EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6640
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Exactness of the Functors HomA (X,−), HomA (−, X),
HomComp(A )(X,−), HomComp(A )(−, X) and the

Homological Functors H̃n(X,−) and H̃n(−, X) in a
Balanced Abelian Category

Ablaye Diallo1,∗, Mohamed Ben Faraj Ben Maaouia1, Mamadou Sanghare2

1 Applied Mathematics, UFR des Sciences Appliquées et Technologie, Université Gaston
Berger, Saint-Louis, Senegal
2 Faculté des Sciences et Techniques, Université Cheikh Anta Diop (UCAD), Dakar, Senegal

Abstract. This article presents several results concerning the exactness of covariant and con-
travariant Hom functors and their derived functors in a balanced abelian category A . In particu-
lar:

(i) The functors HomA (X,−) and HomA (−, X) are left exact, and become exact if and only if
X is projective (resp. injective).

(ii) The functors HomComp(A )(X,−) and HomComp(A )(−, X) on the category of complexes Comp(A )
preserve this behavior.

(iii) The homological functors H̃n(X,−) and H̃n(−, X) are constructed for all n ∈ Z.
(iv) For projective X, the connecting morphism λn : H̃n(X,−)((T, γ)) → H̃n+1(X,−)((Y, α))

allows H̃n(X,−) to send short exact sequences in Comp(A ) into long exact sequences in Ab.

(v) Similarly, for injective X, the morphism δn : H̃n(−, X)((Y, α)) → H̃n+1(−, X)((T, γ)) shows
that H̃n(−, X) also preserves long exact sequences.

2020 Mathematics Subject Classifications: 16E30, 20J05

Key Words and Phrases: Abelian category, balanced category, homological functors, projective
object, injective object, category of abelian groups

Introduction

The main objective of this article is to study the exactness of the functors HomA (X,−)
,HomA (−, X) : A −→ Ab,HomComp(A )(X,−),HomComp(A )(−, X) : Comp(A ) −→ Comp(Ab)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6640

Email addresses: diallo.ablaye@ugb.edu.sn (A. Diallo),
mohamed-ben.maaouia@ugb.edu.sn (M. B. F. B. Maaouia),
mamadou.sanghare@ucad.edu.sn (M. Sanghare)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 2 of 28

and the homological functors H̃n(X,−), H̃n(−, X) : Comp(A ) −→ Ab where A is a bal-
anced abelian category, Ab the category of abelian groups, and n an integer in Z. This
study is motivated by extending the fundamental concepts of homological algebra from
the category of modules to an arbitrary balanced abelian category. This generalization
is not straightforward, as evidenced by the proofs of the various results mentioned in the
abstract. Furthermore, this work is inspired by the exactness of the functors HomA(X,−),
HomA(−, X) and the homological functor Hn : Comp(A-Mod) −→ Ab in the category of
left A-modules A-Mod (resp. right A-modules Mod-A) and complexes of left A-modules
(Comp(A-Mod)) (resp. complexes of right A-modules Comp(Mod-A)): ”The functor and
its relationship with homological functor” [1], ”Localization, Isomorphisms and Adjoint
Isomorphism in the Category Comp(A- Mod”[2],”Localization of hopfian and cohopfian
objects in the categories of a-mod, agr (a-mod) and comp(agr(a-mod))” [3], ”Adjunction
and localization in the category a-alg of a-algebras” [4], ”Modules and rings” [5], ”Notes
on homological algebras” [6], ”Abelian categories” [7], ”Des catégories abéliennes” [8],
”An Introduction to Homological Algebra” [9],”An introduction to homological algebra”
[10] and other important results on abelian category and homological functors by the
autors: ElHadjOusseynou [11], Joseph .J Rotman [12], [13], Bassirou DEMBELE [14],
Ahmed OULD CHBIH [15], Charles A weibel [16], Ahmed OULD CHBIH et al [17],[18]
and Moussa Thiaw [19]. Thus, the article is structured as follows:
In Section 1 titled preliminary results, we provided the following definitions: abelian
category, balanced category, Comp(A ), exact sequence in Comp(A ). And we presented
some preliminary results.
In Section 2, titled the exactness of the functors HomA (X,−) and HomA (−, X), where
A is a balanced abelian category and X is an object in A . The following results have
been shown:

(i) Let A be a balanced abelian category and X an object in A . Then the covariant
functor denoted HomA (X,−) : A −→ Ab, defined by:

(a) ∀Y ∈ Ob(A ), HomA (X,−)(Y ) = HomA (X,Y ) ∈ Ob(Ab);

(b) ∀f ∈ HomA (Y,Z),

HomA (X,−)(f) = HomA (X, f) = f∗ : HomA (X,Y ) −→ HomA (X,Z)
ϕ 7−→ f ◦ ϕ

; is additive, left-exact functor and it is exact if and only if X is a projective object
in A .

(ii) Let A be a balanced abelian category and X an object in A . Then the contravariant
functor denoted HomA (−, X) : A −→ Ab, defined by:

(a) ∀Y ∈ Ob(A ), HomA (−, X)(Y ) = HomA (Y,X) ∈ Ob(Ab)

(b) ∀f ∈ HomA (Y,Z),

HomA (f,X) = f∗ : HomA (Z,X) −→ HomA (Y,X)
ϕ 7−→ ϕ ◦ f



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 3 of 28

is additive, left-exact functor and it is exact if and only if X is an injective object
in A .

In Section 3, titled the exactness of the functors HomComp(A )(X,−) and HomComp(A )(−, X)
where A is a balanced abelian category and X is an object in A . We proved the following
results:

(i) Let A be a balanced abelian category and X an object of A . Then:

(a) the functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab) is a covariant, addi-
tive, and left-exact functor;

(b) the functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab) is exact if and only
if X is a projective object in A .

(ii) Let A be a balanced abelian category and X an object of A . Then:

(a) the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is a contravariant,
additive, and left-exact functor;

(b) the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is exact if and only
if X is an injective object in A .

In Section 4, we studied the homological functors of degree n: H̃n(X,−), H̃n(−, X) :
Comp(A ) → Ab where A is a balanced abelian category and n is an integer in Z. We
proved the following results:

(i) The functor H̃n(X,−) is a covariant additive functor.

(ii) Let (0) // (Y, α)
f
// (Z, β)

g
// (T, γ) // (0) be a short exact sequence

of morphisms in Comp(A ), where X is a projective object in A and A is a balanced
abelian category. Then:

(a) We call the morphism of connection associated to the functor H̃n(X,−), de-
noted by λn, the morphism defined by:

λn : H̃n(X,−)((T, γ)) −→ H̃n+1(X,−)((Y, α))

kn+1 7−→ f∗−1
n+2(β

∗
n+1(g

∗−1
n+1(kn+1)))

∀n ∈ Z

(b) The sequence

· · · // H̃n(X,−)((Y, α))
H̃n(X,−)(f)

// H̃n(X,−)((Z, β))
H̃n(X,−)(g)

// H̃n(X,−)((T, γ))
λn //

H̃n+1(X,−)((Y, α))
H̃n+1(X,−)(f)

// H̃n+1(X,−)((Z, β))
H̃n+1(X,−)(g)

// H̃n+1(X,−)((T, γ))
λn+1

// · · ·
is a long exact sequence of abelian group morphisms. That is, for all n ∈ Z:

Im(H̃n(X,−)(f)) = Ker(H̃n(X,−)(g))

Im(H̃n(X,−)(g)) = Ker(λn)

Im(λn) = Ker(H̃n+1(X,−)(f))



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 4 of 28

(iii) The functor H̃n(−, X) is a contravariant additive functor.

(iv) Let (0) // (Y, α)
f
// (Z, β)

g
// (T, γ) // (0) be a short exact sequence

in Comp(A ), where A is a balanced abelian category and X is an injective object
in A . Then:

(a) We call the morphism of connection associated to the functor H̃n(−, X), de-
noted by λn, the morphism defined by:

δn :
H̃n(−, X)((Y, α)) −→ H̃n+1(−, X)((T, γ))

kn+1 7−→ g∗−1
n+2(β

∗
n+1(f

∗−1
n+1(kn+1)))

, ∀n ∈ Z

(b) The sequence

· · · // H̃n(−, X)((T, γ))
H̃n(−,X)(g)

// H̃n(−, X)((Z, β))
H̃n(−,X)(f)

// H̃n(−, X)((Y, α))
δn //

H̃n+1(−, X)((T, γ))
H̃n+1(−,X)(g)

// H̃n+1(−, X)((Z, β))
H̃n+1(−,X)(f)

// H̃n+1(−, X)((Y, α))
δn+1

// · · ·
is a long exact sequence in Ab. That is (for all n ∈ Z):

Im(H̃n(−, X)(g)) = Ker(H̃n(−, X)(f))

Im(H̃n(−, X)(f)) = Ker(δn)

Im(δn) = Ker(H̃n+1(−, X)(g)).

