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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 294812, 8 pages
http://dx.doi.org/10.1155/2010/294812
Research Article

On Strong Monomorphisms and Strong Epimorphisms

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 10 March 2010; Accepted 5 May 2010

Academic Editor: Mihai Putinar

Copyright © 2010 M. A. Al Shumrani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1] 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

294812.fig.001
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 [1] and for more details see [2].

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

294812.fig.002

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

294812.fig.001
one may find and representatives of and of such that
294812.fig.004
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 [1].

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

294812.fig.001
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

294812.fig.001
is a commutative diagram in pro- with , objects in . By Lemma 2.1, we may find and representatives of and of such that
294812.fig.004
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

294812.fig.001
is a commutative diagram in pro- with , objects in . By Lemma 2.1, we may find and representatives of and of such that
294812.fig.004
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

294812.fig.005
is commutative in pro- with , objects in . We may find , , and representative of such that the following diagram
294812.fig.006
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

294812.fig.005
is commutative in pro- with , objects in . We may find , , and representative of such that the following diagram
294812.fig.006
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

294812.fig.007
is commutative in pro- with , objects in . Note that the following diagram
294812.fig.008
is commutative in pro- . Thus,
294812.fig.009
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 [1].

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 [1].

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 [3]. Therefore, is a strong epimorphism and a monomorphism. Hence, is isomorphism of for each by Theorem 2.6.

References

  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet