\mathscr{F},\tau)}(a+v_{x},x)=(\eta_{x}(a+v_{x}),\rho_{X}(x)).$

Thus, $\Phi\rho_{E(\mathscr{F},\sigma)}=\rho_{E(\mathscr{F},\tau)}\Phi$ if and
only if $\eta_{x}(v_{x})=v_{\rho_{X}(x)}$.

Next, we prove $(1)\Leftrightarrow(3)$. We shall use $(1)\Leftrightarrow(2)$
proved above. Obviously, $\sigma$ is the factor set corresponding to the
section
$s:(X,\rho_{X})\rightarrow(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$
given by $s(x)=(0,x)$. Define
$t:(X,\rho_{X})\rightarrow(E(\mathscr{F},\sigma),\rho_{E(\mathscr{F},\sigma)})$
by $t(x)=(v_{x},x)$. Since

$t(\rho_{X}(x))=(v_{\rho_{X}(x)},\rho_{X}(x))=(\eta_{x}(v_{x}),\rho_{X}(x))=\rho_{E(\mathscr{F},\sigma)}(t(x)),$

it follows that $t$ is a section. Note that, we have
$\tau_{x,y}=\sigma_{x,y}+\phi_{x,y}(v_{x})+\psi_{x,y}(v_{y})-v_{x\ast y}$.
Then, we see that

$\displaystyle t(x)\tilde{\star}t(y)$ $\displaystyle=$
$\displaystyle(v_{x},x)\tilde{\star}(v_{y},y)$ $\displaystyle=$
$\displaystyle(\phi_{x,y}(v_{x})+\psi_{x,y}(v_{y})+\sigma_{x,y},x*y)$
$\displaystyle=$ $\displaystyle(\tau_{x,y}+v_{x*y},x*y)$ $\displaystyle=$
$\displaystyle\tau_{x,y}\cdot(v_{x*y},x*y)$ $\displaystyle=$
$\displaystyle\tau_{x,y}\cdot t(x*y).$

Thus, $\tau$ is the factor set corresponding to $t$.

Conversely, suppose that $\sigma$ and $\tau$ are factor sets of some extension
$(E,\rho_{E})$ of $(X,\rho_{X})$ by $\mathscr{F}$ corresponding to sections
$s:(X,\rho_{X})\rightarrow(E,\rho_{E})$ and
$t:(X,\rho_{X})\rightarrow(E,\rho_{E})$, respectively. Then, for each $x\in
X$, we have $s(x)=v_{x}\cdot t(x)$ for a unique $v_{x}\in A_{x}$. Define
$\Phi:E(\mathscr{F},\sigma)\rightarrow E(\mathscr{F},\tau)$ as
$\Phi\big{(}(a,x)\big{)}=(a+v_{x},x)$. Then, a direct check implies that
$\Phi$ is an equivalence of extensions. ∎

######  Corollary 5.6.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle) and
$\mathscr{F}=(A,\phi,\psi,\eta)$ be an $(X,\rho_{X})$-module. Given an
extension $f:(E,\rho_{E})\rightarrow(X,\rho_{X})$ of $(X,\rho_{X})$ by
$\mathscr{F}$, the following statements are equivalent:

  1. (1)

There exists a symmetric-rack homomorphism
$s:(X,\rho_{X})\rightarrow(E,\rho_{E})$ such that $f\,s=\mathrm{id}_{X}$ and
$s\,\rho_{X}=\rho_{E}\,s.$

  2. (2)

There exists a section such that its corresponding factor set is trivial.

  3. (3)

For any section and the corresponding factor set $\sigma$, there exists a set
of elements $\\{v_{x}\in A_{x}\mid x\in X\\}$ such that

$\sigma_{x,y}=\phi_{x,y}(v_{x})+\psi_{x,y}(v_{y})-v_{x\ast
y}\quad\textrm{and}\quad\eta_{x}(v_{x})=v_{\rho_{X}(x)}$

for all $x,y\in X$.

Extensions satisfying any of (1), (2) or (3) in Corollary 5.6 are called split
extensions.

######  Definition 5.7.

Given a symmetric rack (respectively quandle) $(X,\rho_{X})$ and an
$(X,\rho_{X})$-module $\mathscr{F}$, a map $\sigma:X\times
X\rightarrow\bigsqcup_{x\in X}A_{x}$ is called a coboundary if there exists a
map $v:X\rightarrow\bigsqcup_{x\in X}A_{x}$ such that

$\sigma_{x,y}=\phi_{x,y}(v_{x})+\psi_{x,y}(v_{y})-v_{x\ast
y}\quad\textrm{and}\quad\eta_{x}(v_{x})=v_{\rho_{X}(x)}$

for all $x,y\in X$.

######  Theorem 5.8.

Let $(X,\rho_{X})$ be a symmetric rack and $\mathscr{F}=(A,\phi,\psi,\eta)$ be
an $(X,\rho_{X})$-module. Then there is an abelian group
$\mathcal{H}^{2}_{\mathrm{SR}}(X,\mathscr{F})$ whose elements are in bijective
correspondence with the set of equivalence classes of extensions of
$(X,\rho_{X})$ by $\mathscr{F}$.

###### Proof.

Let $Z^{2}(X,\mathscr{F})$ consists of all factor sets $\sigma:X\times
X\rightarrow\bigsqcup_{x\in X}A_{x}$. Define the binary operation

$(\sigma+\tau)_{x,y}:=\sigma_{x,y}+\tau_{x,y}$

for all $x,y\in X$. In view of Theorem 5.4, it follows that the conditions
(F1)-(F3) are satisfied by $\sigma+\tau$, and hence this binary operation
turns $Z^{2}(X,\mathscr{F})$ into an abelian group with the trivial factor set
as the identity element. Further, the set $B^{2}(X,\mathscr{F})$ of all
coboundaries forms a subgroup of $Z^{2}(X,\mathscr{F})$. Define
$\mathcal{H}^{2}_{\mathrm{SR}}(X,\mathscr{F}):=Z^{2}(X,\mathscr{F})/B^{2}(X,\mathscr{F})$.
Then, by Theorem 5.4 and Proposition 5.5,
$\mathcal{H}^{2}_{\mathrm{SR}}(X,\mathscr{F})$ is in bijective correspondence
with the set of equivalence classes of extensions of $(X,\rho_{X})$ by
$\mathscr{F}$. ∎

Using Theorem 5.4 and [11, Proposition 2.5], we obtain an analogous result for
symmetric quandles.

######  Theorem 5.9.

Let $(X,\rho_{X})$ be a symmetric quandle and $\mathscr{F}=(A,\phi,\psi,\eta)$
be an $(X,\rho_{X})$-module. Then there is an abelian group
$\mathcal{H}^{2}_{\mathrm{SQ}}(X,\mathscr{F})$ whose elements are in bijective
correspondence with the set of equivalence classes of extensions of
$(X,\rho_{X})$ by $\mathscr{F}$.

##  6\. Generalized (co)homology for symmetric racks and symmetric quandles

In this section, we formulate a (co)homology theory for symmetric racks
(respectively quandles) $(X,\rho_{X})$ with coefficients derived from
homogeneous $(X,\rho_{X})$-modules. To initiate this formulation, we first
extract the fundamental axioms that define $(X,\rho_{X})$-modules.