1. Preliminary Results

[Abelian Category]
An abelian category is a category A that satisfies the following conditions:

(i) the category A has a zero object;

(ii) for all objects X and Y in A , the set HomA (X,Y ) is endowed with an abelian group
structure whose composition law is denoted additively ;

(iii) the composition in A is bilinear with respect to the additions;

(iv) every finite family of objects in A has a coproduct;

(v) every morphism in A has a kernel and a cokernel;

(vi) every monomorphism in A is the kernel of its cokernel;

(vii) every epimorphism in A is the cokernel of its kernel.

[Balanced Category]
A category C is balanced if:



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 5 of 28

(i) every monomorphism of C is retractable;

(ii) and every epimorphism of C is splittable.

[Left Exact Sequence in A ]

Let (S): 0 → X
f−→ Y

g−→ Z be a short sequence of morphisms in A . Then (S) is called
left exact if:
g ◦ f = eX,Z , where eX,Z is the zero morphism of the abelian group HomA (X,Z)

N(f) = (0, e0,X) where e0,X : 0 → X is the zero morphism of the abelian group HomA (0, X)

coN(h) = (0, eK,0) where eK,0 : K → 0 is the zero morphism of the abelian group HomA (K, 0)

with h the morphism satisfying i ◦ h = f and Ker g = (K, i).

K>>
h

i
��

0 // X
f
// Y

g
// Z

[Right Exact Sequence in A ]

Let (S): X
f−→ Y

g−→ Z → 0 be a short sequence of morphisms in A . Then (S) is called
right exact if:
g ◦ f = eX,Z , where eX,Z is the zero morphism of the abelian group HomA (X,Z)

CoN (g) = (0, eZ,0) where eZ,0 : Z → 0 is the zero morphism of the abelian group HomA (Z, 0)

coN(h) = (0, eK,0) where eK,0 : K → 0 is the zero morphism of the abelian group HomA (K, 0)

with h the morphism satisfying i ◦ h = f and Ker g = (K, i).

K>>
h

i
��

X
f
// Y

g
// Z // 0

[Exact Sequence in A ]

Let (S): 0 → X
f−→ Y

g−→ Z → 0 be a short sequence of morphisms in A . Then (S) is
called exact if it is both left exact and right exact. That is:

g ◦ f = eX,Z , where eX,Z is the zero morphism of the abelian group HomA (X,Z)

N(f) = (0, e0,X) where e0,X : 0 → X is the zero morphism of the abelian group HomA (0, X)

coN (g) = (0, eZ,0) where eZ,0 : Z → 0 is the zero morphism of the abelian group HomA (Z, 0)

coN(h) = (0, eK,0) where eK,0 : K → 0 is the zero morphism of the abelian group HomA (K, 0)

with h the morphism satisfying i ◦ h = f and Ker g = (K, i).

K>>
h

i
��

0 // X
f
// Y

g
// Z // 0



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 6 of 28

Remark 1.
0 denotes the zero object of the category A .

[Comp(A )]
The category of complexes of an abelian category A , denoted Comp(A ), is defined by:

(i) The objects are complex sequence in A .
A complex sequence in A is a sequence of morphisms in A (αn : Xn → Xn+1)n∈Z,
denoted (X,α), such that αn+1 ◦αn = eXn,Xn+2 ∀n ∈ Z, where eXn,Xn+2 is the zero
morphism of the abelian group HomA (Xn, Xn+2).

(ii) The morphisms (arrows) are complex chains in A .
Let (X,α) = (αn : Xn → Xn+1)n∈Z and (Y, β) = (βn : Yn → Yn+1)n∈Z be two
complex sequences in A . A complex chain (fn : Xn −→ Yn)n∈Z, denoted f :
(X,α) → (Y, β), is a sequence of morphisms in A such that: fn+1 ◦ αn = βn ◦
fn ∀n ∈ Z.

[Right Exact Sequence in Comp(A )]

Let (S) : (Y, α)
f

// (Z, β)
g

// (T, γ) // (0) be a short sequence of mor-

phisms in Comp(A ) where A is an abelian category. Then we say that (S) is right exact

if for every integer n in Z the sequence Yn
fn

// Zn
gn

// Tn // 0 is
a right exact short sequence of morphisms in A . [Left Exact Sequence in Comp(A )]

Let (S) : (0) // (Y, α)
f

// (Z, β)
g

// (T, γ) be a short sequence of mor-

phisms in Comp(A ) where A is an abelian category. Then we say that (S) is left exact

if for every integer n in Z the sequence 0 // Yn
fn

// Zn
gn

// Tn
is a left exact short sequence of morphisms in A . [Exact Sequence in Comp(A )]

Let (S) : (0) // (Y, α)
f

// (Z, β)
g

// (T, γ) // (0) be a short sequence

of morphisms in Comp(A ) where A is an abelian category. Then we say that (S) is exact if

for every integer n in Z the sequence 0 // Yn
fn

// Zn
gn

// Tn // 0
is a short exact sequence of morphisms in A . Let A be an abelian category. Then
Comp(A ) is an abelian category.

Proof. See Page 319 [9].

Let A be a balanced abelian category. Then Comp(A ) is a balanced abelian category.
Proof.

Let A be a balanced abelian category, which means: every monomorphism in A is a
retraction and every epimorphism in A is a section.

• Let f : (X,α) → (Y, β) be a monomorphism in Comp(A ).
Since f is a monomorphism, it follows that for every n ∈ Z, fn : Xn −→ Yn is a
monomorphism in A . And since A is balanced, for every n ∈ Z, fn : Xn −→ Yn is
a retraction. Therefore, f is a retraction.



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 7 of 28

• Let f : (X,α) → (Y, β) be an epimorphism in Comp(A ). Since f is an epimorphism,
it follows that for every n ∈ Z, fn : Xn −→ Yn is an epimorphism in A . And since
A is balanced, for every n ∈ Z, fn : Xn −→ Yn is a section. Therefore, f is a section.
Hence, every monomorphism in Comp(A ) is a retraction and every epimorphism of
Comp(A ) is a section.

Thus, Comp(A ) is balanced. According to the proposition 1, Comp(A ) is balanced
abelian category.

2. Exactness of the Functors HomA (X,−) and HomA (−, X)

Let A be an abelian category, X,Y ∈ Ob(A ), and f : X → Y a morphism of A .
Then:

(i) The kernel of f , N(f) is zero if and only if f is a monomorphism.

(ii) The cokernel of f , coN(f) is zero if and only if f is an epimorphism.

Proof.

[label=)](⇒) Suppose that N(f) is zero and let us show that f is a monomorphism.
Let u, v : Z → X be two morphisms in A such that f ◦ u = f ◦ v.
We have:

f ◦ u = f ◦ v ⇒ f ◦ (u− v) = eZ,Y .

By the definition of the kernel of f , we have: N(f) = (K, i) implies that f ◦i = eK,Y .
If N(f) = (K, i) is zero, then i = eK,X and thus f ◦ i = eK,Y ⇒ f ◦ eK,X = eK,Y .
There exists a unique h : Z → K such that eK,X ◦ h = u− v. Hence,
∀i ∈ HomA (K,X), with eK,X being the neutral element of the abelian group
HomA (K,X), we have:

(i+ eK,X) ◦ h = (eK,X + i) ◦ h = i ◦ h⇒ i ◦ h+ eK,X ◦ h = eK,X ◦ h+ i ◦ h = i ◦ h

⇒

{
i ◦ h+ eK,X ◦ h = i ◦ h+ eZ,X = i ◦ h
eK,X ◦ h+ i ◦ h = eZ,X + i ◦ h = i ◦ h

⇒ eK,X ◦ h = eZ,X

Thus, we have:

eK,X ◦ h = u− v = eZ,X ⇒ u− v = eZ,X

⇒ u− v + v = eZ,X + v

⇒ u+ eZ,X = v

⇒ u = v.

Thus, f is a monomorphism.
(⇐) Suppose that f is a monomorphism and let us show that N(f) is zero.



