EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 4, 2014, 412-418
ISSN 1307-5543 – www.ejpam.com

Some Relations Between Crossed Modules and Simplicial
Objects in Categories of Interest

Yaşar Boyacı1,∗, Osman Avcıoğlu 2

1 Dumlupınar University, Faculty of Education, Kütahya, Turkey
2 Uşak University, Faculty of Arts and Sciences, Uşak, Turkey

Abstract. We introduce a simplicial object in a category of interest and determine relations between
crossed modules and simplicial objects in a category of interest.

2010 Mathematics Subject Classifications: 18B99, 18G30, 18G50,18G55

Key Words and Phrases: Category of interest, Simplicial object, Crossed module

1. Introduction

Categories of interest were introduced in order to study properties of different algebraic
categories and different algebras simultaneously. Roughly speaking, category of interest can be
seen as a gadget which unifies many algebraic constructions. The idea comes from P.G. Higgins
[10] and the definition is due to M. Barr and G. Orzech [11]. The categories of groups, modules
over a ring, vector spaces, associative algebras, associative commutative algebras, Lie algebras
and Leibniz algebras are categories of interest [11]. The categories of crossed modules and
precrossed modules in the category of groups, respectively, are equivalent to the categories of
interests (see e.g. [3, 4]).

The functorial relation between crossed modules and simplicial objects with Moore com-
plex of length 1 in groups, commutative algebras, Lie algebras, Leibniz n-algebras were given
in [1, 2, 5, 8, 9]. In this paper, we will define simplicial objects in categories of interest and
unify the stated results under the name of categories of interest.

2. Category of Interest

We will have the main definitions and the statements given for category of interest in
[4, 7, 11].

∗Corresponding author.

Email addresses: yasar.boyaci@dpu.edu.tr (Y. Boyacı), osman.avcioglu@usak.edu.tr (O. Avcıoğlu)

http://www.ejpam.com 412 c© 2014 EJPAM All rights reserved.



Y. Boyacı, O. Avcıoğlu / Eur. J. Pure Appl. Math, 7 (2014), 412-418 413

Let C be a category of groups with a set of operations Ω and with a set of identities E,
such that E includes the group laws and the following conditions hold. If Ωi is the set of i-ary
operations in Ω, then:

(a) Ω = Ω0 ∪Ω1 ∪Ω2;

(b) the group operations (written additively : 0,−,+) are elements of Ω0, Ω1 and Ω2 respec-
tively. Let Ω′2 = Ω2 \ {+}, Ω′1 = Ω1 \ {−}. Assume that if ∗ ∈ Ω2, then Ω′2 contains ∗◦

defined by x ∗◦ y = y ∗ x and assume Ω0 = {0};

(c) for each ∗ ∈ Ω′2, E includes the identity x ∗ (y + z) = x ∗ y + x ∗ z;

(d) for each ω ∈ Ω′1 and ∗ ∈ Ω′2, E includes the identities ω(x + y) = ω(x) + ω(y) and
ω(x ∗ y) =ω(x) ∗ y .

Let C be an object of C and x1, x2, x3 ∈ C:

Axiom 1: x1 + (x2 ∗ x3) = (x2 ∗ x3) + x1, for each ∗ ∈ Ω′2.

Axiom 2: For each ordered pair (∗,∗) ∈ Ω′2 ×Ω
′
2 there is a word W such that

(x1 ∗ x2)∗x3 =W (x1(x2 x3), x1(x3 x2), (x2 x3)x1,

(x3 x2)x1, x2(x1 x3), x2(x3 x1), (x1 x3)x2, (x3 x1)x2),

where each juxtaposition represents an operation in Ω′2.

Definition 1. A category of groups with operations satisfying Axiom 1 and Axiom 2 is called a
category of interest by Orzech [11].

Example 1. Some examples of categories of interest that are given in [4]: In the example of
groups Ω′2 = ∅. In the case of associative algebras with multiplication represented by ∗, we have
Ω′2 = {∗,∗

◦}. For Lie algebras Ω′2 = ([, ], [, ]
◦) (where [a, b]◦ = [b, a] = −[a, b]). For Leibniz

algebras Ω′2 = ([, ], [, ]
◦) (here [a, b]◦ = [b, a]).

Definition 2. Let C ∈ C. A subobject of C is called an ideal if it is the kernel of some morphism.

Theorem 1. Let A be a subobject of B in C. Then A is an ideal of B if and only if the following
conditions hold:

i) A is a normal subgroup of B;

ii) a ∗ b ∈ A, for all a ∈ A, b ∈ B and ∗ ∈ Ω′2.

Proof. Follows from Theorem 1.7 given in [11].

Definition 3. Let A, B ∈ C. An extension of B by A is a sequence

0 // A i // E
p // B // 0 (1)

in which p is surjective and i is the kernel of p. We say that an extension is split if there is a
morphism s : B −→ E such that ps = 1B.



