On Strong Monomorphisms and Strong Epimorphisms

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 𝑓∶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 diagram

with 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].

Let 𝒞 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=(𝑋_𝛼,𝑝^𝛽_𝛼,𝐴), consisting of a directed set 𝐴, called the index set (from now onward it will be denoted by 𝐼(𝑋)), of 𝒞 objects 𝑋_𝛼 for each 𝛼∈𝐼(𝑋), called the terms of 𝑋, and of 𝒞 morphisms 𝑝^𝛽_𝛼∶𝑋_𝛽→𝑋_𝛼 for each related pair 𝛼<𝛽, called the bonding morphisms of 𝑋 (from now onward it will be denoted by 𝑝^𝛽_𝛼(𝑋)).

If 𝑃 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(𝑋_𝛼,𝑃), 𝛼∈𝐼(𝑋), and so 𝑓 can be represented by 𝑔∶𝑋_𝛼→𝑃. Note that the morphism from 𝑋 to 𝑋_𝛼 represented by the identity 𝑋_𝛼→𝑋_𝛼 is called the projection morphism and denoted by 𝑝(𝑋)_𝛼.

If 𝑋 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 diagram

commutes.

Recall that an object 𝑋 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∶𝑋_𝛽→𝑋 satisfying 𝑝(𝑋)_𝛼∘𝑟=𝑝(𝑋)^𝛽_𝛼 where 𝑝(𝑋)_𝛼∶𝑋→𝑋_𝛼 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 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{𝛼∈𝐼(𝑋)} is a level morphism of pro-𝒞. For any commutative diagram with 𝑃, 𝑄 objects in 𝒞

one may find 𝛼∈𝐼(𝑋) and representatives 𝑎_𝛼∶𝑋_𝛼→𝑃 of 𝑎 and 𝑏_𝛼∶𝑌_𝛼→𝑄 of 𝑏 such that

is commutative.

Proof. Choose representatives 𝑢∶𝑋_𝛽→𝑃 of 𝑎 and 𝑣∶𝑌_𝛽→𝑄 of 𝑏. Since 𝑔∘𝑢∘𝑝(𝑋)_𝛽=𝑔∘𝑎=𝑏∘𝑓=𝑣∘𝑝(𝑌)_𝛽∘𝑓=𝑣∘𝑓_𝛽∘𝑝(𝑋)_𝛽, there is 𝛼>𝛽 such that 𝑔∘𝑢∘𝑝(𝑋)^𝛼_𝛽=𝑣∘𝑓_𝛽∘𝑝(𝑋)^𝛼_𝛽. Put 𝑎_𝛼=𝑢∘𝑝(𝑋)^𝛼_𝛽 and 𝑏_𝛼=𝑣∘𝑝(𝑌)^𝛼_𝛽.

Recall that a morphism 𝑓∶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 morphism 𝑓∶𝑋→𝑌 in pro-𝒞 is called a strong monomorphism (strong epimorphism, resp.) if for every commutative diagram

with 𝑃, 𝑄 objects in 𝒞, there is a morphism ℎ∶𝑌→𝑃 such that ℎ∘𝑓=𝑎 (𝑔∘ℎ=𝑏, resp.).

Note that if 𝑋 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. If 𝑓 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. If 𝑔∘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. Let 𝑓∶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. Let 𝑓∶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 that 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{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 each 𝛼∈𝐼(𝑋), there is a morphism 𝑢_𝛼∶𝑌→𝑋_𝛼 such that 𝑢_𝛼∘𝑓=𝑝(𝑋)_𝛼 (𝑓_𝛼∘𝑢_𝛼=𝑝(𝑌)_𝛼, resp.).(iii)For each 𝛼∈𝐼(𝑋), there is 𝛽∈𝐼(𝑋), 𝛽>𝛼 and a morphism 𝑔_𝛼,𝛽∶𝑌_𝛽→𝑋_𝛼 such that 𝑔_𝛼,𝛽∘𝑓_𝛽=𝑝(𝑋)^𝛽_𝛼 (𝑓_𝛼∘𝑔_𝛼,𝛽=𝑝(𝑌)^𝛽_𝛼, 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 that 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{𝛼∈𝐼(𝑋)} is a level morphism of pro-𝒞. If each 𝑓_𝛼 is a strong monomorphism of 𝒞 for each 𝛼, then 𝑓 is a strong monomorphism of pro-𝒞.

Proof. Suppose that 𝑓_𝛼 is a strong monomorphism of 𝒞 for each 𝛼. Suppose that

is a commutative diagram in pro-𝒞 with 𝑃, 𝑄 objects in 𝒞. By Lemma 2.1, we may find 𝛼∈𝐼(𝑋) and representatives 𝑎_𝛼∶𝑋_𝛼→𝑃 of 𝑎 and 𝑏_𝛼∶𝑌_𝛼→𝑄 of 𝑏 such that

is commutative, where for 𝛼>𝛽, 𝑎_𝛼=𝑢∘𝑝(𝑋)^𝛼_𝛽, 𝑢∶𝑋_𝛽→𝑃, and 𝑎=𝑢∘𝑝(𝑋)_𝛽. Thus, there is ℎ∶𝑌_𝛼→𝑃 such that ℎ∘𝑓_𝛼=𝑎_𝛼 since 𝑓_𝛼 is a strong monomorphism of 𝒞. There is 𝑐=ℎ∘𝑝(𝑌)_𝛼∶𝑌→𝑃. But 𝑐∘𝑓=ℎ∘𝑝(𝑌)_𝛼∘𝑓=ℎ∘𝑓_𝛼∘𝑝(𝑋)_𝛼=𝑎_𝛼∘𝑝(𝑋)_𝛼=𝑢∘𝑝(𝑋)^𝛼_𝛽∘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 that 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{𝛼∈𝐼(𝑋)} is a level morphism of pro-𝒞. If each 𝑓_𝛼 is a strong epimorphism of 𝒞 for each 𝛼, then 𝑓 is a strong epimorphism of pro-𝒞.

