# On Strong Monomorphisms and Strong Epimorphisms

Authors: M. A. Al Shumrani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

---

## Abstract

J. Dydak and F. R. Ruiz del Portal defined strong monomorphism and strong epimorphism in procategories. They obtained a useful characterization of them and some results. In this paper, we aim to study these notions further and obtain some properties of them.


---

## Body

## 1. Introduction

Dydak and Ruiz del Portal in [1] studied isomorphisms in procategories and obtained the following characterization of isomorphisms in procategories.Proposition 1.1.
Letf:X→Y be a morphism in pro-𝒞 where 𝒞 is an arbitrary category. f is an isomorphism if and only if for any commutative diagramwith P, Q objects in 𝒞, there is h:Y→P such that g∘h=b and h∘f=a.This characterization led them to introduce the notions of strong monomorphism and strong epimorphism in procategories. They studied them and obtained some results and a useful characterization of them.In this paper, we study some properties of strong monomorphisms and strong epimorphisms in procategories.


## 2. Preliminaries

First we recall some basic facts about procategories. The main reference is [1] and for more details see [2].Let𝒞 be an arbitrary category. Loosely speaking, the pro-category pro-𝒞 of 𝒞 is the universal category with inverse limits containing 𝒞 as a full subcategory. An object of pro-𝒞 is an inverse system in 𝒞, denoted by X=(Xα,pαβ,A), consisting of a directed set A, called the index set (from now onward it will be denoted by I(X)), of 𝒞 objects Xα for each α∈I(X), called the terms of X, and of 𝒞 morphisms pαβ:Xβ→Xα for each related pair α<β, called the bonding morphisms of X (from now onward it will be denoted by pαβ(X)).IfP is an object of 𝒞 and X is an object of pro-𝒞, then a morphism f:X→P in pro-𝒞 is the direct limit of Mor(Xα,P), α∈I(X), and so f can be represented by g:Xα→P. Note that the morphism from X to Xα represented by the identity Xα→Xα is called the projection morphism and denoted by p(X)α.IfX and Y are two objects in pro-𝒞 with identical index sets, then a morphism f:X→Y is called a level morphism if, for each α<β, the following diagramcommutes.Recall that an objectX of pro-𝒞 is uniformly movable if every α∈I(X) admits a β>α (a uniform movability index of α) such that there is a morphism r:Xβ→X satisfying p(X)α∘r=p(X)αβ where p(X)α:X→Xα is the projection morphism.The following lemma is important and will be used later. Therefore, we include its proof for completeness.Lemma 2.1.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞. For any commutative diagram with P, Q objects in 𝒞one may find α∈I(X) and representatives aα:Xα→P of a and bα:Yα→Q of b such thatis commutative.Proof.
Choose representativesu:Xβ→P of a and v:Yβ→Q of b. Since g∘u∘p(X)β=g∘a=b∘f=v∘p(Y)β∘f=v∘fβ∘p(X)β, there is α>β such that g∘u∘p(X)βα=v∘fβ∘p(X)βα. Put aα=u∘p(X)βα and bα=v∘p(Y)βα.Recall that a morphismf:X→Y of a category 𝒞 is called a monomorphism if f∘g=f∘h implies g=h for any two morphisms g,h:Z→X. A morphism f:X→Y of a category 𝒞 is called an epimorphism if g∘f=h∘f implies g=h for any two morphisms g,h:Y→Z.Next, we recall definitions of strong monomorphism and strong epimorphism and state some of their basic results obtained. The main reference is [1].Definition 2.2.
A morphismf:X→Y in pro-𝒞 is called a strong monomorphism (strong epimorphism, resp.) if for every commutative diagramwith P, Q objects in 𝒞, there is a morphism h:Y→P such that h∘f=a (g∘h=b, resp.).Note that ifX and Y are objects of 𝒞, then f:X→Y is a strong monomorphism (strong epimorphism, resp.) if and only if f has a left inverse (a right inverse, resp.).The following result presents the relation between monomorphisms and strong monomorphisms and between epimorphisms and strong epimorphisms.Lemma 2.3.
Iff is a strong monomorphism (strong epimorphism, resp.) of pro-𝒞, then f is a monomorphism (epimorphism, resp.) of pro-𝒞.The following lemma is very useful.Lemma 2.4.
Ifg∘f is a strong monomorphism (strong epimorphism, resp.), then f is a strong monomorphism (g is a strong epimorphism, resp.).The following theorems are characterizations of isomorphisms in pro-𝒞 in terms of strong monomorphisms and strong epimorphisms.Theorem 2.5.
Letf:X→Y be a morphism in pro-𝒞. The following statements are equivalent.(i)
f is an isomorphism.(ii)
f is a strong monomorphism and an epimorphism.Theorem 2.6.
Letf:X→Y be a morphism in pro-𝒞 where 𝒞 is a category with direct sums. The following statements are equivalent.(i)
f is an isomorphism.(ii)
f is a strong epimorphism and a monomorphism.The following useful characterization of strong monomorphisms and strong epimorphisms was obtained.Proposition 2.7.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞. The following statements are equivalent.(i)
f is a strong monomorphism (strong epimorphism, resp.).(ii)
For eachα∈I(X), there is a morphism uα:Y→Xα such that uα∘f=p(X)α (fα∘uα=p(Y)α, resp.).(iii)
For eachα∈I(X), there is β∈I(X), β>α and a morphism gα,β:Yβ→Xα such that gα,β∘fβ=p(X)αβ (fα∘gα,β=p(Y)αβ, resp.).The immediate consequence of this characterization is that both notions are preserved by functors pro-F if F:𝒞→𝒟.