######  Definition 6.1.

The symmetric rack algebra of a symmetric rack $(X,\rho_{X})$ is the
associative $\mathbb{Z}$-algebra, denoted by $\mathbb{Z}(X,\rho_{X})$, which
is generated by the set $\\{\phi_{x,y}^{\pm 1},\,\psi_{x,y},\,\eta_{x}\mid
x,y\in X\\}$ and admit the following defining relations:

  1. (A1)

$\phi_{x,y}\phi_{x,y}^{-1}=\phi_{x,y}^{-1}\phi_{x,y}=1$,

  2. (A2)

$\phi_{x\ast y,z}\phi_{x,y}=\phi_{x\ast z,y\ast z}\phi_{x,z}$,

  3. (A3)

$\phi_{x\ast y,z}\psi_{x,y}=\psi_{x\ast z,y\ast z}\phi_{y,z}$,

  4. (A4)

$\eta_{\rho_{X}(x)}\eta_{x}=1$,

  5. (A5)

$\phi_{\rho_{X}(x),y}\eta_{x}=\eta_{x\ast y}\phi_{x,y}$,

  6. (A6)

$\psi_{\rho_{X}(x),y}=\eta_{x\ast y}\psi_{x,y}$,

  7. (A7)

$\phi_{x\ast\rho_{X}(y),y}\phi_{x,\rho_{X}(y)}=1$,

  8. (A8)

$\psi_{x\ast y,z}=\phi_{x\ast z,y\ast z}\psi_{x,z}+\psi_{x\ast z,y\ast
z}\psi_{y,z}$,

  9. (A9)

$\phi_{x\ast\rho_{X}(y),y}\psi_{x,\rho_{X}(y)}\eta_{y}=-\psi_{x\ast\rho_{X}(y),y}$,

for all $x,y,z\in X.$

######  Definition 6.2.

The symmetric quandle algebra of a symmetric quandle $(X,\rho_{X})$ is the
symmetric rack algebra of the underlying symmetric rack which satisfies the
additional relation

  1. (A10)

$\phi_{x,x}+\psi_{x,x}=1$

for all $x\in X.$

######  Remark 6.3.

Let $(X,\rho_{X})$ be a symmetric rack (respectively quandle). Then the
category of homogeneous $(X,\rho_{X})$-modules is equivalent to the category
of modules over $\mathbb{Z}(X,\rho_{X})$. Thus, these are abelian with enough
injective and projective objects.

Let $(X,\rho_{X})$ be a symmetric rack. For each $n\geq 0$, define
$C_{n}(X,\rho_{X}):=\mathbb{Z}(X,\rho_{X})X^{n},~{}\mathrm{i.e.}$, the free
left $\mathbb{Z}(X,\rho_{X})$-module with basis $X^{n}$, where $X^{0}=\\{p\\}$
for some fixed $p\in X$. For the sake of convenience, we fix the notation

$[x_{1}\cdots x_{n}]:=(((x_{1}\ast x_{2})\ast x_{3})\ast\cdots\ast x_{n})$

for any $(x_{1},x_{2},\ldots,x_{n})\in X^{n}$. For each $n\geq 2$, let
$\partial_{n}:C_{n}(X,\rho_{X})\rightarrow C_{n-1}(X,\rho_{X})$ be the
$\mathbb{Z}(X,\rho_{X})$-linear map defined on the basis as

$\displaystyle\partial_{n}\big{(}(x_{1},\ldots,x_{n})\big{)}$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}(x_{1}\ast
x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})$ $\displaystyle+$
$\displaystyle(-1)^{n}\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{n}),$

and

$\partial_{1}(x)=-\psi_{x\ast^{-1}p,p}(p).$

Using conditions (A2), (A3), and (A8), it has been proved in [1, Lemma 4.2]
that $\partial_{n-1}\,\partial_{n}=1$ for all $n\geq 1$. Thus,
$(C_{n}(X,\rho_{X}),\partial)$ is a chain complex. We next define a subcomplex
of this chain complex.

For each $n\geq 1$, consider the left $\mathbb{Z}(X,\rho_{X})$-submodule
$D_{n}^{SR}(X,\rho_{X})$ of $C_{n}(X,\rho_{X})$ generated by $U_{n}\cup
V_{n}$, where

$U_{n}=\big{\\{}\eta_{[x_{1}\cdots
x_{n}]}(x_{1},\ldots,x_{n})-(\rho_{X}(x_{1}),x_{2},\ldots,x_{n})\mid(x_{1},x_{2},\ldots,x_{n})\in
X^{n}\big{\\}},$ $\displaystyle V_{n}$ $\displaystyle=$
$\displaystyle\bigcup_{i=2}^{n}\big{\\{}\big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}(x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n})+(x_{1},\ldots,x_{n})\big{)}\mid$
$\displaystyle(x_{1},x_{2},\ldots,x_{n})\in X^{n}\big{\\}}.$

for $n\geq 2$ and $V_{1}=\emptyset$.

######  Lemma 6.4.

Let $(X,\rho_{X})$ be a symmetric rack. Then
$\partial_{n}\big{(}U_{n}\big{)}\subset D_{n-1}^{SR}(X,\rho_{X})$ for each
$n\geq 2$.

###### Proof.

Using the fact that $\rho_{X}$ is a good involution and the rack identity

$[x_{1}\cdots\widehat{x_{j}}\cdots x_{n}]\ast[x_{j}\cdots x_{n}]=[x_{1}\cdots
x_{n}]$ (6.0.1)

for each $(x_{1},x_{2},\ldots,x_{n})\in X^{n}$, it follows from the condition
(A5) that

$\eta_{\underbrace{[x_{1}\cdots x_{n}]}_{x\ast
y}}\phi_{\underbrace{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}]}_{x},\underbrace{[x_{j}\cdots
x_{n}]}_{y}}=\phi_{\underbrace{[\rho_{X}(x_{1})\cdots\widehat{x_{j}}\cdots
x_{n}]}_{\rho_{X}(x)},\underbrace{[x_{j}\cdots
x_{n}]}_{y}}\eta_{\underbrace{[x_{1}\cdots\widehat{x_{j}}\cdots x_{n}]}_{x}}.$
(6.0.2)

Similarly, it follows from (A6) that

$\eta_{\underbrace{[x_{1}\cdots x_{n}]}_{x\ast
y}}\psi_{\underbrace{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}]}_{x},\underbrace{[x_{2}x_{3}\cdots
x_{n}]}_{y}}=\psi_{\underbrace{[\rho_{X}(x_{1})\widehat{x_{2}}x_{3}\cdots
x_{n}]}_{\rho_{X}(x)},\underbrace{[x_{2}x_{3}\cdots x_{n}]}_{y}}.$ (6.0.3)

Then, we have

$\displaystyle\partial_{n}\big{(}\eta_{[x_{1}\cdots
x_{n}]}(x_{1},\ldots,x_{n})-(\rho_{X}(x_{1}),x_{2},\ldots,x_{n})\big{)}$
$\displaystyle=$ $\displaystyle\eta_{[x_{1}\cdots
x_{n}]}\partial_{n}(x_{1},\ldots,x_{n})-\partial_{n}(\rho_{X}\big{(}x_{1}),x_{2},\ldots,x_{n}),\quad\textrm{since
$\partial_{n}$ is $\mathbb{Z}(X,\rho_{X})$-linear}$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\eta_{[x_{1}\cdots
x_{n}]}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})$ $\displaystyle-$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\eta_{[x_{1}\cdots
x_{n}]}(x_{1}\ast x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})$
$\displaystyle+$ $\displaystyle(-1)^{n}\eta_{[x_{1}\cdots
x_{n}]}\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots x_{n}],[x_{2}x_{3}\cdots
x_{n}]}(x_{2},\ldots,x_{n})$ $\displaystyle-$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\phi_{[\rho_{X}(x_{1})\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(\rho_{X}(x_{1}),\ldots,\widehat{x_{j}},\ldots,x_{n})$ $\displaystyle+$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}(\rho_{X}(x_{1})\ast
x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})$ $\displaystyle-$
$\displaystyle(-1)^{n}\psi_{[\rho_{X}(x_{1})\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\Big{(}\eta_{[x_{1}\cdots
x_{n}]}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})$ $\displaystyle-$
$\displaystyle\phi_{[\rho_{X}(x_{1})\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(\rho_{X}(x_{1}),\ldots,\widehat{x_{j}},\ldots,x_{n})\Big{)}$
$\displaystyle-$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\Big{(}\eta_{[x_{1}\cdots
x_{n}]}(x_{1}\ast x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})$
$\displaystyle+$ $\displaystyle(\rho_{X}(x_{1})\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{n})\Big{)}$ $\displaystyle+$
$\displaystyle(-1)^{n}\Big{(}\eta_{[x_{1}\cdots
x_{n}]}\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots x_{n}],[x_{2}x_{3}\cdots
x_{n}]}(x_{2},\ldots,x_{n})-\psi_{[\rho_{X}(x_{1})\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{n})\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\phi_{[\rho_{X}(x_{1})\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots x_{n}]}\Big{(}\eta_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})-(\rho_{X}(x_{1}),\ldots,\widehat{x_{j}},\ldots,x_{n})\Big{)}$
$\displaystyle-$
$\displaystyle\sum_{j=2}^{n}(-1)^{j}(-1)^{n}\Big{(}\eta_{[x_{1}\cdots
x_{n}]}(x_{1}\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{n})+(\rho_{X}(x_{1})\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{n})\Big{)},$ $\displaystyle\quad\textrm{using
\eqref{lemma eq 1} and \eqref{lemma eq 2}}.$

Hence, $\partial_{n}\big{(}U_{n}\big{)}$ lies in $D_{n-1}^{SR}(X,\rho_{X})$. ∎

######  Lemma 6.5.

Let $(X,\rho_{X})$ be a symmetric rack. Then
$\partial_{n}\big{(}V_{n}\big{)}\subset D_{n-1}^{SR}(X,\rho_{X})$ for each
$n\geq 2$.