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 8 of 28

Let i : K → X be a morphism such that f ◦ i = eK,X .
But eK,Y = f ◦ eK,X . Indeed: ∀i ∈ HomA (K,X), with eK,X being the neutral
element of the abelian group HomA (K,X), we have:

f ◦ (i+ eK,X) = f ◦ (eK,X + i) = f ◦ i⇒ f ◦ i+ f ◦ eK,X = f ◦ eK,X + f ◦ i = f ◦ i
⇒ f ◦ eK,X = eK,Y

Since f is a monomorphism, i = eK,X . Thus, the kernel of f , N(f), is zero. (⇒)
Suppose that coN(f) is zero and let us show that f is an epimorphism.
Let g, h : Y → Z be two morphisms such that g ◦ f = h ◦ f . We have:

g ◦ f = h ◦ f ⇒ (g − h) ◦ f = eX,Z .

Since coN(f) = (Y, p) implies p ◦ f = eX,T , and since coN(f) is zero, we have
p = eY,T , and thus:

k ◦ eY,T = g − h = eY,Z

Thus, we have:

k ◦ eY,T = g − h = eY,Z ⇒ g − h = eZ,X

⇒ g − h+ h = eY,Z + h

⇒ g + eY,Z = h

⇒ g = h.

Thus, f is an epimorphism.
(⇐) Suppose that f is an epimorphism and let us show that coN(f) is zero.
Let p : Y → T be a morphism such that p ◦ f = eX,T .
Now, eX,T = eX,T ◦ f. Indeed:
∀p ∈ HomA (Y, T ), where eY,T is the neutral element of the abelian group HomA (Y, T ),
we have:

(p+ eY,T ) ◦ f = (eY,T + p) ◦ f = p ◦ f ⇒ p ◦ f + eY,T ◦ f = eY,T ◦ f + p ◦ f = p ◦ f

⇒

{
p ◦ f + eY,T ◦ f = p ◦ f + eX,T = p ◦ f
eY,T ◦ f + p ◦ f = eX,T + p ◦ f = p ◦ f

⇒ eY,T ◦ f = eX,T

We have:
p ◦ f = eY,T ◦ f = eX,T .

Since f is an epimorphism, p = eY,T . Therefore, the cokernel of f , coN(f), is zero.

Let A be a balanced abelian category and X an object in A . Then the functor denoted
by HomA (X,−) : A −→ Ab defined by:



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 9 of 28

(i)(ii)(i) ∀Y ∈ Ob(A ), HomA (X,−)(Y ) = HomA (X,Y ) ∈ Ob(Ab);

(ii) ∀f ∈ HomA (Y,Z),

HomA (X,−)(f) = HomA (X, f) = f∗ : HomA (X,Y ) −→ HomA (X,Z)
ϕ 7−→ f ◦ ϕ

is covariant, additive, left exact functor, and it is exact if and only if X is a projective
object in A .

Proof.

• It’s evident that HomA (X,−) : A −→ Ab is an additive covariant functor.

• Let us show that HomA (X,−) : A −→ Ab is a left-exact functor.
Consider the following short left-exact sequence of morphisms in A :

0 // Y
f
// Z

g
// T

We will show that

{eX,0} // HomA (X,Y )
HomA (X,f)=f∗

// HomA (X,Z)
HomA (X,g)=g∗

// HomA (X,T )

is a left-exact sequence. It suffices to show that Ker(f∗) = {eX,Y }, where eX,Y is the
neutral element (the zero morphism) of HomA (X,Y ), and that Im(f∗) = Ker(g∗).
• Show that Ker(f∗) = {eX,Y }. Since f∗ is a morphism of abelian groups, the kernel
of f∗ is:

Ker(f∗) = {ϕ ∈ HomA (X,Y ) : f∗(ϕ) = f ◦ ϕ = eX,Z},

where eX,Z is the neutral element (the zero morphism) of the abelian group HomA (X,Z).
Let ϕ ∈ Ker(f∗).
We have:

ϕ ∈ Ker(f∗) ⇒ f∗(ϕ) = f ◦ ϕ = eX,Z .

Now, f ◦ eX,Y = eX,Z . Indeed, for all ϕ ∈ HomA (X,Y ), where eX,Y is the neutral
element of the abelian group HomA (X,Y ), we have:

f ◦ (ϕ+ eX,Y ) = f ◦ (eX,Y + ϕ) = f ◦ ϕ⇒ f ◦ ϕ+ f ◦ eX,Y = f ◦ eX,Y + f ◦ ϕ = f ◦ ϕ

⇒

{
f ◦ ϕ+ f ◦ eX,Y = f ◦ ϕ+ eX,Z = f ◦ ϕ
f ◦ eX,Y + f ◦ ϕ = eX,Z + f ◦ ϕ = f ◦ ϕ

⇒ f ◦ eX,Y = eX,Z .

Thus, we have:
f ◦ ϕ = f ◦ eX,Y = eX,Z .



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 10 of 28

Or by hypothesis, the kernel of f is zero N(f) = (0, eX,0). Bylemma 2, f is a
monomorphism. Hence,

f ◦ ϕ = f ◦ eX,Y ⇒ ϕ = eX,Y .

Thus, Ker(f∗) = {eX,Y }.

• Show that Im(f∗) = Ker(g∗).
- First, show that Im(f∗) ⊂ Ker(g∗). It suffices to show that g∗◦f∗ = eHomA (X,Y ),HomA (X,T ),
where eHomA (X,Y ),HomA (X,T ) is the zero morphism of the abelian group
HomAb(HomA (X,Y ),HomA (X,T )). We have:

g∗ ◦ f∗ : HomA (X,Y ) → HomA (X,T ).

Let ϕ ∈ HomA (X,Y ). We have:

g∗ ◦ f∗(ϕ) = g∗(f∗(ϕ)) = g∗(f ◦ ϕ) = g ◦ (f ◦ ϕ) = (g ◦ f) ◦ ϕ
= eY,T ◦ ϕ (since by hypothesis, g ◦ f = eY,T )

= eX,T because

for all ψ ∈ HomA (Y, T ), where eY,T is the neutral element of the abelian group
HomA (Y, T ), we have:

(ψ + eY,T ) ◦ ϕ = (eY,T + ψ) ◦ ϕ = ψ ◦ ϕ⇒ ψ ◦ ϕ+ eY,T ◦ ϕ = eY,T ◦ ϕ+ ψ ◦ ϕ = ψ ◦ ϕ

⇒

{
ψ ◦ ϕ+ eY,T ◦ ϕ = ψ ◦ ϕ+ eX,T = ψ ◦ ϕ
eY,T ◦ ϕ+ ψ ◦ ϕ = eX,T + ψ ◦ ϕ = ψ ◦ ϕ

⇒ eY,T ◦ ϕ = eX,T .

Thus, g∗ ◦ f∗ = eHomA (X,Y ),HomA (X,T ), and therefore Im(f∗) ⊂ Ker(g∗).
- Now, show that Ker(g∗) ⊂ Im(f∗).
We have:

Ker(g∗) = {ϕ ∈ HomA (X,Z) : g∗(ϕ) = g ◦ ϕ = eX,T }.

Let ψ ∈ Ker(g∗) = Ker(HomA (X, g)). Show that ψ ∈ Im(f∗). Consider the left-exact
short sequence of morphisms in A :

0 → Y
f−→ Z

g−→ T,

which implies:

– N(f) = (0, e0,Y ),

– g ◦ f = eY,T ,



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 11 of 28

– coN(h) = (0, eK,0), where h : Y → K such that i ◦h = f , with i : K → Z being
the kernel of g. That is, the following diagram commutes:

K99
h

i
��

0 // Y
f

// Z
g

// T

We have:
ψ ∈ Ker(g∗) ⇒ g∗(ψ) = eX,T

⇒ g ◦ ψ = eX,T .

Since coN(h) = (0, eK,0), by Lemma 2, h is an epimorphism. Since A is a balanced
category, every epimorphism is split. That is, there exists h′ : K → Y such that
h ◦ h′ = 1K . Thus, the following diagram commutes:

K99
h

i
��

oo
ψ′

X
ψ

yy
0 // Y99

h′

f
// Z

g
// T

K

We have:
i ◦ h = f ⇒ i ◦ h ◦ h′ = f ◦ h′

⇒ i ◦ 1K = f ◦ h′

⇒ i = f ◦ h′

⇒ i ◦ ψ′ = f ◦ h′ ◦ ψ′

⇒ ψ = f ◦ (h′ ◦ ψ′)

⇒ ψ = f∗(h′ ◦ ψ′).