Y. Boyacı, O. Avcıoğlu / Eur. J. Pure Appl. Math, 7 (2014), 412-418 414

Definition 4. For A, B ∈ C we will say that we have a set of actions of B on A, whenever there is
a map f∗ : A× B −→ A, for each ∗ ∈ Ω2.

Definition 5. A split extension of B by A induces an action of B on A corresponding to the opera-
tions in C. For a given split extension (1), we have

b · a =s(b) + a− s(b), (2)

b ∗ a =s(b) ∗ a, (3)

for all b ∈ B, a ∈ A and ∗ ∈ Ω2
′.

Actions defined by (2) and (3) are called derived actions of B on A. Given an action of B on
A, the semidirect product Ao B is a universal algebra whose underlying set is A× B and the
operations are defined by

ω(a, b) =(ω (a) ,ω (b)),

(a′, b′) + (a, b) =(a′ + b′ · a, b′ + b),

(a′, b′) ∗ (a, b) =(a′ ∗ a+ a′ ∗ b+ b′ ∗ a, b′ ∗ b),

for all a, a′ ∈ A, b, b′ ∈ B.

Definition 6. A precrossed module in C is a triple (C1, C0,∂ ), where C0, C1 ∈ C, the object C0
has a derived action on C1 or shortly C0 acts on C1 and ∂ : C1 −→ C0 is a morphism in C with
the conditions:

CM 1) ∂ (c0 · c1) = c0+ ∂ (c1)− c0, ∂ (c0 ∗ c1) = c0 ∗ ∂ (c1), for all c0 ∈ C0, c1 ∈ C1, and ∗ ∈ Ω2
′.

In addition, if ∂ : C1 −→ C0 satisfies the conditions

CM 2) ∂ (c1) · c′1 = c1 + c′1 − c1, ∂ (c1) ∗ c′1 = c1 ∗ c′1,

for all c1, c′1 ∈ C1, and ∗ ∈ Ω′2, then the triple (C1, C0,∂ ) is called a crossed module in C.

Definition 7. A morphism between two crossed modules (C1, C0,∂ ) −→ (C ′1, C ′0,∂ ′) is a pair of
morphisms (µ1,µ0) in C, µ0 : C0 −→ C ′0, µ1 : C1 −→ C ′1, such that

i) µ0∂ (c) = ∂ ′µ1(c),

ii) µ1(r · c) = µ0(r) ·µ1(c),

iii) µ1(r ∗ c) = µ0(r) ∗µ1(c),

for all r ∈ C0, c ∈ C1 and ∗ ∈ Ω2
′.

With this definition, we have a category whose objects are crossed modules and morphisms
are morphisms of crossed modules defined above.

The category of crossed modules will be denoted by Xmod(C).



Y. Boyacı, O. Avcıoğlu / Eur. J. Pure Appl. Math, 7 (2014), 412-418 415

3. Simplicial Objects in a Category of Interest

Let 4 be the category of finite ordinals. A simplicial object in a category of interest C is
a functor from the opposite category 4op to C. In other words, a simplicial object C in C is a
sequence

C= {C0, C1, . . . , Cn, . . .}

together with face and degeneracy maps

dn
i : Cn −→ Cn−1, 0≤ i ≤ n (n 6= 0)

sn
i : Cn −→ Cn+1, 0≤ i ≤ n

which are homomorphisms of objects in C satisfying the following simplicial identities;

did j = d j−1di for i < j

dis j =







s j−1di
id

s jdi−1

for i < j
for i = j or i = j + 1
for i > j + 1

sis j = s j+1si for i ≤ j

for 0≤ i ≤ n (Here the superscripts of maps are dropped for shortness).

3.1. The Moore Complex

The Moore complex NC of a simplicial object C in a category of interest C is the complex

NC : · · · −→ NCn
∂n−→ NCn−1

∂n−1−→ · · ·
∂2−→ NC1

∂1−→ NC0

where NC0 = C0, NCn =
n−1
⋂

i=0
Kerdi and ∂n is the restriction of dn to NCn.

We say that the Moore complex NC of a simplicial object C is of length k if NCn = 0,
for all n ≥ k + 1. Now define a category whose objects are simplicial objects with Moore
complex of length k and the morphisms are families of homomorphisms compatible with face
and degeneracy maps. We denote this category by Simp≤k(C).

3.2. Truncated Simplicial Objects

The following terminology is adapted from [6]. Details of the group case can be found in
[6]. For each k ≥ 0 we have a subcategory of 4, denoted by 4≤k obtained by the objects [ j]
of 4 with j ≤ k. A k-truncated simplicial object is a functor from 4op

≤k to C. Consequently, a
k-truncated simplicial object is a family of objects {C0, C1, . . . , Ck} and homomorphism
di : Cn −→ Cn−1, si : Cn −→ Cn+1, for each 0≤ i ≤ n which satisfy the simplicial identities. We
denote the category of k-truncated simplicial objects by TrkSimp(C). There is a truncation
functor t rk from the category Simp(C) to the category TrkSimp(C) given by restrictions. This



Y. Boyacı, O. Avcıoğlu / Eur. J. Pure Appl. Math, 7 (2014), 412-418 416

truncation functor has a left adjoint stk and a right adjoint costk called as k-skeleton and
k-coskeleton respectively. These adjoints can be pictured as follows;

TrkSimp(C)
t rk←−
−→
costk

Simp(C)
t rk−→
←−
stk

TrkSimp(C).