###### Proof.

For a fixed $2\leq i\leq n$, we have

$\displaystyle\partial_{n}\big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}(x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n})\big{)}$ $\displaystyle=$
$\displaystyle\underbrace{\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\phi_{[(x_{1}\ast x_{i})\cdots\widehat{(x_{j}\ast
x_{i})}\cdots(x_{i-1}\ast x_{i})\rho_{X}(x_{i})x_{i+1}\cdots
x_{n}],[(x_{j}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\rho_{X}{(x_{i})}x_{i+1}\cdots x_{n}]}}_{1A}$
$\displaystyle\underbrace{\Big{(}x_{1}\ast x_{i},\ldots,\widehat{x_{j}\ast
x_{i}},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n}\Big{)}}_{1A}$ $\displaystyle+$
$\displaystyle\underbrace{(-1)^{i}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\phi_{[(x_{1}\ast x_{i})\cdots\cdots(x_{i-1}\ast
x_{i})\widehat{\rho_{X}(x_{i})}x_{i+1}\cdots
x_{n}],[\rho_{X}{(x_{i})}x_{i+1}\cdots x_{n}]}}_{1B}$
$\displaystyle\underbrace{\Big{(}x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\widehat{\rho_{X}(x_{i})},x_{i+1},\ldots,x_{n}\Big{)}}_{1B}$
$\displaystyle+$
$\displaystyle\underbrace{\sum_{j=i+1}^{n}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\phi_{[(x_{1}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\rho_{X}(x_{i})x_{i+1}\cdots\widehat{x_{j}}\cdots x_{n}],[(x_{j}\cdots
x_{n}]}}_{1C}$ $\displaystyle\underbrace{\Big{(}x_{1}\ast
x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,\widehat{x_{j}},\ldots,x_{n}\Big{)}}_{1C}$
$\displaystyle-$
$\displaystyle\underbrace{\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}}_{1D}$ $\displaystyle\underbrace{\Big{(}(x_{1}\ast
x_{i})\ast(x_{j}\ast x_{i}),\ldots,(x_{j-1}\ast x_{i})\ast(x_{j}\ast
x_{i}),\widehat{x_{j}\ast x_{i}},x_{j+1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n}\Big{)}}_{1D}$ $\displaystyle-$
$\displaystyle\underbrace{(-1)^{i}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}}_{1E}$
$\displaystyle\underbrace{\Big{(}(x_{1}*x_{i})*\rho_{X}(x_{i}),\ldots,(x_{i-1}*x_{i})*\rho_{X}(x_{i}),\widehat{\rho_{X}(x_{i})},x_{i+1},\ldots,x_{n}\Big{)}}_{1E}$
$\displaystyle-$
$\displaystyle\underbrace{\sum_{j=i+1}^{n}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}}_{1F}$ $\displaystyle\underbrace{\Big{(}(x_{1}\ast
x_{i})\ast x_{j},\ldots,(x_{i-1}\ast x_{i})\ast x_{j},\rho_{X}(x_{i}\ast
x_{j}),x_{i+1}\ast x_{j},\ldots,{x_{j-1}\ast
x_{j}},\widehat{x_{j}},x_{j+1},\ldots,x_{n}\Big{)}}_{1F}$ $\displaystyle+$
$\displaystyle\underbrace{(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\psi_{[(x_{1}\ast x_{i})\widehat{(x_{2}\ast
x_{i})}\cdots(x_{i-1}\ast x_{i})\rho_{X}(x_{i})x_{i+1}\cdots
x_{n}],[(x_{2}\ast x_{i})\cdots\cdots(x_{i-1}\ast
x_{i})\rho_{X}(x_{i})x_{i+1}\cdots x_{n}]}}_{1G}$
$\displaystyle\underbrace{\Big{(}x_{2}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n}\Big{)}}_{1G}$

and

$\displaystyle\partial_{n}\big{(}(x_{1},\ldots,x_{n})\big{)}$ $\displaystyle=$
$\displaystyle\underbrace{\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})}_{2A}$
$\displaystyle+$
$\displaystyle\underbrace{(-1)^{i}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots
x_{n}]}(x_{1},\ldots,x_{i-1},\widehat{x_{i}},x_{i+1},\ldots,x_{n})}_{2B}$
$\displaystyle+$
$\displaystyle\underbrace{\sum_{j=i+1}^{n}(-1)^{j}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})}_{2C}$
$\displaystyle-$
$\displaystyle\underbrace{\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}(x_{1}\ast
x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})}_{2D}$ $\displaystyle-$
$\displaystyle\underbrace{(-1)^{i}(-1)^{n}(x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},x_{i+1},\ldots,x_{n})}_{2E}$ $\displaystyle-$
$\displaystyle\underbrace{\sum_{j=i+1}^{n}(-1)^{j}(-1)^{n}(x_{1}\ast
x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})}_{2F}$ $\displaystyle+$
$\displaystyle\underbrace{(-1)^{n}\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{n})}_{2G}.$

We see that

$\displaystyle 1A+2A$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\phi_{[(x_{1}\ast x_{i})\cdots\widehat{(x_{j}\ast
x_{i})}\cdots(x_{i-1}\ast x_{i})\rho_{X}(x_{i})x_{i+1}\cdots
x_{n}],[(x_{j}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\rho_{X}{(x_{i})}x_{i+1}\cdots x_{n}]}$ $\displaystyle(x_{1}\ast
x_{i},\ldots,\widehat{x_{j}\ast x_{i}},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n})+\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})\Big{)}.$

Then, using (6.0.1), the fact that $\rho_{X}$ is a good involution and the
condition (A2), we see that

$\displaystyle\phi_{\underbrace{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}]}_{x\ast y},\underbrace{[x_{i}\cdots x_{n}]}_{z}}$
$\displaystyle\phi_{\underbrace{[(x_{1}\ast x_{i})\cdots\widehat{(x_{j}\ast
x_{i})}\cdots(x_{i-1}\ast x_{i})\rho_{X}(x_{i})x_{i+1}\cdots
x_{n}]}_{x},\underbrace{[(x_{j}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\rho_{X}{(x_{i})}x_{i+1}\cdots x_{n}]}_{y}}$ $\displaystyle=$
$\displaystyle\phi_{\underbrace{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}]}_{x\ast z},\underbrace{[x_{j}\cdots x_{n}]}_{y\ast
z}}\phi_{\underbrace{[x_{1}\cdots\widehat{x_{j}}\cdots\widehat{x_{i}}\cdots
x_{n}]}_{x},\underbrace{[x_{i}\cdots x_{n}]}_{z}}.$

Thus, the term 1A + 2A lies in $D_{n-1}^{SR}(X,\rho_{X})$. Next, we consider

$\displaystyle 1B+2E$ $\displaystyle=$
$\displaystyle(-1)^{i}(-1)^{n}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\phi_{[(x_{1}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\widehat{\rho_{X}(x_{i})}x_{i+1}\cdots
x_{n}],[\rho_{X}{(x_{i})}x_{i+1}\cdots x_{n}]}$ $\displaystyle(x_{1}\ast
x_{i},\ldots,x_{i-1}\ast
x_{i},\widehat{\rho_{X}(x_{i})},x_{i+1},\ldots,x_{n})-(-1)^{i}(-1)^{n}(x_{1}\ast
x_{i},\ldots,x_{i-1}\ast x_{i},x_{i+1},\ldots,x_{n}).$