Thus, ψ ∈ Im(f∗). Hence, Ker(g∗) ⊂ Im(f∗).

Therefore, HomA (X,−) : A −→ Ab is a covariant, additive, and left-exact functor.

• Show that HomA (X,−) : A −→ Ab is exact if and only if X is a projective object
in A .

• Suppose X is a projective object in A and show that HomA (X,−) : A −→ Ab is
an exact functor.
Consider the following short exact sequence of morphisms in A :

0 // Y
f
// Z

g
// T // 0



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 12 of 28

We will show that

{eX,0} // HomA (X,Y )
HomA (X,f)=f∗

// HomA (X,Z)
HomA (X,g)=g∗

// HomA (X,T ) // {eX,0}

is exact. By part (i), we know that

{eX,0} // HomA (X,Y )
HomA (X,f)=f∗

// HomA (X,Z)
HomA (X,g)=g∗

// HomA (X,T )

is left-exact. It remains to show that HomA (X, g) is an epimorphism.
Since X is projective, for every epimorphism g : Z ↠ T in A and every morphism
f : X → T in A , there exists a morphism ϕ : X → Z in A such that g ◦ ϕ = f .
This means the following diagram commutes:

X
ϕ

~~

f
��

Z g
// // T // 0

That is, for every f ∈ HomA (X,T ), there exists ϕ ∈ HomA (X,Z) such that
g ◦ ϕ = f = g∗(ϕ). Hence, HomA (X, g) = g∗ is an epimorphism.
• Conversely, suppose HomA (X,−) : A −→ Ab is an exact functor and show that
X is a projective object in A . We have:
The exactness of HomA (X,−) implies that for every short exact sequence of mor-
phisms in A :

0 // Y
f
// Z

g
// T // 0 ,

the sequence

{eX,0} // HomA (X,Y )
HomA (X,f)=f∗

// HomA (X,Z)
HomA (X,g)=g∗

// HomA (X,T ) // {eX,0}

is exact. This means g∗ is an epimorphism. Therefore, for every epimorphism
g : Z ↠ T in A and every morphism f : X → T in A , there exists a morphism
ϕ : X → Z in A such that g ◦ ϕ = g∗(ϕ) = f . This means the following diagram
commutes:

X
ϕ

~~

f
��

Z g
// // T // 0

Hence, X is a projective object in A .

Let A be a balanced abelian category and X an object in A . Then the functor denoted
by HomA (X,−) : A −→ Ab defined by:



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 13 of 28

(i) ∀Y ∈ Ob(A ), HomA (−, X)(Y ) = HomA (Y,X) ∈ Ob(Ab)

(ii) ∀f ∈ HomA (Y,Z),

HomA (−, X)(f) = HomA (f,X) = f∗ : HomA (X,Y ) −→ HomA (X,Z)
ϕ 7−→ ϕ ◦ f

is a contravariant, additive, left exact functor, and it is exact if and only if X is a injective
object in A .

Proof.

• It is evident that HomA (−, X) : A −→ Ab is a contravariant additive functor.

• Let us show that HomA (−, X) : A −→ Ab is a left-exact functor.
Consider the right short exact sequence of morphisms in A :

Y
f
// Z

g
// T // 0

We must show that the sequence:

{e0,X} // HomA (T,X)
HomA (g,X)=g∗

// HomA (Z,X)
HomA (f,X)=f∗

// HomA (Y,X)

is left-exact. It suffices to show that the kernel of g∗, Ker g∗, is zero and that
Im g∗ = Ker f∗.
• Let us show that the kernel of g∗, Ker(HomA (g,X)) = Ker g∗, is zero. By Lemma
2, it suffices to show that HomA (g,X) = g∗ is a monomorphism. By definition of
HomA (g,X) = g∗, we have:

HomA (g,X) = g∗ : HomA (T,X) −→ HomA (Z,X)
ϕ 7−→ ϕ ◦ g

Let ϕ1, ϕ2 ∈ HomA (T,X) such that g∗(ϕ1) = g∗(ϕ2), i.e., ϕ1 ◦ g = ϕ2 ◦ g. We have:

ϕ1 ◦ g = ϕ2 ◦ g =⇒ ϕ1 = ϕ2,

since by hypothesis, the cokernel of g, N(g), is zero, and by Lemma 2, g is an epi-
morphism. Thus, HomA (g,X) = g∗ is a monomorphism.
• Let us show that Im g∗ = Ker f∗.
- First, we show that Im g∗ ⊂ Ker f∗. It suffices to show that f∗◦g∗ = eHomA (T,X),HomA (Y,X),
where eHomA (T,X),HomA (Y,X) is the neutral element (zero morphism) of the abelian
group
HomAb(HomA (T,X),HomA (Y,X)). We have:

f∗ ◦ g∗ : HomA (T,X) → HomA (Y,X).



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 14 of 28

Let ϕ ∈ HomA (T,X). Then:

f∗ ◦ g∗(ϕ) = f∗(g∗(ϕ)) = f∗(ϕ ◦ g) = (ϕ ◦ g) ◦ f
= ϕ ◦ (g ◦ f) (since by hypothesis, g ◦ f = eY,T )

= ϕ ◦ eY,T = eY,X , because

for all u ∈ HomA (Y, T ), where eY,T is the neutral element of the abelian group
HomA (Y, T ), we have:

ϕ ◦ (u+ eY,T ) = ϕ ◦ (eY,T + u) = ϕ ◦ u =⇒ ϕ ◦ u+ ϕ ◦ eY,T = ϕ ◦ eY,T + ϕ ◦ u = ϕ ◦ u

=⇒

{
ϕ ◦ u+ ϕ ◦ eY,T = ϕ ◦ u+ eY,X = ϕ ◦ u,
ϕ ◦ eY,T + ϕ ◦ u = eY,X + ϕ ◦ u = ϕ ◦ u,

=⇒ ϕ ◦ eY,T = eY,X .

Thus, f∗ ◦ g∗ = eHomA (T,X),HomA (Y,X), and therefore Im g∗ ⊂ Ker f∗.

- Now, we show that Ker(f∗) ⊂ Im g∗. We have:

Ker(f∗) = {ϕ ∈ HomA (Z,X) : f∗(ϕ) = ϕ ◦ f = eY,X}.

Let ϕ ∈ Ker(f∗) = KerHomA (f,X). We show that ϕ ∈ Im g∗. Since ϕ ∈ Ker(f∗),
we have:

ϕ ∈ Ker(f∗) =⇒ f∗(ϕ) = eY,X

=⇒ ϕ ◦ f = eY,X .

Since the sequence:

Y
f
// Z

g
// T // 0

is right short exact, we have g ◦ f = eY,T . Let coN(f) = (j, P ), and by definition of
the cokernel of f , we have j◦f = eY,P and there exists a unique morphism h1 : P → T
such that h1 ◦ j = g. Thus, h1 is an epimorphism. Indeed, let u, v : R → P such
that u ◦ h1 = v ◦ h1. Then:

u ◦ h1 = v ◦ h1 =⇒ (u ◦ h1) ◦ j = (v ◦ h1) ◦ j
=⇒ u ◦ (h1 ◦ j) = v ◦ (h1 ◦ j)
=⇒ u ◦ g = v ◦ g
=⇒ u = v (since g is an epimorphism).

Thus, h1 is an epimorphism. Since A is a balanced category, h1 is a split epimor-
phism, i.e., there exists a unique h′1 : T → P such that:

h1 ◦ h′1 = 1T .

Moreover, h′1 ◦ g = j.



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 15 of 28

Since ϕ ◦ f = eY,X , by the definition of coN(f), we have j ◦ f = eY,P , and there
exists a unique h2 : P → X such that h2 ◦ j = ϕ. That is, the following diagram
commutes:

T

h′1
��

Y
f

// Z

g
99

j
//

ϕ
��

P

h1

OO

h2
yy

X

Thus, we have:
ϕ = h2 ◦ j
= h2 ◦ (h′1 ◦ g) (since h′1 ◦ g = j)

= (h2 ◦ h′1) ◦ g
ϕ = g∗(h2 ◦ h′1).

Hence, ϕ ∈ Im(g∗). Therefore, Ker(f∗) ⊂ Im g∗. Thus, HomA (−, X) : A −→ Ab is
a contravariant, additive, and left-exact functor.

• We now show that HomA (−, X) : A −→ Ab is an exact functor if and only if X is
an injective object in A .
• Suppose X is an injective object in A and show that HomA (−, X) : A −→ Ab is
exact. Consider the short exact sequence of morphisms in A :

0 // Y
f
// Z

g
// T // 0

We must show that the sequence:

{e0,X} // HomA (T,X)
HomA (g,X)=g∗

// HomA (Z,X)
HomA (f,X)=f∗

// HomA (Y,X) // {e0,X}

is exact. By part 1, the sequence:

{e0,X} // HomA (T,X)
HomA (g,X)=g∗

// HomA (Z,X)
HomA (f,X)=f∗

// HomA (Y,X)

is left-exact. It remains to show that HomA (f,X) = f∗ is an epimorphism. Since
X is injective, for every monomorphism f : Y ↪→ Z in A and every morphism
h : Y → X in A , there exists a morphism ϕ : Z → X in A such that ϕ ◦ f = h.
This means the following diagram commutes:

X

0 // Y

h

OO

f
// Z

ϕ
``



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 16 of 28

In other words, for every h ∈ HomA (Y,X), there exists ϕ ∈ HomA (Z,X) such that
ϕ ◦ f = f∗(ϕ) = h. Hence, HomA (f,X) is an epimorphism.
• Conversely, suppose HomA (−, X) : A −→ Ab is exact and show that X is an
injective object in A . Since HomA (X,−) is exact, for every short exact sequence in
A :

0 // Y
f
// Z

g
// T // 0

the sequence:

{e0,X} // HomA (T,X)
HomA (g,X)=g∗

// HomA (Z,X)
HomA (f,X)=f∗

// HomA (Y,X) // {e0,X}

is exact. This means f∗ is an epimorphism. Thus, for every monomorphism f : Y ↪→
Z in A and every morphism h : Y → X in A , there exists a morphism ϕ : Z → X
in A such that ϕ ◦ f = f∗(ϕ) = h. This means the following diagram commutes:

X

0 // Y

h

OO

f
// Z

ϕ
``

Hence, X is an injective object in A . Thus, HomA (−, X) : A −→ Ab is an exact
functor if and only if X is an injective object in A .

3. Exactness of Functors HomComp(A )(X,−) and HomComp(A )(−, X)

Let A be a balanced abelian category and X an object in A . Then:

[label=.]The functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab) is a covariant,
additive, and left-exact functor. The functor HomComp(A )(X,−) : Comp(A ) →
Comp(Ab) is exact if and only if X is a projective object in A .

Proof.

[label=.]Let us show that the functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab)
is covariant, additive, and left-exact. It is evident that HomComp(A )(X,−) is covari-
ant and additive. Let us show that the functor HomComp(A )(X,−) : Comp(A ) →
Comp(Ab) is left-exact.
Consider the left short exact sequence of morphisms in Comp(A ):

(0) // (Y, α)
f
// (Z, β)

g
// (T, θ)

where A is a balanced abelian category. We must show that

HomComp(A )(X,−)((0)) // HomComp(A )(X,−)((Y, α))
f∗

//

HomComp(A )(X,−)((Z, β))
g∗

// HomComp(A )(X,−)((T, θ))



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 17 of 28

is a left short exact sequence of morphisms in Comp(Ab). We have the following
diagram:

HomComp(A )(X,−)((0)) : ... //

��

0HomComp(A )(X,0)
//

��

0HomComp(A )(X,0)
//

��

0HomComp(A )(X,0)
//

��

...

HomComp(A )(X,−)((Y, α)) : ... //

f∗
��

HomComp(A )(X,Yn)
α∗

n//

f∗
n ��

HomComp(A )(X,Yn+1)
α∗

n+1
//

f∗
n+1 ��

HomComp(A )(X,Yn+2) //

f∗
n+2 ��

...

HomComp(A )(X,−)((Z, β)) : ... //

g∗
��

HomComp(A )(X,Zn)
β∗
n//

g∗
n ��

HomComp(A )(X,Zn+1)
β∗
n+1
//

g∗
n+1 ��

HomComp(A )(X,Zn+2)
g∗
n+2 ��

// ...

HomComp(A )(X,−)((T, θ)) : ... // HomComp(A )(X,Tn)
θ∗
n// HomComp(A )(X,Tn+1)

θ∗
n+1
// HomComp(A )(X,Tn+2) // ...

By Theorem 2, for every integer n ∈ Z, the sequence

0HomComp(A )(X,0)
// HomComp(A )(X,Yn)

HomA (X,f)=f∗
n// HomComp(A )(X,Zn)

HomA (X,g)=g∗
n// HomComp(A )(X,Tn)

is a left short exact sequence of morphisms in Comp(Ab).
Hence,

HomComp(A )(X,−)((0)) // HomComp(A )(X,−)((Y, α))
f

//

HomComp(A )(X,−)((Z, β))
g

// HomComp(A )(X,−)((T, θ))

is a left short exact sequence of morphisms in Comp(Ab).
Items i.a and i.b imply that the functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab)
is covariant, additive, and left-exact. Let us show that the functor HomComp(A )(X,−) :
Comp(A ) → Comp(Ab) is exact if and only ifX is a projective object in A . Suppose
that the functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab) is exact and show that
X is a projective object in A . We have:
HomComp(A )(X,−) : Comp(A ) → Comp(Ab) being exact implies that for every

short exact sequence of morphisms in Comp(A ), (0) // (Y, α)
f
// (Z, β)

g
// (T, θ) // (0)

the sequence

HomComp(A )(X,−)((0)) // HomComp(A )(X,−)((Y, α))
f∗

//

HomComp(A )(X,−)((Z, β))
g∗

// HomComp(A )(X,−)((T, θ)) // HomComp(A )(X,−)((0))

is a short exact sequence of morphisms in Comp(Ab). This means the following di-



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 18 of 28

agram commutes:

HomComp(A )(−, X)((0)) : ... //

��

0HomComp(A )(X,0)
//

��

0HomComp(A )(X,0)
//

��

0HomComp(A )(X,0)
//

��

...

HomComp(A )(X,−)((Y, α)) : ... //

f∗
��

HomComp(A )(X,Yn)
α∗

n//

f∗
n ��

HomComp(A )(X,Yn+1)
α∗

n+1
//

f∗
n+1 ��

HomComp(A )(X,Yn+2) //

f∗
n+2 ��

...

HomComp(A )(X,−)((Z, β)) : ... //

g∗
��

HomComp(A )(X,Zn)
β∗
n//

g∗
n ��

HomComp(A )(X,Zn+1)
β∗
n+1
//

g∗
n+1 ��

HomComp(A )(X,Zn+2)
g∗
n+2 ��

// ...

HomComp(A )(X,−)((T, θ)) : ... //

��

HomComp(A )(X,Tn)
θ∗
n//

��

HomComp(A )(X,Tn+1)
θ∗
n+1
//

��

HomComp(A )(X,Tn+2) //

��

...

HomComp(A )(−, X)((0)) : ... // 0HomComp(A )(X,0)
// 0HomComp(A )(X,0)

// 0HomComp(A )(X,0)
// ...

We have for every integer n ∈ Z, g∗n is an epimorphism. Therefore, for every
epimorphism gn : Zn ↠ Tn in A and every morphism fn : X → Tn in A , there
exists a morphism ϕn : X → Zn in A such that gn ◦ ϕn = g∗(ϕn) = fn. This means
the following diagram commutes:

X
ϕn

}}

fn
��

Zn gn
// // Tn // 0

Hence, X is a projective object in A . Suppose that X is a projective object of A
and let us show that the functor HomComp(A )(X,−) : Comp(A ) → Comp(Ab) is an
exact functor. Let the following short exact sequence of morphisms in A :

(0) // (Y, α)
f
// (Z, β)

g
// (T, θ) // (0)

We show that

HomComp(A )(X,−)((0)) // HomComp(A )(X,−)((Y, α))
f∗

//

HomComp(A )(X,−)((Z, β))
g∗

// HomComp(A )(X,−)((T, θ)) // HomComp(A )(X,−)((0))

is a short exact sequence of morphisms in Comp(Ab). By i., we have:

HomComp(A )(X,−)((0)) // HomComp(A )(X,−)((Y, α))
f∗

//

HomComp(A )(X,−)((Z, β))
g∗

// HomComp(A )(X,−)((T, θ))

is a left short exact sequence of morphisms in Comp(Ab). Thus, it remains to show
that for every n, HomA (X, gn) is an epimorphism. SinceX is projective, by Theorem
2, for every epimorphism gn : Zn ↠ Tn in A and every morphism fn : X → Tn in



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 19 of 28

A , there exists a morphism ϕn : X → Zn in A such that gn ◦ ϕn = fn. This means
that the following diagram commutes:

X
ϕn

}}

fn
��

Zn gn
// // Tn // 0

That is, for every fn ∈ HomComp(A )(X,Tn), there exists ϕn ∈ HomComp(A )(X,Zn)
such that gn ◦ ϕn = fn = g∗n(ϕn). Hence, HomComp(A )(X, gn) = g∗n is an epi-
morphism for every integer n in Z. Therefore, the functor HomComp(A )(X,−) :
Comp(A ) → Comp(Ab) is exact. Thus, (ii.1) and (ii.2) imply that the functor
HomComp(A )(X,−) : Comp(A ) → Comp(Ab) is exact if and only if X is a projec-
tive object in A .