See [6] for details about the functors costk and stk.

Theorem 2. The category Xmod(C) of crossed modules is naturally equivalent to the category
Simp≤1(C) of simplicial objects with Moore complex of length 1.

Proof. Let C be a simplicial object with Moore complex of length 1. Take G = ker d0 and ∂
is the restriction of d1 to G. Define the actions of C0 on G by

c0 · g =s0(c0) + g − s0(c0),

c0 ∗ g =s0(c0) ∗ g,

for all c0 ∈ C0 and g ∈ G. By using this action ∂ : G −→ C0 is a crossed module. Indeed,

CM 1: Since d1s0 = id, we have

∂ (c0 · g) =∂ (s0(c0) + g − s0(c0))

=c0 + ∂ (g)− c0,

∂ (c0 ∗ g) =∂ (s0(c0) ∗ g)

=c0 ∗ ∂ (g),

for all c0 ∈ C0 and g ∈ G.

CM 2: Since s0d1 = d2s0, d2s1 = id, we have

∂ (g ′) ∗ g =s0d1(g
′) ∗ g

=(s0d1(g
′)− g ′ + g ′) ∗ g

=
�

s0d1(g
′)− g ′

�

∗ g + g ′ ∗ g

=
�

d2s0 g ′ − d2s1 g ′
�

∗
�

d2s1 g
�

+ g ′ ∗ g

=d2

��

s0 g ′ − s1 g ′
�

∗
�

s1 g
��

+ g ′ ∗ g

=g ′ ∗ g,

for all g, g ′ ∈ G.

By a similar way, we have
∂ (g ′) · g = g ′ + g − g ′

for all g, g ′ ∈ G.
So we obtain the functor

N1 : Simp≤1(C) −→ Xmod(C).



REFERENCES 417

Conversely, let ∂ : G −→ H be a crossed module. By using the action of H on G, we can form
the semi-direct product C1 := G oH = {(g, h) : h ∈ H, g ∈ G}. We have the homomorphisms

d0 :G oH −→ H

(g, h) 7−→ h

d1 :G oH −→ H

(g, h) 7−→ ∂ (g) + h

s0 :H −→ G oH

h 7−→ (0, h)

which satisfy the simplicial identities. Finally

C1

d1,d0
�
s0

C0

is a 1-truncated simplicial object. Thus we have the functor

s1 : Xmod(C) −→ Tr1Simp(C).

By using the functor stk from the category of k-truncated simplicial objects to that of simplicial
objects with Moore complex of length 1, we have

M : Xmod(C) −→Simp≤1(C)

defined as the composition of s1 and st1. Finally we have the natural equivalence between the
category of simplicial objects with Moore complex of length 1 and that of crossed modules in
a category of interest C.

The main result of the paper can be diagramized as

Simp≤1(C)
N
�
M

Xmod(C).

References

[1] Z Arvasi and İ Akça. Simplicial and crossed lie algebras. Homology, Homotopy and
Applications, 4(1):43–57, 2002.

[2] Z Arvasi and T Porter. Higher dimensional peiffer elements in simplicial commutative
algebras. Theory and Applications of Categories, 3(1):1–23, 1997.

[3] J M Casas, T Datuashvili, and M Ladra. Actor of a precrossed module. Communication
in Algebra, 37(12):4516–4541, 2009.



REFERENCES 418

[4] J M Casas, T Datuashvili, and M Ladra. Universal strict general actors and actors in
categories of interest. Applied Categorical Structures, 18(1):85–114, 2010.

[5] J M Casas, E Khmaladze, and M Ladra. Crossed modules for leibniz n- algebras. Forum
Math, 20:841–858, 2008.

[6] E B Curtis. Simplicial homotopy theory. Advances in Mathematics, 6(2):107–209, 1971.

[7] T Datuashvili. Cohomologically trivial internal categories in categories of groups with
operations. Applied Categorical Structures, 3(3):221–237, 1995.

[8] G J Ellis. Higher dimensional crossed modules of algebras. Journal of Pure and Applied
Algebra, 52(3):277–282, 1988.

[9] G J Ellis. Homotopical aspects of lie algebras. Journal of The Australian Mathematical
Society, 54(3):393–419, 1993.

[10] P J Higgins. Groups with multiple operators. Proceedings of The London Mathematical
Society, 6(3):366–416, 1956.

[11] G Orzech. Obstruction theory in algebraic categories 1 and 2. Journal of Pure and Applied
Algebra, 2(4):287–314 and 315–340, 1972.