Due to condition (A7), we have

$\phi_{\underbrace{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}]}_{x\ast\rho_{X}(y)},\underbrace{[x_{i}\cdots
x_{n}]}_{y}}\phi_{\underbrace{[(x_{1}\ast x_{i})\cdots\cdots(x_{i-1}\ast
x_{i})\widehat{\rho_{X}(x_{i})}x_{i+1}\cdots
x_{n}]}_{x},\underbrace{[\rho_{X}{(x_{i})}x_{i+1}\cdots
x_{n}]}_{\rho_{X}(y)}}=\mathrm{id},$

and hence $1B+2E=0$. Consider the term

$\displaystyle 1C+2C$ $\displaystyle=$
$\displaystyle\sum_{j=i+1}^{n}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\phi_{[(x_{1}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\rho_{X}(x_{i})x_{i+1}\cdots\widehat{x_{j}}\cdots x_{n}],[x_{j}\cdots
x_{n}]}$ $\displaystyle(x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,\widehat{x_{j}},\ldots,x_{n})+\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{n})\Big{)}.$

Using the condition (A2), we see that

$\displaystyle\phi_{\underbrace{[x_{1}\cdots\widehat{x_{i}}\cdots x_{j}\cdots
x_{n}]}_{x\ast z},\underbrace{[x_{i}\cdots x_{j}\cdots x_{n}]}_{y\ast
z}}\phi_{\underbrace{[(x_{1}\ast x_{i})\cdots(x_{i-1}\ast
x_{i})\rho_{X}(x_{i})x_{i+1}\cdots\widehat{x_{j}}\cdots
x_{n}]}_{x},\underbrace{[x_{j}\cdots x_{n}]}_{z}}$ $\displaystyle=$
$\displaystyle\phi_{\underbrace{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}]}_{x\ast y},\underbrace{[x_{j}\cdots
x_{n}]}_{z}}\phi_{\underbrace{[x_{1}\cdots\widehat{x_{i}}\cdots\widehat{x_{j}}\cdots
x_{n}]}_{x},\underbrace{[x_{i}\cdots\widehat{x_{j}}\cdots x_{n}]}_{y}}.$

Thus, 1C + 2C lies in $D_{n-1}^{SR}(X,\rho_{X})$. Next, we consider

$\displaystyle 1D+2D$ $\displaystyle=$
$\displaystyle-\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}$ $\displaystyle((x_{1}\ast x_{i})\ast(x_{j}\ast
x_{i}),\ldots,(x_{j-1}\ast x_{i})*(x_{j}*x_{i}),\widehat{x_{j}\ast
x_{i}},x_{j+1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n})$ $\displaystyle+$
$\displaystyle(x_{1}\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{n})\Big{)}.$

Since $\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots x_{n}],[x_{i}\cdots
x_{n}]}=\phi_{[(x_{1}\ast x_{j})\cdots(x_{j-1}\ast
x_{j})x_{j+1}\cdots\widehat{x_{i}}\cdots x_{n}],[x_{i}\cdots x_{n}]}$, it
follows that 1D + 2D lies in $D_{n-1}^{SR}(X,\rho_{X})$. Next, we have

$\displaystyle 1E+2B$ $\displaystyle=$
$\displaystyle(-1)^{i}(-1)^{n}\Big{(}-\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots
x_{n}]}(x_{1},\ldots,x_{i-1},\widehat{\rho_{X}(x_{i})},x_{i+1},\ldots,x_{n})$
$\displaystyle+$ $\displaystyle\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots
x_{n}]}(x_{1},\ldots,x_{i-1},\widehat{x_{i}},x_{i+1},\ldots,x_{n})\Big{)}$
$\displaystyle=$ $\displaystyle 0.$

Consider the term

$\displaystyle 1F+2F$ $\displaystyle=$
$\displaystyle-\sum_{j=i+1}^{n}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}$ $\displaystyle(x_{1}\ast x_{i}\ast
x_{j},\ldots,x_{i-1}\ast x_{i}\ast x_{j},\rho_{X}(x_{i}\ast x_{j}),x_{i+1}\ast
x_{j},\ldots,{x_{j-1}\ast x_{j}},\widehat{x_{j}},x_{j+1},\ldots,x_{n})$
$\displaystyle+$ $\displaystyle(x_{1}\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{n})\Big{)}.$

Since $\phi_{[(x_{1}\ast x_{j})\cdots\widehat{(x_{i}\ast
x_{j})}\cdots(x_{j-1}\ast x_{j})x_{j+1}\cdots x_{n})],[(x_{i}\ast
x_{j})\cdots(x_{j-1}\ast x_{j})x_{j+1}\cdots
x_{n}]}=\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots x_{n}],[x_{i}\cdots x_{n}]}$,
it follows that 1F + 2F lies in $D_{n-1}^{SR}(X,\rho_{X})$. Next, consider the
term

$\displaystyle 1G+2G$ $\displaystyle=$
$\displaystyle(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}],[x_{i}\cdots x_{n}]}\psi_{[(x_{1}\ast x_{i})\widehat{(x_{2}\ast
x_{i})}\cdots(x_{i-1}\ast x_{i})\rho_{X}(x_{i})x_{i+1}\cdots
x_{n}],[(x_{2}\ast x_{i})\cdots\cdots(x_{i-1}\ast
x_{i})\rho_{X}(x_{i})x_{i+1}\cdots x_{n}]}$ $\displaystyle(x_{2}\ast
x_{i},\ldots,x_{i-1}\ast
x_{i},\rho_{X}(x_{i}),x_{i+1},\ldots,x_{n})+\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{n})\Big{)}$

If $i>2$, then using the condition (A3), we get

$\displaystyle\phi_{\underbrace{[x_{1}\cdots\widehat{x_{i}}\cdots
x_{n}]}_{x\ast y},\underbrace{[x_{i}\cdots x_{n}]}_{z}}$
$\displaystyle\psi_{\underbrace{[(x_{1}\ast x_{i})\widehat{(x_{2}\ast
x_{i})}\cdots(x_{i-1}\ast x_{i})\rho_{X}(x_{i})x_{i+1}\cdots
x_{n}]}_{x},\underbrace{[(x_{2}\ast x_{i})\cdots\cdots(x_{i-1}\ast
x_{i})\rho_{X}(x_{i})x_{i+1}\cdots x_{n}]}_{y}}$ $\displaystyle=$
$\displaystyle\psi_{\underbrace{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}]}_{x\ast z},\underbrace{[x_{2}x_{3}\cdots x_{n}]}_{y\ast
z}}\phi_{\underbrace{[x_{2}\cdots\widehat{x_{i}}\cdots
x_{n}]}_{y},\underbrace{[x_{i}\cdots x_{n}]}_{z}}.$

Thus, in this case, 1G + 2G lies in $D_{n-1}^{SR}(X,\rho_{X})$. If $i=2$, then
the condition (A9) gives

$\displaystyle-\psi_{\underbrace{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}]}_{x\ast\rho_{X}(y)},\underbrace{[x_{2}x_{3}\cdots x_{n}]}_{y}}$
$\displaystyle=$
$\displaystyle\phi_{\underbrace{[x_{1},\widehat{x_{2}},x_{3},\cdots
x_{n}]}_{x\ast\rho_{X}(y)},\underbrace{[x_{2},x_{3},\cdots
x_{n}]}_{y}}\psi_{\underbrace{[x_{1}\cdots
x_{n}]}_{x},\underbrace{[\rho_{X}(x_{2})x_{3}\cdots
x_{n}]}_{\rho_{X}(y)}}\eta_{\underbrace{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}]}_{y}}.$

Hence, 1G + 2G lies in $D_{n-1}^{SR}(X,\rho_{X})$, and we have proved that
$\partial_{n}(V_{n})\subset D_{n-1}^{SR}(X,\rho_{X})$. ∎

######  Proposition 6.6.

Let $(X,\rho_{X})$ be a symmetric rack. Then
$\partial_{n}(D_{n}^{SR}(X,\rho_{X}))\subset D_{n-1}^{SR}(X,\rho_{X})$ for
each $n\geq 1$.

###### Proof.

In view of Lemma 6.4 and Lemma 6.5, it remains to verify that
$\partial_{1}(D_{1}^{SR}(X,\rho_{X}))={0}$. Note that,
$D_{1}^{SR}(X,\rho_{X})$ is generated by $\\{\eta_{x}(x)-\rho_{X}(x)\mid x\in
X\\}$. It follows from the condition (A6) that

$\partial_{1}(\eta_{x}(x)-\rho_{X}(x))=\eta_{x}\partial_{1}(x)-\partial_{1}\rho_{X}(x)=-\eta_{x}\psi_{x\ast^{-1}p,p}(p)+\psi_{\rho_{X}(x)\ast^{-1}p,p}(p)=0$

for all $x\in X$. Hence, $\partial_{n}(D_{n}^{SR}(X,\rho))\subset
D_{n-1}^{SR}(X,\rho)$ for all $n\geq 0$. ∎

Lemma 6.4, Lemma 6.5 and Proposition 6.6 lead to a (co)homology theory for
symmetric racks. Let $(X,\rho_{X})$ be a symmetric rack and $A$ be a right
$\mathbb{Z}(X,\rho_{X})$-module. Define

$C_{n}\big{(}(X,\rho_{X}),A\big{)}:=A\otimes_{\mathbb{Z}(X,\rho_{X})}C_{n}(X,\rho_{X})$

and

$C_{n}^{SR}\big{(}(X,\rho_{X}),A\big{)}:=\big{(}A\otimes_{\mathbb{Z}(X,\rho_{X})}C_{n}(X,\rho_{X})\big{)}/\big{(}A\otimes_{\mathbb{Z}(X,\rho_{X})}D_{n}^{SR}(X,\rho_{X})\big{)}.$

Let $\partial_{n}:C_{n}^{SR}((X,\rho_{X}),A)\rightarrow
C_{n-1}^{SR}((X,\rho_{X}),A)$ be the induced $\mathbb{Z}(X,\rho_{X})$-linear
map. Then $\big{(}C_{*}^{SR}((X,\rho_{X}),A),\partial_{*}\big{)}$ is a chain
complex, yielding homology groups $H_{n}^{SR}((X,\rho_{X}),A)$. Let us set
$C_{n}^{SR}(X,\rho_{X})=C_{n}(X,\rho_{X})/D_{n}^{SR}(X,\rho_{X})$. If $A$ is a
left $\mathbb{Z}(X,\rho)$-module, then we define

$C^{n}_{SR}\big{(}(X,\rho_{X}),A\big{)}:=\operatorname{Hom}_{\mathbb{Z}(X,\rho_{X})}\big{(}C_{n}^{SR}(X,\rho_{X}),A\big{)}$

and define $\delta^{n}:C^{n}_{SR}((X,\rho_{X}),A)\to
C^{n+1}_{SR}((X,\rho_{X}),A)$ to be the induced coboundary map. Then,
$\big{(}C^{*}_{SR}((X,\rho_{X}),A),\delta^{*}\big{)}$ is a cochain complex,
leading to cohomology groups $H^{n}_{SR}((X,\rho_{X}),A)$.

######  Remark 6.7.

In view of Remark 6.3, for a homogeneous $(X,\rho_{X})$-module
$\mathscr{F}=(A,\phi,\psi,\eta)$, the cohomology group
$\mathcal{H}^{2}_{\mathrm{SR}}(X,\mathscr{F})$ defined in Theorem 5.8
coincides with the cohomology group $H^{2}_{SR}((X,\rho_{X}),A)$ defined
above.

We can modify the preceding construction for quandles as well. Let
$(X,\rho_{X})$ be a symmetric quandle. Define $D_{n}^{SQ}(X,\rho_{X})$ to be
the submodule of $C_{n}(X,\rho_{X})$ generated by the $U_{n}\cup V_{n}\cup
W_{n}$, where

$W_{n}:=\bigcup_{i=1}^{n-1}\big{\\{}(x_{1},\ldots,x_{n})\in X^{n}\mid
x_{i}=x_{i+1}\big{\\}}.$

######  Proposition 6.8.

Let $(X,\rho_{X})$ be a symmetric quandle. Then
$\partial_{n}(D^{SQ}_{n}(X,\rho_{X}))\subset D^{SQ}_{n-1}(X,\rho_{X})$ for
each $n\geq 2$.

###### Proof.

It suffices to prove that
$\partial_{n}(x_{1},\ldots,x_{i-1},x_{i},x_{i},x_{i+2},\ldots,x_{n})\in
D^{Q}_{n-1}(X,\rho_{X})$ for each $1\leq i\leq n-1$. For $i=1$, we have

$\displaystyle\partial_{n}\big{(}(x_{1},x_{1},x_{3},\ldots,x_{n})\big{)}$
$\displaystyle=$ $\displaystyle(-1)^{n}\phi_{[x_{1}x_{3}\cdots
x_{n}],[x_{1}x_{3}\cdots x_{n}]}(x_{1},x_{3},\ldots,x_{n})$ $\displaystyle+$
$\displaystyle\sum_{j=3}^{n}(-1)^{j}(-1)^{n}\phi_{[x_{1}x_{1}x_{3}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},x_{1},x_{3},\ldots,\widehat{x_{j}},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle(-1)^{n}(x_{1}\ast x_{1},x_{3},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle\sum_{j=3}^{n}(-1)^{j}(-1)^{n}(x_{1}\ast
x_{j},x_{1}\ast x_{j},x_{3}\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{n})$ $\displaystyle+$
$\displaystyle(-1)^{n}\psi_{[x_{1}\widehat{x_{1}}x_{3}\cdots
x_{n}],[x_{1}x_{3}\cdots x_{n}]}(x_{1},x_{3},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle\sum_{j=3}^{n}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}x_{1}x_{3}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},x_{1},x_{3},\ldots,\widehat{x_{j}},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{j},x_{1}\ast x_{j},x_{3}\ast
x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})\Big{)}\in
D^{Q}_{n-1}(X,\rho_{X}),$

where the last equality follows due to

$\displaystyle\phi_{[x_{1}x_{3}\cdots x_{n}],[x_{1}x_{3}\cdots
x_{n}]}(x_{1},x_{3},\ldots,x_{n})+\psi_{[x_{1}\widehat{x_{1}}x_{3}\cdots
x_{n}],[x_{1}x_{3}\cdots x_{n}]}(x_{1},x_{3}\ldots,x_{n})$ $\displaystyle=$
$\displaystyle(\phi_{[x_{1}x_{3}\cdots x_{n}],[x_{1}x_{3}\cdots
x_{n}]}+\psi_{[x_{1}\widehat{x_{1}}x_{3}\cdots x_{n}],[x_{1}x_{3}\cdots
x_{n}]})(x_{1},x_{3},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle(x_{1},x_{3},\ldots,x_{n}),$

which itself is a consequence of the condition (A10).

For $i\geq 2$, we have

$\displaystyle\partial_{n}\big{(}(x_{1},\ldots,x_{i-1},x_{i},x_{i},x_{i+2},\ldots,x_{n})\big{)}$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{i},x_{i},\ldots x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{i},x_{i},\ldots x_{n})\Big{)}$ $\displaystyle+$
$\displaystyle(-1)^{i}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots
x_{i-1}\widehat{x_{i}}x_{i}x_{i+2}\cdots x_{n}],[x_{i}x_{i}x_{i+2}\cdots
x_{n}]}(x_{1},\ldots,x_{i-1},\widehat{x_{i}},x_{i},x_{i+2},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},x_{i},x_{i+2},\ldots,x_{n})\Big{)}$ $\displaystyle+$
$\displaystyle(-1)^{i+1}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots
x_{i-1}x_{i}\widehat{x_{i}}x_{i+2}\cdots x_{n}],[x_{i}x_{i+2}\cdots
x_{n}]}(x_{1},\ldots,x_{i-1},x_{i},\widehat{x_{i}},x_{i+2},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{i},\ldots,x_{i-1}\ast
x_{i},x_{i}\ast x_{i},x_{i+2},\ldots,x_{n})\Big{)}$ $\displaystyle+$
$\displaystyle\sum_{j=i+2}^{n}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,x_{i},x_{i},\ldots\widehat{x_{j}},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{j},\ldots x_{i}\ast
x_{j},x_{i}\ast x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})\Big{)}$
$\displaystyle+$ $\displaystyle(-1)^{n}\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{i},x_{i},\ldots,x_{n})$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{i-1}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,\widehat{x_{j}},\ldots,x_{i},x_{i},\ldots x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{j},\ldots,x_{j-1}\ast
x_{j},x_{j+1},\ldots,x_{i},x_{i},\ldots x_{n})\Big{)}$ $\displaystyle+$
$\displaystyle\sum_{j=i+2}^{n}(-1)^{j}(-1)^{n}\Big{(}\phi_{[x_{1}\cdots\widehat{x_{j}}\cdots
x_{n}],[x_{j}\cdots
x_{n}]}(x_{1},\ldots,x_{i},x_{i},\ldots\widehat{x_{j}},\ldots,x_{n})$
$\displaystyle-$ $\displaystyle(x_{1}\ast x_{j},\ldots x_{i}\ast
x_{j},x_{i}\ast x_{j},\ldots,x_{j-1}\ast x_{j},x_{j+1},\ldots,x_{n})\Big{)}$
$\displaystyle+$ $\displaystyle(-1)^{n}\psi_{[x_{1}\widehat{x_{2}}x_{3}\cdots
x_{n}],[x_{2}x_{3}\cdots x_{n}]}(x_{2},\ldots,x_{i},x_{i},\ldots,x_{n})\in
D^{Q}_{n-1}(X,\rho_{X}),$

where the $i$-th and the $(i+1)$-th terms cancel out due to the equality

$\phi_{[x_{1}\cdots x_{i-1}\widehat{x_{i}}x_{i}x_{i+2}\cdots
x_{n}],[x_{i}x_{i}x_{i+2}\cdots x_{n}]}=\phi_{[x_{1}\cdots
x_{i-1}x_{i}\widehat{x_{i}}x_{i+2}\cdots x_{n}],[x_{i}x_{i+2}\cdots x_{n}]}.$

Hence, $\partial_{n}(D^{SQ}_{n}(X,\rho_{X}))\subset D^{SQ}_{n-1}(X,\rho_{X})$,
and the proof is complete. ∎

We can now define the homology and cohomology of symmetric quandles. Let
$(X,\rho_{X})$ be a symmetric quandle and $M$ be a right
$\mathbb{Z}(X,\rho_{X})$-module. We define

$C_{n}^{SQ}\big{(}(X,\rho_{X}),M\big{)}:=\big{(}M\otimes_{\mathbb{Z}(X,\rho_{X})}C_{n}(X,\rho_{X})\big{)}/\big{(}M\otimes_{\mathbb{Z}(X,\rho_{X})}D_{n}^{SQ}(X,\rho_{X})\big{)}$

and the boundary map to be the same as in the case of symmetric racks. Then
$\big{(}C_{*}^{SQ}((X,\rho_{X}),M),\partial_{*}\big{)}$ is a chain complex
leading to homology groups $H_{n}^{SQ}((X,\rho_{X}),M)$. Similarly, if $M$ is
a left $\mathbb{Z}(X,\rho)$-module, then define

$C^{n}_{SQ}\big{(}(X,\rho_{X}),M\big{)}:=\operatorname{Hom}_{\mathbb{Z}(X,\rho_{X})}\big{(}C_{n}^{SQ}(X,\rho_{X}),M\big{)}$

and take $\delta^{n}$ to be the same as in the case of symmetric racks. Then
$\big{(}C^{*}_{SQ}((X,\rho_{X}),M),\delta^{*}\big{)}$ is a cochain complex
yielding cohomology group $H^{n}_{SQ}((X,\rho_{X}),M)$.

######  Remark 6.9.

We can equip the algebra $\mathbb{Z}(X,\rho_{X})$ with an augmentation
$\epsilon:\mathbb{Z}(X,\rho_{X})\rightarrow\mathbb{Z}$, defined as
$\epsilon(\phi_{x,y})=1,~{}\epsilon(\psi_{x,y})=0$ and
$\epsilon(\eta_{x})=-1$. Also, any abelian group $A$ can be considered as
trivial homogeneous $(X,\rho_{X})$-module by setting

$A_{x}=A,\quad\phi_{x,y}=\mathrm{id}_{A},\quad\psi_{x,y}=0\quad\textrm{and}\quad\eta_{x}=-\mathrm{id}_{A}$

for all $x,y\in X$. In view of Remark 6.3, our (co)homology is a module-
theoretic generalisation of the (co)homology defined in [16], and also
recovers it as a special case. Applications of this module-theoretic
cohomology of symmetric racks (respectively quandles) to knot theory will be
explored in a forthcoming work.

######  Remark 6.10.

By definition, a map $\kappa:C_{2}(X,\rho_{X})\rightarrow A$, where $A$ is a
$\mathbb{Z}(X,\rho_{X})$-module, is a symmetric rack 2-cocycle if and only if
$\kappa$ satisfies the following conditions:

  1. (1)

for any $(x_{1},x_{2},x_{3})\in X^{3},$ we have

$-\phi_{[x_{1}x_{3}],[x_{2}x_{3}]}\kappa(x_{1},x_{3})+\phi_{[x_{1}x_{2}],x_{3}}\kappa(x_{1},x_{2})+\kappa(x_{1}*x_{2},x_{3})-\kappa(x_{1}*x_{3},x_{2}*x_{3})-\psi_{[x_{1}x_{3}],[x_{2}x_{3}]}\kappa(x_{2},x_{3})=0.$

  2. (2)

for any $(x_{1},x_{2})\in X^{2},$ we have

$\eta_{[x_{1}x_{2}]}\kappa(x_{1},x_{2})-\kappa(\rho_{X}(x_{1}),x_{2})=0\text{
and
}\phi_{x_{1},x_{2}}\kappa(x_{1}*x_{2},\rho_{X}(x_{2}))+\kappa(x_{1},x_{2})=0$

Furthermore, $\kappa$ is a symmetric quandle 2-cocycle if it additionally
satisfies $\kappa(x,x)=0$ for all $x\in X.$

##  7\. Relation between cohomology of symmetric racks and cohomology of
associated groups

In [7, Section 5], an isomorphism between the second rack cohomology of a rack
and the first cohomology of its associated group has been established. In this
section, we establish an analogous relation between the second cohomology of a
symmetric rack and the first cohomology of its associated group.

######  Theorem 7.1.

Let $(X,\rho_{X})$ be a symmetric rack and $A$ be an abelian group viewed as a
trivial homogeneous $(X,\rho_{X})$-module. Let $A$ also be viewed as a
symmetric rack with the trivial rack operation and with $a\mapsto-a$ as the
good involution, and let $\operatorname{Hom}(X,A)$ be the abelian group of
symmetric rack homomorphisms with the point-wise addition. Let
$G_{(X,\rho_{X})}$ act on $\operatorname{Hom}(X,A)$ from the left by
$(e_{x}\cdot f)(z)=f(z*x)$. Then there is a group isomorphism

$H^{2}_{SR}\big{(}(X,\rho_{X}),A\big{)}\cong
H^{1}\big{(}G_{(X,\rho_{X})},\operatorname{Hom}(X,A)\big{)}.$

###### Proof.

For $x,y,z\in X$ and $f\in\operatorname{Hom}(X,A)$, we see that

$(e_{x}e_{y}\cdot f)(z)=(e_{y}\cdot
f)(z*x)=f((z*x)*y)=f((z*y)*(x*y))=(e_{x*y}\cdot f)(z*y)=(e_{y}e_{x*y}\cdot
f)(z)$

and

$(e_{x}^{-1}\cdot f)(z)=f(z*^{-1}x)=f(z*\rho_{X}(x))=(e_{\rho_{X}(x)}\cdot
f)(z).$

Thus, the left action of $G_{(X,\rho_{X})}$ on $\operatorname{Hom}(X,A)$ is
well-defined.

Let $f:G_{(X,\rho_{X})}\rightarrow\operatorname{Hom}(X,A)$ be a group
$1$-cocycle. Define $\theta_{f}:X\times X\rightarrow A$ by

$\theta_{f}((x,y))=f(e_{y})(x).$

We claim that $\theta_{f}$ is a symmetric rack 2-cocycle. For all $x,y,z\in
X$, we have

$\displaystyle\theta_{f}(x,z)+\theta_{f}(x*z,y*z)$ $\displaystyle=$
$\displaystyle f(e_{z})(x)+f(e_{y*z})(x*z)$ $\displaystyle=$ $\displaystyle
f(e_{z})(x)+(e_{z}\cdot f(e_{y*z}))(x)$ $\displaystyle=$ $\displaystyle
f(e_{z})(x)+(f(e_{z}e_{y*z})-f(e_{z}))(x),\quad\text{since $f$ is a group
1-cocycle}$ $\displaystyle=$ $\displaystyle f(e_{z}e_{y*z})(x)$
$\displaystyle=$ $\displaystyle f(e_{y}e_{z})(x)$ $\displaystyle=$
$\displaystyle(f(e_{y}e_{z})-f(e_{y}))(x)+f(e_{y})(x)$ $\displaystyle=$
$\displaystyle(e_{y}\cdot f(e_{z}))(x)+f(e_{y})(x),\quad\text{since $f$ is a
group 1-cocycle}$ $\displaystyle=$ $\displaystyle f(e_{z})(x*y)+f(e_{y})(x)$
$\displaystyle=$ $\displaystyle\theta_{f}(x*y,z)+\theta_{f}(x,y).$

Further, since $f(e_{y})$ is a symmetric rack homomorphism, we have

$\theta_{f}(x,y)+\theta_{f}(\rho_{X}(x),y)=f(e_{y})(x)+f(e_{y})(\rho_{X}(x))=f(e_{y})(x)-f(e_{y})(x)=0.$

Since $f$ is a group 1-cocycle, we get

$\displaystyle\theta_{f}(x,y)+\theta_{f}(x*y,\rho_{X}(y))$ $\displaystyle=$
$\displaystyle f(e_{y})(x)+f(e_{\rho_{X}(y)})(x\ast y)$ $\displaystyle=$
$\displaystyle f(e_{y})(x)+(e_{y}\cdot f(e_{\rho_{X}(y)}))(x)$
$\displaystyle=$ $\displaystyle
f(e_{y})(x)+(f(e_{y}e_{\rho_{X}(y)})-f(e_{y}))(x)$ $\displaystyle=$
$\displaystyle 0,\quad\text{since $f(1)=0$},$

and hence $\theta_{f}$ is a symmetric rack 2-cocycle. This gives a map
$Z^{1}\big{(}G_{(X,\rho_{X})},\operatorname{Hom}(X,A)\big{)}\rightarrow
H^{2}_{SR}\big{(}(X,\rho_{X}),A\big{)}$ defined by $f\mapsto[\theta_{f}]$.
Since the group structure in both the domain and the codomain is the point-
wise addition of maps, it follows that $f\mapsto[\theta_{f}]$ is a group
homomorphism. If $f\in B^{1}(G_{(X,\rho_{X})},\operatorname{Hom}(X,A)),$ then
for all $a\in G_{(X,\rho_{X})}$ and $x\in X$, we have

$f(a)(x)=(a\cdot\phi-\phi)(x)$

for some $\phi\in\operatorname{Hom}(X,A)$. Thus, for all $x,y\in X$, we have

$\theta_{f}(x,y)=f(e_{y})(x)=(e_{y}\cdot\phi-\phi)(x)=(e_{y}\cdot\phi)(x)-\phi(x)=\phi(x*y)-\phi(x).$

This shows that $\theta_{f}\in B^{2}((X,\rho_{X});A),$ and hence we obtain a
map

$\Psi:H^{1}\big{(}G_{(X,\rho_{X})},\operatorname{Hom}(X,A)\big{)}\rightarrow
H^{2}_{SR}\big{(}(X,\rho_{X}),A\big{)}.$

For the reverse direction, let $\sigma\in Z_{SR}^{2}((X,\rho_{X}),A)$. Then
$\sigma$ satisfies the symmetric rack 2-cocycle conditions

$\sigma(x,z)+\sigma(x*z,y*z)=\sigma(x,y)+\sigma(x*y,z),$ (7.0.1)
$\sigma(x,y)+\sigma(\rho_{X}(x),y)=0,$ (7.0.2)
$\sigma(x,y)+\sigma(x*y,\rho_{X}(y))=0,$ (7.0.3)

for all $x,y,z\in X.$ Let us define
$\sigma^{\prime}:X\rightarrow\operatorname{Hom}(X,A)$ by

$\sigma^{\prime}(x)(y)=\sigma(y,x)$

for all $x,y\in X$. Since
$\sigma^{\prime}(x)(\rho_{X}(y))=\sigma(\rho_{X}(y),x)=-\sigma(x,y)$ by the
cocycle condition (7.0.2), and hence
$\sigma^{\prime}(x)\in\operatorname{Hom}(X,A)$. Identifying $X$ with the set
$\\{e_{x}\mid x\in X\\}$, we have a map $\tau_{\sigma}:X\rightarrow
G_{(X,\rho_{X})}\ltimes\operatorname{Hom}(X,A)$ defined by

$\tau_{\sigma}(e_{x})=(e_{x},\sigma^{\prime}(x))$

for all $x\in X$. Note that

$\tau_{\sigma}(e_{x})\tau_{\sigma}(e_{\rho_{X}(x)})=(e_{x},\sigma^{\prime}(x))(e_{\rho_{X}(x)},\sigma^{\prime}(\rho_{X}(x)))=\big{(}e_{x}e_{\rho_{X}(x)},\sigma^{\prime}(x)+e_{x}\cdot\sigma^{\prime}(\rho_{X}(x))\big{)}.$

Further, the symmetric rack 2-cocycle condition (7.0.3) gives

$\displaystyle\big{(}\sigma^{\prime}(x)+e_{x}\cdot\sigma^{\prime}(\rho_{X}(x))\big{)}(y)$
$\displaystyle=$
$\displaystyle\sigma^{\prime}(x)(y)+(e_{x}\cdot\sigma^{\prime}(\rho_{X}(x)))(y)$
$\displaystyle=$
$\displaystyle\sigma^{\prime}(x)(y)+\sigma^{\prime}(\rho_{X}(x))(y*x)$
$\displaystyle=$ $\displaystyle\sigma(y,x)+\sigma(y*x,\rho_{X}(x))$
$\displaystyle=$ $\displaystyle 0.$

Thus, $\sigma^{\prime}(x)+e_{x}\cdot\sigma^{\prime}(\rho_{X}(x))$ is the
trivial map, and consequently
$\tau_{\sigma}(e_{x})\tau_{\sigma}(e_{\rho_{X}(x)})=(e,0),$ where $e$ is the
identity of $G_{(X,\rho_{X})}.$ Similarly, we can prove that
$\tau_{\sigma}(e_{\rho_{X}(x)})\tau_{\sigma}(e_{x})=(e,0)$. Next, we consider

$\displaystyle\tau_{\sigma}(e_{y})\,\tau_{\sigma}(e_{x*y})\,\tau_{\sigma}(e_{y})^{-1}\,\tau_{\sigma}(e_{x})^{-1}$
$\displaystyle=$
$\displaystyle(e_{y},\sigma^{\prime}(y))\,(e_{x*y},\sigma^{\prime}({x*y}))\,(e_{y},\sigma^{\prime}(y))^{-1}\,(e_{x},\sigma^{\prime}(x))^{-1}$
$\displaystyle=$
$\displaystyle(e_{y},\sigma^{\prime}(y))\,(e_{x*y},\sigma^{\prime}({x*y}))\,(e_{y}^{-1},e_{y}^{-1}\cdot(-\sigma^{\prime}(y)))\,(e_{x}^{-1},e_{x}^{-1}\cdot(-\sigma^{\prime}(x)))$
$\displaystyle=$
$\displaystyle\big{(}e_{y}e_{x*y},\sigma^{\prime}(y)+e_{y}\cdot\sigma^{\prime}({x*y})\big{)}\,\big{(}e_{y}^{-1}e_{x}^{-1},e_{y}^{-1}\cdot(-\sigma^{\prime}(y))+e_{y}^{-1}e_{x}^{-1}\cdot(-\sigma^{\prime}(x))\big{)}$
$\displaystyle=$
$\displaystyle(e,\sigma^{\prime}(y)+e_{y}\cdot\sigma^{\prime}({x*y}))+(e_{y}e_{x*y})\cdot(e_{y}^{-1}\cdot(-\sigma^{\prime}(y))+e_{y}^{-1}e_{x}^{-1}\cdot(-\sigma^{\prime}(x))).$