Let A be a balanced abelian category and X an object in A . Then:

[label=.]the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is a contravari-
ant, additive, and left-exact functor; the functor HomComp(A )(−, X) : Comp(A ) →
Comp(Ab) is exact if and only if X is a injective object in A .

Proof.
i. Let us show that the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is con-
travariant, additive, and left exact.

(i)(ii)ii.ai.b(i)ii.1ii.2(i)(ii)i.1 It is evident that HomComp(A )(−, X) is contravariant and additive.

i.2 Let us show that the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is left
exact.
Let

(Y, α)
f
// (Z, β)

g
// (T, θ) // (0)

be a right short exact sequence of morphisms in Comp(A ), where A is a balanced
abelian category. We show that

HomComp(A )(−, X)((0)) // HomComp(A )(−, X)((T, θ))
g∗

//

HomComp(A )(−, X)((Z, β))
f∗

// HomComp(A )(−, X)((Y, α))

is a left short exact sequence of morphisms in Comp(Ab). Consider the following



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 20 of 28

diagram:

HomComp(A )(−, X)((0)) : ... //

��

0HomComp(A )(0,X)
//

��

0HomComp(A )(0,X)
//

��

0HomComp(A )(0,X)
//

��

...

HomComp(A )(−, X)((T, θ)) : ... //

g∗
��

HomComp(A )(Tn, X)
θ∗
n//

g∗
n ��

HomComp(A )(Tn+1, X)
θ∗
n+1
//

g∗
n+1 ��

HomComp(A )(Tn+2, X) //

g∗
n+2 ��

...

HomComp(A )(−, X)((Z, β)) : ... //

f∗
��

HomComp(A )(Zn, X)
β∗
n//

f∗
n ��

HomComp(A )(Zn+1, X)
β∗
n+1
//

f∗
n+1 ��

HomComp(A )(Zn+2, X)
f∗
n+2 ��

// ...

HomComp(A )(−, X)((Y, α)) : ... // HomComp(A )(Yn, X)
α∗

n// HomComp(A )(Yn+1, X)
α∗

n+1
// HomComp(A )(Yn+2, X) // ...

Now, by Theorem 2, for every integer n in Z, the sequence

HomComp(A )(0, X) // HomComp(A )(Tn, X)
g∗
n // HomComp(A )(Zn, X)

f∗
n // HomComp(A )(Yn, X)

is a left short exact sequence of morphisms in Comp(Ab). Hence

HomComp(A )(−, X)((0)) // HomComp(A )(−, X)((T, θ))
g∗

//

HomComp(A )(−, X)((Z, β))
f∗

// HomComp(A )(−, X)((Y, α))

is a left short exact sequence of morphisms in Comp(Ab). Therefor i.1 and i.2 imply
that the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is contravariant,
additive, and left exact.

ii Let us show that the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is exact
if and only if X is an injective object in A .

ii.1 Suppose that the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is exact and
show that X is an injective object in A . We have:
HomComp(A )(X,−) : Comp(A ) → Comp(Ab) being exact implies that for every

short exact sequence of morphisms in Comp(A ), (0) // (Y, α)
f
// (Z, β)

g
// (T, θ) // (0)

then

HomComp(A )(−, X)((0)) // HomComp(A )(−, X)((T, θ))
g∗

//

HomComp(A )(−, X)((Z, β))
f∗

// HomComp(A )(−, X)((Y, α)) // HomComp(A )(−, X)((0))

is a short exact sequence of morphisms in Comp(Ab). This means that the following



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 21 of 28

diagram is commutative:

HomComp(A )(−, X)((0)) : ... //

��

0HomComp(A )(0,X)
//

��

0HomComp(A )(0,X)
//

��

0HomComp(A )(0,X)
//

��

...

HomComp(A )(−, X)((T, θ)) : ... //

g∗
��

HomComp(A )(Tn, X)
θ∗
n//

g∗
n ��

HomComp(A )(Tn+1, X)
θ∗
n+1
//

g∗
n+1 ��

HomComp(A )(Tn+2, X) //

g∗
n+2 ��

...

HomComp(A )(−, X)((Z, β)) : ... //

f∗
��

HomComp(A )(Zn, X)
β∗
n//

f∗
n ��

HomComp(A )(Zn+1, X)
β∗
n+1
//

f∗
n+1 ��

HomComp(A )(Zn+2, X)

f∗
n+2 ��

// ...

HomComp(A )(−, X)((Y, α)) : ... //

��

HomComp(A )(Yn, X)
α∗

n//

��

HomComp(A )(Yn+1, X)
α∗

n+1
//

��

HomComp(A )(Yn+2, X) //

��

...

HomComp(A )(−, X)((0)) : ... // 0HomComp(A )(0,X)
// 0HomComp(A )(0,X)

// 0HomComp(A )(0,X)
// ...

We have for every integer n in Z, f∗n is an epimorphism. Therefore, for every
monomorphism fn : Yn ↪→ Zn in A and every morphism hn : Yn → X in A , there
exists a morphism ϕn : Zn → X in A such that ϕn ◦ fn = f∗n(ϕn) = hn. This means
that the following diagram commutes:

X

O // Yn

hn

OO

fn
// Zn

ϕn

aa

Hence, X is an injective object of A .

ii.2 Suppose thatX is an injective object in A and show that the functor HomComp(A )(−, X) :
Comp(A ) → Comp(Ab) is exact.

Let the short exact sequence of morphisms in A (0) // (Y, α)
f
// (Z, β)

g
// (T, θ) // (0)

and show that

HomComp(A )(−, X)((0)) // HomComp(A )(−, X)((T, θ))
g∗

//

HomComp(A )(−, X)((Z, β))
f∗

// HomComp(A )(−, X)((Y, α)) // HomComp(A )(−, X)((0))

is a short exact sequence of morphisms in Comp(Ab). By (i), we have:

HomComp(A )(−, X)((0)) // HomComp(A )(−, X)((T, θ))
g∗

//

HomComp(A )(−, X)((Z, β))
f∗

// HomComp(A )(−, X)((Y, α))

is a left short exact sequence of morphisms in Comp(Ab). Thus, it remains to show
that for every n, HomComp(A )(f,X) = f∗ is an epimorphism. By Theorem 2, for
every integer n in Z, since X is injective, for every monomorphism fn : Yn ↪→ Zn in
A and every morphism hn : Yn → X in A , there exists a morphism ϕn : Zn → X



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 22 of 28

in A such that ϕn ◦ fn = hn. This means that the following diagram commutes:

X

O // Yn

hn

OO

fn
// Zn

ϕn
aa

That is, for every hn ∈ HomComp(A )(Yn, X), there exists ϕn ∈ HomComp(A )(Zn, X)
such that ϕn ◦fn = f∗n(ϕn) = hn. Hence, HomA (f,X) is an epimorphism. Therefore,
the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is exact. Thus, (ii.1) and
(ii.2) imply that the functor HomComp(A )(−, X) : Comp(A ) → Comp(Ab) is exact
if and only if X is an injective object in A .

