Research Article | Open Access

Volume 2010 |Article ID 294812 | https://doi.org/10.1155/2010/294812

M. A. Al Shumrani, "On Strong Monomorphisms and Strong Epimorphisms", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 294812, 8 pages, 2010. https://doi.org/10.1155/2010/294812

# On Strong Monomorphisms and Strong Epimorphisms

Accepted05 May 2010
Published19 Jul 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.

#### 1. Introduction

Dydak and Ruiz del Portal in  studied isomorphisms in procategories and obtained the following characterization of isomorphisms in procategories.

Proposition 1.1. Let be a morphism in pro- where is an arbitrary category. is an isomorphism if and only if for any commutative diagram with , objects in , there is such that and .

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  and for more details see .

Let be an arbitrary category. Loosely speaking, the pro-category pro- of is the universal category with inverse limits containing as a full subcategory. An object of pro- is an inverse system in , denoted by , consisting of a directed set , called the index set (from now onward it will be denoted by ), of objects for each , called the terms of , and of morphisms for each related pair , called the bonding morphisms of (from now onward it will be denoted by ).

If is an object of and is an object of pro-, then a morphism in pro- is the direct limit of Mor, , and so can be represented by . Note that the morphism from to represented by the identity is called the projection morphism and denoted by .

If and are two objects in pro- with identical index sets, then a morphism is called a level morphism if, for each , the following diagram commutes.

Recall that an object of pro- is uniformly movable if every admits a (a uniform movability index of ) such that there is a morphism satisfying where 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 that is a level morphism of pro-. For any commutative diagram with , objects in one may find and representatives of and of such that is commutative.

Proof. Choose representatives of and of . Since , there is such that . Put and .

Recall that a morphism of a category is called a monomorphism if implies for any two morphisms . A morphism of a category is called an epimorphism if implies for any two morphisms .

Next, we recall definitions of strong monomorphism and strong epimorphism and state some of their basic results obtained. The main reference is .

Definition 2.2. A morphism in pro- is called a strong monomorphism (strong epimorphism, resp.) if for every commutative diagram with , objects in , there is a morphism such that (, resp.).

Note that if and are objects of , then is a strong monomorphism (strong epimorphism, resp.) if and only if 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. If is a strong monomorphism (strong epimorphism, resp.) of pro-, then is a monomorphism (epimorphism, resp.) of pro-.

The following lemma is very useful.

Lemma 2.4. If is a strong monomorphism (strong epimorphism, resp.), then is a strong monomorphism ( is a strong epimorphism, resp.).

The following theorems are characterizations of isomorphisms in pro- in terms of strong monomorphisms and strong epimorphisms.

Theorem 2.5. Let be a morphism in pro-. The following statements are equivalent.(i) is an isomorphism.(ii) is a strong monomorphism and an epimorphism.

Theorem 2.6. Let be a morphism in pro- where is a category with direct sums. The following statements are equivalent.(i) is an isomorphism.(ii) is a strong epimorphism and a monomorphism.

The following useful characterization of strong monomorphisms and strong epimorphisms was obtained.

Proposition 2.7. Suppose that is a level morphism of pro-. The following statements are equivalent.(i) is a strong monomorphism (strong epimorphism, resp.).(ii)For each , there is a morphism such that (, resp.).(iii)For each , there is , and a morphism such that (, resp.).

The immediate consequence of this characterization is that both notions are preserved by functors pro- if .

#### 3. Properties of Strong Monomorphisms and Strong Epimorphisms

Theorem 3.1. Suppose that is a level morphism of pro-. If each is a strong monomorphism of for each , then is a strong monomorphism of pro-.

Proof. Suppose that is a strong monomorphism of for each . Suppose that is a commutative diagram in pro- with , objects in . By Lemma 2.1, we may find and representatives of and of such that is commutative, where for , , , and . Thus, there is such that since is a strong monomorphism of . There is . But . Hence, is a strong monomorphism of pro-.

Similarly, we have the following result.

Theorem 3.2. Suppose that is a level morphism of pro-. If each is a strong epimorphism of for each , then is a strong epimorphism of pro-.

Proof. Suppose that is a strong epimorphism of for each . Suppose that is a commutative diagram in pro- with , objects in . By Lemma 2.1, we may find and representatives of and of such that is commutative, where for , , , and . Thus, there is such that since is a strong epimorphism of . There is . But . Hence, is a strong epimorphism of pro-.