The symmetric rack 2-cocycle condition (7.0.1) gives

$\displaystyle(\sigma^{\prime}(y)+e_{y}\cdot\sigma^{\prime}({x*y}))+(e_{y}e_{x*y})\cdot(e_{y}^{-1}\cdot(-\sigma^{\prime}(y))+e_{y}^{-1}e_{x}^{-1}\cdot(-\sigma^{\prime}(x)))(z)$
$\displaystyle=$
$\displaystyle\sigma(z,y)+\sigma(z*y,x*y)+(e_{y}e_{x*y})\cdot(e_{y}^{-1}\cdot(-\sigma^{\prime}(y))+e_{y}^{-1}e_{x}^{-1}\cdot(-\sigma^{\prime}(x)))(z)$
$\displaystyle=$
$\displaystyle\sigma(z,y)+\sigma(z*y,x*y)+(e_{x}\cdot(-\sigma^{\prime}(y))+(-\sigma^{\prime}(x)))(z)$
$\displaystyle=$
$\displaystyle\sigma(z,y)+\sigma(z*y,x*y)+(-\sigma^{\prime}(y))(z*x)-\sigma^{\prime}(x)(z)$
$\displaystyle=$
$\displaystyle\sigma(z,y)+\sigma(z*y,x*y)-\sigma(z*x,y)-\sigma(z,x)$
$\displaystyle=$ $\displaystyle 0.$

It follows that
$\tau_{\sigma}(e_{y})\tau_{\sigma}(e_{x*y})\tau_{\sigma}(e_{y})^{-1}\tau_{\sigma}(e_{x})^{-1}=(e,0)$
for all $x,y\in X$, and hence we can extend $\tau_{\sigma}$ to a group
homomorphism $\tau_{\sigma}:G_{(X,\rho_{X})}\rightarrow
G_{(X,\rho_{X})}\ltimes\operatorname{Hom}(X,A)$.

Let us define $\chi_{\sigma}:G_{(X,\rho_{X})}\to\operatorname{Hom}(X,A)$ to be
the composition

$\chi_{\sigma}=p_{2}\,\tau_{\sigma}$

where
$p_{2}:(G_{(X.\rho_{X})}\ltimes\operatorname{Hom}(X,A))\rightarrow\operatorname{Hom}(X,A)$
is the projection onto the second factor. Since $\tau_{\sigma}$ is a group
homomorphism, it follows that $\chi_{\sigma}$ is a group 1-cocycle. This gives
a map $Z^{2}_{SR}\big{(}(X,\rho_{X}),A\big{)}\rightarrow
H^{1}\big{(}G_{(X,\rho_{X})},\operatorname{Hom}(X,A)\big{)}$ defined by
$\sigma\mapsto[\chi_{\sigma}]$. If $\pi\in B^{2}_{SR}((X,\rho_{X}),A)$, then
$\pi=\delta^{1}(\tilde{\pi})$ for some $\tilde{\pi}\in
C^{1}_{SR}((X,\rho_{X}),A)$, that is,

$\pi(x,y)=\tilde{\pi}(x)-\tilde{\pi}(x*y)$

for all $x,y\in X$. If $g\in G_{(X,\rho_{X})}$, then
$g=e_{x_{1}}^{\epsilon_{1}}e_{x_{2}}^{\epsilon_{2}}\cdots
e_{x_{n}}^{\epsilon_{n}}$ for some $x_{i}\in X$ and
$\epsilon_{i}\in\\{1,-1\\}$. Thus, for all $y\in X$, we have

$\displaystyle\chi_{\pi}(g)(y)$ $\displaystyle=$ $\displaystyle
p_{2}(\tau_{\pi}(g))(y)$ $\displaystyle=$ $\displaystyle
p_{2}(\tau_{\pi}(e_{x_{1}}^{\epsilon_{1}}e_{x_{2}}^{\epsilon_{2}}\cdots
e_{x_{n}}^{\epsilon_{n}}))(y)$ $\displaystyle=$ $\displaystyle
p_{2}(\tau_{\pi}(e_{x_{1}})^{\epsilon_{1}}\tau_{\pi}(e_{x_{2}})^{\epsilon_{2}}\cdots\tau_{\pi}(e_{x_{n}})^{\epsilon_{n}})(y)$
$\displaystyle=$ $\displaystyle
p_{2}\big{(}(e_{x_{1}},\pi^{\prime}(x_{1}))^{\epsilon_{1}}(e_{x_{2}},\pi^{\prime}(x_{2}))^{\epsilon_{2}}\cdots(e_{x_{n}},\pi^{\prime}(x_{n}))^{\epsilon_{n}}\big{)}(y)$
$\displaystyle=$ $\displaystyle
p_{2}((e_{x_{1}}^{\epsilon_{1}}e_{x_{2}}^{\epsilon_{2}}\cdots
e_{x_{n}}^{\epsilon_{n}},\,t(x_{1})+e_{x_{1}}^{\epsilon_{1}}\cdot
t(x_{2}))+\cdots+e_{x_{1}}^{\epsilon_{1}}e_{x_{2}}^{\epsilon_{2}}\cdots
e_{x_{n}}^{\epsilon_{n}}\cdot t(x_{n})))(y)$ $\displaystyle=$
$\displaystyle\big{(}t(x_{1})+e_{x_{1}}^{\epsilon_{1}}\cdot
t(x_{2})+\cdots+e_{x_{1}}^{\epsilon_{1}}e_{x_{2}}^{\epsilon_{2}}\cdots
e_{x_{n-1}}^{\epsilon_{n-1}}\cdot t(x_{n})\big{)}(y),$

where $t(x_{i})=\pi^{\prime}(x_{i})$ for $\epsilon_{i}=1$ and
$t(x_{i})=e_{x_{i}}^{-1}\cdot(-\pi^{\prime}(x_{i}))$ for $\epsilon_{i}=-1$.
For instance, if all $\epsilon_{i}=1$, then

$\displaystyle\chi_{\pi}(g)(y)$ $\displaystyle=$
$\displaystyle\pi(y,x_{1})+\pi(y*x_{1},x_{2})+\cdots+\pi(((y*x_{1})*x_{2})\cdots*x_{n-1},x_{n})$
$\displaystyle=$
$\displaystyle\tilde{\pi}(y)-\tilde{\pi}(y*x_{1})+\tilde{\pi}(y*x_{1})-\tilde{\pi}((y*x_{1}*)x_{2})+\cdots+$
$\displaystyle\tilde{\pi}(((y*x_{1})*x_{2})\cdots*x_{n-1})-\tilde{\pi}(((y*x_{1})*x_{2})\cdots
x_{n})$ $\displaystyle=$
$\displaystyle\tilde{\pi}(y)-\tilde{\pi}((y*x_{1})*x_{2})\cdots x_{n})$
$\displaystyle=$
$\displaystyle\tilde{\pi}(y)-(e_{x_{1}}^{\epsilon_{1}}e_{x_{2}}^{\epsilon_{2}}\cdots
e_{x_{n}}^{\epsilon_{n}})\cdot\tilde{\pi}(y)$ $\displaystyle=$
$\displaystyle\tilde{\pi}(y)-g\cdot\tilde{\pi}(y)=(\tilde{\pi}-g\cdot\tilde{\pi})(y).$

The cases when some $\epsilon_{i}=-1$ can be dealt with in a similar way
through a routine computation. Thus,
$\chi_{\pi}(g)=\tilde{\pi}-g\cdot\tilde{\pi}$, which shows that $\chi_{\pi}\in
B^{1}(G_{(X,\rho_{X})},\operatorname{Hom}(X,A))$. Hence, we obtain a map

$\Psi:H^{2}_{SR}\big{(}(X,\rho_{X}),A\big{)}\rightarrow
H^{1}\big{(}G_{(X,\rho_{X})},\operatorname{Hom}(X,A)\big{)}$

and it is easy to see that $\Psi$ and $\Phi$ are inverses to each other, and
hence $H^{2}_{SR}\big{(}(X,\rho_{X}),A\big{)}\cong
H^{1}\big{(}G_{(X,\rho_{X})},\operatorname{Hom}(X,A)\big{)}$, which is
desired. ∎

######  Acknowledgement.

BK thanks CSIR for the PhD research fellowship. DS thanks IISER Mohali for the
PhD research fellowship. MS is supported by the SwarnaJayanti Fellowship
grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04.

##  8\. Declaration

The authors declare that there is no data associated to this paper and that
there are no conflicts of interests.

## References

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