4. Exactness of Homological Functors of Degree n:
H̃n(X,−) and H̃n(−, X)

Consider the homological functor Hn : Comp(Ab) = Comp(Z −Mod) −→ Ab which is a
special case of the homological functor Hn : Comp(A-Mod) −→ Ab for all n ∈ Z. That is,
Hn is a covariant additive functor. [H̃n(X,−)]
Let A be a balanced abelian category and X a projective object in A . Then homological
functor of degree n (n ∈ Z), denoted H̃n(X,−) = Hn ◦ HomComp(A )(X,−) where Hn :

Comp(Ab) −→ Ab, H̃n(X,−) : Comp(A ) −→ Ab is defined by:

(i) for any complex sequence in Comp(A ) (βn : Yn → Yn+1)n∈Z denoted (Y, β), we
associate
H̃n(X,−)((Y, β)) = (Hn ◦HomComp(A )(X,−))((Y, β)) = Kerβ∗n+1/Imβ

∗
n ∀n ∈ Z;

(ii) for any complex sequence (Y, β) = (βn : Yn → Yn+1)n∈Z in Comp(A ), any complex
sequence (Z,α) = (αn : Zn → Zn+1)n∈Z in Comp(A ), and any complex chain
f : (Y, β) → (Z,α) = (fn : Yn −→ Zn)n∈Z in Comp(A ) denoted f : (Y, β) −→ (Z,α),
we associate:

H̃n(X,−)(f) : (Hn ◦HomComp(A )(X,−))((Y, β)) −→ (Hn ◦HomComp(A )(X,−))((Z,α))

gn 7−→ fn(gn)

is a covariant additive functor.
Proof.

We know that the homology functor Hn : Comp(A-Mod) −→ Ab is defined by:

(i) for any complex sequence in Comp(A-Mod) (βn : Yn → Yn+1)n∈Z denoted (Y, β), we
associate

Hn(Y, β) = Kerβn+1/Imβn ∀n ∈ Z;



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 23 of 28

(ii) for any complex sequence (Y, β) = (βn : Yn → Yn+1)n∈Z in Comp(A-Mod), any
complex sequence (Z,α) = (αn : Zn → Zn+1)n∈Z in Comp(A-Mod), and any complex
chain f : (Y, β) → (Z,α) = (fn : Yn −→ Zn)n∈Z in Comp(A-Mod) denoted f :
(Y, β) −→ (Z,α), we associate:

Hn(f) : Hn(Y, β) −→ Hn(Z,α)

gn 7−→ fn(gn)

and Hn is a covariant additive functor. Since Comp(Ab) = Comp(Z − Mod) is
a special case of Comp(A-Mod), the functor H̃n is covariant. By Theorem 3,
HomComp(A )(X,−) is covariant. Now, the composition of two covariant func-
tors is covariant, so H̃n(X,−) : Comp(A ) −→ Ab is well-defined and is a covariant
functor. By Proposition 1 and Theorem 2, H̃n(X,−) is an additive functor. Hence,
H̃n(X,−) is a covariant additive functor. And Hn is a covariant additive functor.
Since Comp(Ab) = Comp(Z−Mod) is a special case of Comp(A-Mod), the functor
H̃n(X,−) is well-defined and is a covariant additive functor.

Let (0) // (Y, α)
f
// (Z, β)

g
// (T, γ) // (0) be a short exact sequence of

morphisms in Comp(A ), where X is a projective object in A and A is a balanced abelian
category. Then:

(i) the morphism of connection associated to the covariant functor H̃n(X,−) is defined
by:

λn : H̃n(X,−)((T, γ)) −→ H̃n+1(X,−)((Y, α))

kn+1 7−→ f∗−1
n+2(β

∗
n+1(g

∗−1
n+1(kn+1)))

∀n ∈ Z;

(ii) the sequence

. . . // H̃n(X,−)((Y, α))
H̃n(X,−)(f)

// H̃n(X,−)((Z, β))
H̃n(X,−)(g)

// H̃n((T, γ))
λn //

H̃n+1(X,−)((Y, α))
H̃n+1(X,−)(f)

// H̃n+1(X,−)((Z, β))
H̃n+1(X,−)(g)

// H̃n+1(X,−)((T, γ))
λn+1

// . . .

is a long exact sequence of abelian group morphisms. That is, for all n ∈ Z:
Im(H̃n(X,−)(f)) = Ker(H̃n(X,−)(g))

Im(H̃n(X,−)(g)) = Ker(λn)

Im(λn) = Ker(H̃n+1(X,−)(f)).

Proof.

(i) We know that if the homological functor Hn : Comp(A-Mod) −→ Ab is covariant,
then for every short exact sequence of morphisms in Comp(A-Mod)



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 24 of 28

(0) // (Y, α)
f
// (Z, β)

g
// (T, γ) // (0) the connecting morphism is defined

by
λn : Hn((T, γ)) −→ Hn+1((Y, α))

kn+1 7−→ f−1
n+2(βn+1(g

−1
n+1(kn+1)))

∀n ∈ Z.

Since Comp(Ab) = Comp(Z−Mod) is a special case of Comp(A-Mod), by Theorem
3 if X is a projective object in A where A is a balanced abelian category, then

λn : H̃n(X,−)((T, γ)) −→ ˜Hn+1(X,−)((Y, α))

kn+1 7−→ f∗−1
n+2(β

∗
n+1(g

∗−1
n+1(kn+1)))

∀n ∈ Z

is well-defined with Hn : Comp(Ab) −→ Ab.

(ii) According to Theorem 3, if X is a projective object in A , where A is a bal-
anced abelian category, then HomComp(A )(X,−) transforms any complex sequence
in Comp(A ) into a complex sequence in Comp(Ab). However, Hn transforms
every short exact sequence in Comp(A-Mod) into a long exact sequence of mor-
phisms in Ab. In particular, Hn transforms any complex sequence of morphisms
in Comp(Ab) = Comp(Z − Mod) into a long exact sequence of morphisms in Ab.
Now, H̃n(X,−) = Hn ◦ HomComp(A )(X,−), where Hn : Comp(Ab) −→ Ab, and

H̃n(X,−) : Comp(A ) −→ Ab. Thus, the sequence

. . . // H̃n(X,−)((Y, α))
H̃n(X,−)(f)

// H̃n(X,−)((Z, β))
H̃n(X,−)(g)

// H̃n((T, γ))
λn //

H̃n+1(X,−)((Y, α))
H̃n+1(X,−)(f)

// H̃n+1(X,−)((Z, β))
H̃n+1(X,−)(g)

// H̃n+1(X,−)((T, γ))
λn+1(−,X)

// . . .

is a long exact sequence of abelian group morphisms.

[H̃n(−, X)]
Let A be a balanced abelian category and X an injective object in A . Then the homo-
logical functor of degree n (n ∈ Z), denoted H̃n(−, X) = Hn ◦ HomComp(A )(−, X), where

Hn : Comp(Ab) −→ Ab, H̃n(−, X) : Comp(A ) −→ Ab is defined by:

(i) For any complex (Y, β) = (βn : Yn → Yn+1)n∈Z in Comp(A ), we associate:

H̃n(−, X)((Y, β)) = Hn(HomComp(A )(−, X))((Y, β)) = Kerβ∗n+1/Imβ
∗
n ∀n ∈ Z

(ii) For any complex morphism f : (Y, β) → (Z,α) = (fn : Yn → Zn)n∈Z, we associate:

H̃n(f,X) : H̃n(−, X)((Z,α)) −→ H̃n(−, X)((Y, β))

gn 7−→ gn(fn)

is a contravariant additive functor.
Proof.

We know that the homology functor Hn : Comp(A-Mod) −→ Ab is defined by:



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 25 of 28

(i) For any complex (Y, β) = (βn : Yn → Yn+1)n∈Z in Comp(A-Mod), we associate:

Hn((Y, β)) = Kerβn+1/Imβn ∀n ∈ Z

(ii) For any complex morphism f : (Y, β) → (Z,α), we associate:

Hn(f) : Hn((Z,α)) −→ Hn((Y, β))

gn 7−→ gn(fn)

Since Hn is a covariant additive functor, and Comp(Ab) = Comp(Z − Mod) is a
special case of Comp(A-Mod), the functor H̃n is covariant. According to Theorem 3,
HomComp(A )(−, X) is contravariant. However, the composition of a covariant functor

and a contravariant functor is contravariant, so H̃n(−, X) : Comp(A ) −→ Ab is well-
defined and is a contravariant functor. By Proposition 1 and Theorem 2, H̃n(−, X)
is an additive functor. Thus, H̃n(−, X) is a contravariant additive functor.