## 3. Properties of Strong Monomorphisms and Strong Epimorphisms

Theorem 3.1.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞. If each fα is a strong monomorphism of 𝒞 for each α, then f is a strong monomorphism of pro-𝒞.Proof.
Suppose thatfα is a strong monomorphism of 𝒞 for each α. Suppose thatis a commutative diagram in pro-𝒞 with P, Q objects in 𝒞. By Lemma 2.1, we may find α∈I(X) and representatives aα:Xα→P of a and bα:Yα→Q of b such thatis commutative, where for α>β, aα=u∘p(X)βα, u:Xβ→P, and a=u∘p(X)β. Thus, there is h:Yα→P such that h∘fα=aα since fα is a strong monomorphism of 𝒞. There is c=h∘p(Y)α:Y→P. But c∘f=h∘p(Y)α∘f=h∘fα∘p(X)α=aα∘p(X)α=u∘p(X)βα∘p(X)α=u∘p(X)β=a. Hence, f is a strong monomorphism of pro-𝒞.Similarly, we have the following result.Theorem 3.2.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞. If each fα is a strong epimorphism of 𝒞 for each α, then f is a strong epimorphism of pro-𝒞.Proof.
Suppose thatfα is a strong epimorphism of 𝒞 for each α. Suppose thatis a commutative diagram in pro-𝒞 with P, Q objects in 𝒞. By Lemma 2.1, we may find α∈I(X) and representatives aα:Xα→P of a and bα:Yα→Q of b such thatis commutative, where for α>β, bα=v∘p(Y)βα, v:Yβ→Q, and b=v∘p(Y)β. Thus, there is h:Yα→P such that g∘h=bα since fα is a strong epimorphism of 𝒞. There is c=h∘p(Y)α:Y→P. But g∘c=g∘h∘p(Y)α=bα∘p(Y)α=v∘p(Y)βα∘p(Y)α=v∘p(Y)β=b. Hence, f is a strong epimorphism of pro-𝒞.Lemma 3.3.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞. If f is a strong epimorphism of pro-𝒞 and each p(Y)α is a strong epimorphism of pro-𝒞 for each α∈I(X), then fα is a strong epimorphism of 𝒞 for each α∈I(X).Proof.
Suppose thatf is a strong epimorphism of pro-𝒞 and each p(Y)α is a strong epimorphism of pro-𝒞 for each α∈I(X). By Proposition 2.7, we have for each α∈I(X), fα∘uα=p(Y)α where uα:Y→Xα. Since p(Y)α is a strong epimorphism, we have that fα is a strong epimorphism of 𝒞 by Lemma 2.4.Corollary 3.4.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞. If each p(X)αβ is a strong monomorphism of 𝒞 and f is a strong monomorphism of pro-𝒞, then fβ is a strong monomorphism of 𝒞 for some β∈I(X).Proof.
Assume that eachp(X)αβ is a strong monomorphism of 𝒞 and f is a strong monomorphism of pro-𝒞. By Proposition 2.7, we have gα,β∘fβ=p(X)αβ for some β∈I(X) where gα,β:Yβ→Xα. Since p(X)αβ is a strong monomorphism, we have that fβ is a strong monomorphism of 𝒞 by Lemma 2.4.Proposition 3.5.
Ifp(X)αβ is a strong monomorphism of 𝒞 for each β>α, then p(X)α is a strong monomorphism of pro-𝒞 for each α∈I(X).Proof.
Assume thatp(X)αβ is a strong monomorphism of 𝒞 for each β>α. Assume that the following diagramis commutative in pro-𝒞 with P, Q objects in 𝒞. We may find β∈I(X), β>α, and representative aβ:Xβ→P of a such that the following diagram is commutative. But p(X)αβ is a strong monomorphism of 𝒞. Thus, there is h:Xα→P such that h∘p(X)αβ=aβ. Therefore, h∘p(X)αβ∘p(X)β=aβ∘p(X)β, that is, h∘p(X)α=a. Hence, p(X)α is a strong monomorphism of pro-𝒞 for each α∈I(X).Proposition 3.6.
LetX be an object of pro-𝒞. Then the following conditions on X are equivalent. (i)