Lemma 3.3. Suppose that is a level morphism of pro-. If is a strong epimorphism of pro- and each is a strong epimorphism of pro- for each , then is a strong epimorphism of for each .

Proof. Suppose that is a strong epimorphism of pro- and each is a strong epimorphism of pro- for each . By Proposition 2.7, we have for each , where . Since is a strong epimorphism, we have that is a strong epimorphism of by Lemma 2.4.

Corollary 3.4. Suppose that is a level morphism of pro-. If each is a strong monomorphism of and is a strong monomorphism of pro-, then is a strong monomorphism of for some .

Proof. Assume that each is a strong monomorphism of and is a strong monomorphism of pro-. By Proposition 2.7, we have for some where . Since is a strong monomorphism, we have that is a strong monomorphism of by Lemma 2.4.

Proposition 3.5. If is a strong monomorphism of for each , then is a strong monomorphism of pro- for each .

Proof. Assume that is a strong monomorphism of for each . Assume that the following diagram is commutative in pro- with , objects in . We may find , , and representative of such that the following diagram is commutative. But is a strong monomorphism of . Thus, there is such that . Therefore, , that is, . Hence, is a strong monomorphism of pro- for each .

Proposition 3.6. Let be an object of pro-. Then the following conditions on are equivalent. (i) is a strong epimorphism of for each .(ii) is a strong epimorphism of pro- for each .

Proof. (i)(ii) Assume that is a strong epimorphism of for each . Assume that the following diagram is commutative in pro- with , objects in . We may find , , and representative of such that the following diagram is commutative. But is a strong epimorphism of . Thus, there is such that . Hence, is a strong epimorphism of pro- for each .
(ii)(i) Assume that is a strong epimorphism of pro- for each . If , then . Hence, is a strong epimorphism of by Lemma 2.4.

Theorem 3.7. Let be a category with inverse limits. Let be an object of and let be a morphism in pro-. If is a strong epimorphism of , then is a strong epimorphism of pro-.

Proof. Suppose that is a strong epimorphism. Suppose that the following diagram is commutative in pro- with , objects in . Note that the following diagram is commutative in pro-. Thus, is commutative in . But is a strong epimorphism. Therefore, there is such that . Hence, is a strong epimorphism of pro-.

Remark 3.8. Note that if is a morphism in pro-, then we must assume that is uniformly movable for this theorem to hold and this result is Corollary .4 in .

Proposition 3.9. Let be an object of pro-. Then the following conditions on are equivalent.(i)There is a strong monomorphism , where is an object of .(ii) is a strong monomorphism of pro- for some .(iii)There is such that is a strong monomorphism of pro- for all .

Proof. (i)(ii) Let be a representative of . Thus, . But is a strong monomorphism. Hence, is a strong monomorphism of pro- by Lemma 2.4.
(ii)(iii) For all , we have . Hence, is a strong monomorphism of pro- by Lemma 2.4.
(iii)(i) Put . Hence, the result holds.

The following result is Proposition .2 in .

Proposition 3.10. Let be a category with inverse limits. Then if is an object of pro-, then the following conditions on are equivalent. (i)There is a strong epimorphism , where is an object of .(ii) is uniformly movable.

Theorem 3.11. Suppose that is a level morphism of pro- where is a category with direct sums such that each is a strong monomorphism of and each is a strong epimorphism of . If is an isomorphism of pro-, then there is such that is isomorphism of for all .

Proof. Assume that is an isomorphism of pro-. By Corollary 3.4, is a strong monomorphism of for some . By Lemma 3.3, is a strong epimorphism of for each . Since is a strong monomorphism of for some , we have that is a monomorphism by Lemma 2.3. Thus, there is such that is a monomorphism of for all by Corollary .9 in . Therefore, is a strong epimorphism and a monomorphism. Hence, is isomorphism of for each by Theorem 2.6.

1. J. Dydak and F. R. Ruiz del Portal, “Isomorphisms in pro-categories,” Journal of Pure and Applied Algebra, vol. 190, no. 1–3, pp. 85–120, 2004.
2. S. Mardešić and J. Segal, Shape Theory, vol. 26 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1982. View at: MathSciNet
3. J. Dydak and F. R. Ruiz del Portal, “Monomorphisms and epimorphisms in pro-categories,” Topology and Its Applications, vol. 154, no. 10, pp. 2204–2222, 2007.

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