Let (0) // (Y, α)
f
// (Z, β)

g
// (T, γ) // (0) be a short exact sequence in

Comp(A ), where A is a balanced abelian category and X an injective object in A . Then:

(i) the morphism of connection associated to the contravariant functor H̃n(−, X) is
defined by:

δn :
H̃n(−, X)((Y, α)) −→ H̃n+1(−, X)((T, γ))

kn+1 7−→ g∗−1
n+2(β

∗
n+1(f

∗−1
n+1(kn+1)))

, ∀n ∈ Z

(ii) The sequence

· · · // H̃n(−, X)((T, γ))
H̃n(−,X)(g)

// H̃n(−, X)((Z, β))
H̃n(−,X)(f)

// H̃n(−, X)((Y, α))

δn// H̃n+1(−, X)((T, γ))
H̃n+1(−,X)(g)

// H̃n+1(−, X)((Z, β))
H̃n+1(−,X)(f)

// H̃n+1(−, X)((Y, α))
δn+1

// · · ·

is a long exact sequence in Ab. That is (∀n ∈ Z):
Im(H̃n(−, X)(g)) = Ker(H̃n(−, X)(f))

Im(H̃n(−, X)(f)) = Ker(δn)

Im(δn) = Ker(H̃n+1(−, X)(g))

Proof.



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 26 of 28

(i) We know that if the homological functor Hn : Comp(A-Mod) −→ Ab is covariant,
then for every short exact sequence of morphisms in Comp(A-Mod)

(0) // (Y, α)
f
// (Z, β)

g
// (T, γ) // (0) the connecting morphism is defined

by
λn : Hn((Y, α)) −→ Hn+1((T, γ))

kn+1 7−→ g−1
n+2(βn+1(f

−1
n+1(kn+1)))

∀n ∈ Z

Since Comp(Ab) = Comp(Z−Mod) is a special case of Comp(A-Mod), by Theorem
3 if X is an injective object in A where A is a balanced abelian category, then by
definition of H̃n(−, X):

δn : H̃n(−, X)((Y, α)) −→ H̃n+1(−, X)((T, γ))

kn+1 7−→ g∗−1
n+2(β

∗
n+1(f

∗−1
n+1(kn+1)))

∀n ∈ Z

is well-defined with Hn : Comp(Ab) −→ Ab.

(ii) According to Theorem 3, if X is an injective object in A , where A is a bal-
anced abelian category, then HomComp(A )(−, X) transforms any complex sequence
in Comp(A ) into a complex sequence in Comp(Ab). However, Hn transforms
every short exact sequence in Comp(A-Mod) into a long exact sequence of mor-
phisms in Ab. In particular, Hn transforms any complex sequence of morphisms
in Comp(Ab) = Comp(Z − Mod) into a long exact sequence of morphisms in Ab.

Now, H̃n(−, X) = Hn ◦ HomComp(A )(−, X), where Hn : Comp(Ab) −→ Ab, and

H̃n(−, X) : Comp(A ) −→ Ab. Thus, the sequence

· · · // H̃n(−, X)((T, γ))
H̃n(−,X)(g)

// H̃n(−, X)((Z, β))
H̃n(−,X)(f)

// H̃n(−, X)((Y, α))

λn// H̃n+1(−, X)((T, γ))
H̃n+1(−,X)(g)

// H̃n+1(−, X)((Z, β))
H̃n+1(−,X)(f)

// H̃n+1(−, X)((Y, α)) // · · ·

is a long exact sequence of abelian group morphisms.

5. Conclusion

In this article, by using the exacteness of we have shown the exactness of the functors
HomA (X,−),
HomA (−, X) : Comp(A ) → Comp(Ab) and the exactness of the functors HomComp(A )(X,−),
HomComp(A )(−, X) : Comp(A ) → Comp(Ab) where A is a balanced abelian category. We

then constructed the additive covariant homological functor: H̃n(X,−) = Hn◦HomComp(A )(X,−)

and the additive contravariant homological functor: H̃n(−, X) = Hn ◦HomComp(A )(−, X)
whereHn : Comp(Ab) −→ Ab and HomComp(A )(X,−), HomComp(A )(−, X) : Comp(A ) −→



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 27 of 28

Comp(Ab) ∀n ∈ Z.Moreover, we constructed the connecting morphism λn : H̃n(X,−)((T, γ)) →
H̃n+1(X,−)((Y, α)) associated to the covariant functor H̃n(X,−), by showing how the
functor H̃n(X,−) transforms a short exact sequence of morphisms in Comp(A ) into a
long exact sequence of morphisms in Ab, where X is an projective object of A and A
is a balanced abelian category. Then we again constructed the connecting morphism
δn : H̃n(−, X)((Y, α)) → H̃n+1(−, X)((T, γ)) associated to the contravariant functor
H̃n(−, X) also showing how the functor H̃n(−, X) transforms a short exact sequence of
morphisms in Comp(A ) into a long exact sequence of morphisms in Ab, where X is an
injective object of A and A is a balanced abelian category.

References

[1] Bassirou Dembele, Mohamed Ben Faraj ben Maaouia, and Mamadou Sanghare. The
functor and its relationship with homological functors torn and extn. In The Moroccan
Andalusian Meeting on Algebras and their Applications, pages 253–271. Springer,
2018.

[2] Bassirou Dembele, Mohamed Ben Faraj Ben Maaouia, and Mamadou Sanghare. Lo-
calization, isomorphisms and adjoint isomorphism in the category comp(a- mod).
Journal of Mathematics Research, 12(4):1–65, 2020.

[3] Seydina Ababacar Balde, Mohamed Ben Faraj Ben Maaouia, and Ahmed Ould Chbih.
Localization of hopfian and cohopfian objects in the categories of a-mod, agr(a-
mod) and comp(agr(a-mod)). European Journal of Pure and Applied Mathematics,
14(2):404–422, 2021.

[4] Moussa Thiaw. Relation entre Foncteur Localisation S−1() et les Foncteurs Ho-
mologiques Ext et Tor dans la catégorie A-Alg(RESP.Alg-A). Thèse, Université Gas-
ton Berger, Saint-Louis, Février 2020.

[5] Friedrich Kasch. Modules and rings, volume 17. Academic press, 1982.
[6] Joseph J Rotman. Notes on homological algebras, university of illinois. Urbana, 1968.
[7] Peter J Freyd. Abelian categories, volume 1964. Harper & Row New York, 1964.
[8] Pierre Gabriel. Des catégories abéliennes. Bulletin de la Société Mathématique de

France, 90:323–448, 1962.
[9] Joseph J Rotman. An Introduction to Homological Algebra. Springer, 2nd edition,

2009.
[10] Charles A Weibel. An introduction to homological algebra, volume 38. Cambridge

university press, 1994.
[11] El Hadji Ousseynou DIALLO. Hopficité et Co-Hopficité dans la catégorie COMP

des Complexes. Thèse, Faculté des Sciences et Techniques, Université Cheikh Anta
DIOP, 2014.

[12] Mathieu Dupont et al. Catégories abéliennes en dimension 2. PhD thesis, Université
catholique de Louvain, 2008 (English version), 2008.

[13] Sebastian Posur. A constructive approach to freyd categories. Applied Categorical
Structures, 29(1):171–211, 2021.

[14] Bassirou Dembele. Foncteur Localisation dans la catégorie Comp(A-Mod) des suites



A. Diallo, M. B. F. B. Maaouia, M. Sanghare / Eur. J. Pure Appl. Math, 18 (4) (2025), 6640 28 of 28

complexes de morphismes de A-modules à gauche et applications sur les dimensions
homologiques et sur les enveloppes et couvertures plates dans Comp(A-Mod). Thèse,
Université Gaston Berger, Saint-Louis, Décembre 2020.

[15] Ahmed Ould Chbih. Graduation et filtration des modules de fractions sur des anneaux
non nécessairement commutatifs. Thèse, Université Gaston Berger, Saint-Louis, Avril
2016.

[16] Charles Weibel and MCR Butler. An introduction to homological algebra. Bulletin
of the London Mathematical Society, 28(132):322–323, 1996.

[17] Ahmed Ould Chbih, MBF Maaouia, and Mamadou Sanghare. Graduation of mod-
ule of fraction on a graded domain ring not necessarily commutative. International
Journal of Algebra, 9(10):457–474, 2015.

[18] Ahmed Ould Chbih, Mohamed Ben Faraj Ben Maaouia, and Mamadou Sanghare.
Localization in the category comp(gr(a-mod)) of complex associated to the category
gr(a-mod) of graded left a-modules over a graded ring. European Journal of Pure and
Applied Mathematics, 16(3):1913–1939, 2023.

[19] Moussa Thiaw and Mohamed Ben Faraj Ben Maaouia. Adjunction and localization in
the category a-alg of a-algebras. European Journal of Pure and Applied Mathematics,
13(3):472–482, 2020.