p(X)αβ is a strong epimorphism of 𝒞 for each β>α.(ii)
p(X)α is a strong epimorphism of pro-𝒞 for each α∈I(X).Proof.
(i)⇒(ii) Assume that p(X)αβ is a strong epimorphism of 𝒞 for each β>α. Assume that the following diagramis commutative in pro-𝒞 with P, Q objects in 𝒞. We may find β∈I(X), β>α, and representative aβ:Xβ→P of a such that the following diagramis commutative. But p(X)αβ is a strong epimorphism of 𝒞. Thus, there is h:Xα→P such that g∘h=b. Hence, p(X)α is a strong epimorphism of pro-𝒞 for each α∈I(X).
(ii)⇒(i) Assume that p(X)α is a strong epimorphism of pro-𝒞 for each α∈I(X). If β>α, then p(X)α=p(X)αβ∘p(X)β. Hence, p(X)αβ is a strong epimorphism of 𝒞 by Lemma 2.4.Theorem 3.7.
Let𝒞 be a category with inverse limits. Let P be an object of 𝒞 and let f:X→P be a morphism in pro-𝒞. If limf is a strong epimorphism of 𝒞, then f is a strong epimorphism of pro-𝒞.Proof.
Suppose thatlimf is a strong epimorphism. Suppose that the following diagramis commutative in pro-𝒞 with Q, W objects in 𝒞. Note that the following diagram is commutative in pro-𝒞. Thus,is commutative in 𝒞. But limf is a strong epimorphism. Therefore, there is h:P→Q such that g∘h=b. Hence, f is a strong epimorphism of pro-𝒞.Remark 3.8.
Note that iff:X→Y is a morphism in pro-𝒞, then we must assume that Y is uniformly movable for this theorem to hold and this result is Corollary 4.4 in [1].Proposition 3.9.
LetX be an object of pro-𝒞. Then the following conditions on X are equivalent.(i)
There is a strong monomorphismf:X→P, where P is an object of 𝒞.(ii)
p(X)α is a strong monomorphism of pro-𝒞 for some α∈I(X).(iii)
There isα∈I(X) such that p(X)β is a strong monomorphism of pro-𝒞 for all β≥α.Proof.
(i)⇒(ii) Let g:Xα→P be a representative of f. Thus, f=g∘p(X)α. But f is a strong monomorphism. Hence, p(X)α is a strong monomorphism of pro-𝒞 by Lemma 2.4.
(ii)⇒(iii) For all β≥α, we have p(X)α=p(X)αβ∘p(X)β. Hence, p(X)β is a strong monomorphism of pro-𝒞 by Lemma 2.4.
(iii)⇒(i) Put f=p(X)β. Hence, the result holds.The following result is Proposition4.2 in [1].Proposition 3.10.
Let𝒞 be a category with inverse limits. Then if X is an object of pro-𝒞, then the following conditions on X are equivalent. (i)
There is a strong epimorphismf:P→X, where P is an object of 𝒞.(ii)
X is uniformly movable.Theorem 3.11.
Suppose thatf={fα:Xα→Yα}α∈I(X) is a level morphism of pro-𝒞 where 𝒞 is a category with direct sums such that each p(X)αβ is a strong monomorphism of 𝒞 and each p(Y)α is a strong epimorphism of 𝒞. If f is an isomorphism of pro-𝒞, then there is α∈I(X) such that fβ is isomorphism of 𝒞 for all β≥α.Proof.
Assume thatf is an isomorphism of pro-𝒞. By Corollary 3.4, fα is a strong monomorphism of 𝒞 for some α∈I(X). By Lemma 3.3, fα is a strong epimorphism of 𝒞 for each α∈I(X). Since fα is a strong monomorphism of 𝒞 for some α∈I(X), we have that fα is a monomorphism by Lemma 2.3. Thus, there is α∈I(X) such that fβ is a monomorphism of 𝒞 for all β≥α by Corollary 2.9 in [3]. Therefore, fβ is a strong epimorphism and a monomorphism. Hence, fβ is isomorphism of 𝒞 for each β≥α by Theorem 2.6.



---

*Source: 294812-2010-07-19.xml*