Proof. Suppose that 𝑓_𝛼 is a strong epimorphism of 𝒞 for each 𝛼. Suppose that

is a commutative diagram in pro-𝒞 with 𝑃, 𝑄 objects in 𝒞. By Lemma 2.1, we may find 𝛼∈𝐼(𝑋) and representatives 𝑎_𝛼∶𝑋_𝛼→𝑃 of 𝑎 and 𝑏_𝛼∶𝑌_𝛼→𝑄 of 𝑏 such that

is commutative, where for 𝛼>𝛽, 𝑏_𝛼=𝑣∘𝑝(𝑌)^𝛼_𝛽, 𝑣∶𝑌_𝛽→𝑄, and 𝑏=𝑣∘𝑝(𝑌)_𝛽. Thus, there is ℎ∶𝑌_𝛼→𝑃 such that 𝑔∘ℎ=𝑏_𝛼 since 𝑓_𝛼 is a strong epimorphism of 𝒞. There is 𝑐=ℎ∘𝑝(𝑌)_𝛼∶𝑌→𝑃. But 𝑔∘𝑐=𝑔∘ℎ∘𝑝(𝑌)_𝛼=𝑏_𝛼∘𝑝(𝑌)_𝛼=𝑣∘𝑝(𝑌)^𝛼_𝛽∘𝑝(𝑌)_𝛼=𝑣∘𝑝(𝑌)_𝛽=𝑏. Hence, 𝑓 is a strong epimorphism of pro-𝒞.

Lemma 3.3. Suppose that 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{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 𝑓_𝛼 is a strong epimorphism of 𝒞 for each 𝛼∈𝐼(𝑋).

Proof. Suppose that 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). By Proposition 2.7, we have for each 𝛼∈𝐼(𝑋), 𝑓_𝛼∘𝑢_𝛼=𝑝(𝑌)_𝛼 where 𝑢_𝛼∶𝑌→𝑋_𝛼. Since 𝑝(𝑌)_𝛼 is a strong epimorphism, we have that 𝑓_𝛼 is a strong epimorphism of 𝒞 by Lemma 2.4.

Corollary 3.4. Suppose that 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{𝛼∈𝐼(𝑋)} is a level morphism of pro-𝒞. If each 𝑝(𝑋)^𝛽_𝛼 is a strong monomorphism of 𝒞 and 𝑓 is a strong monomorphism of pro-𝒞, then 𝑓_𝛽 is a strong monomorphism of 𝒞 for some 𝛽∈𝐼(𝑋).

Proof. Assume that each 𝑝(𝑋)^𝛽_𝛼 is a strong monomorphism of 𝒞 and 𝑓 is a strong monomorphism of pro-𝒞. By Proposition 2.7, we have 𝑔_𝛼,𝛽∘𝑓_𝛽=𝑝(𝑋)^𝛽_𝛼 for some 𝛽∈𝐼(𝑋) where 𝑔_𝛼,𝛽∶𝑌_𝛽→𝑋_𝛼. Since 𝑝(𝑋)^𝛽_𝛼 is a strong monomorphism, we have that 𝑓_𝛽 is a strong monomorphism of 𝒞 by Lemma 2.4.

Proposition 3.5. If 𝑝(𝑋)^𝛽_𝛼 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 that 𝑝(𝑋)^𝛽_𝛼 is a strong monomorphism of 𝒞 for each 𝛽>𝛼. Assume that the following diagram

is commutative in pro-𝒞 with 𝑃, 𝑄 objects in 𝒞. We may find 𝛽∈𝐼(𝑋), 𝛽>𝛼, and representative 𝑎_𝛽∶𝑋_𝛽→𝑃 of 𝑎 such that the following diagram

is commutative. But 𝑝(𝑋)^𝛽_𝛼 is a strong monomorphism of 𝒞. Thus, there is ℎ∶𝑋_𝛼→𝑃 such that ℎ∘𝑝(𝑋)^𝛽_𝛼=𝑎_𝛽. Therefore, ℎ∘𝑝(𝑋)^𝛽_𝛼∘𝑝(𝑋)_𝛽=𝑎_𝛽∘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. Let 𝑋 be an object of pro-𝒞. Then the following conditions on 𝑋 are equivalent. (i)𝑝(𝑋)^𝛽_𝛼 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)^𝛽_𝛼 is a strong epimorphism of 𝒞 for each 𝛽>𝛼. Assume that the following diagram

is commutative in pro-𝒞 with 𝑃, 𝑄 objects in 𝒞. We may find 𝛽∈𝐼(𝑋), 𝛽>𝛼, and representative 𝑎_𝛽∶𝑋_𝛽→𝑃 of 𝑎 such that the following diagram

is commutative. But 𝑝(𝑋)^𝛽_𝛼 is a strong epimorphism of 𝒞. Thus, there is ℎ∶𝑋_𝛼→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)^𝛽_𝛼∘𝑝(𝑋)_𝛽. Hence, 𝑝(𝑋)^𝛽_𝛼 is a strong epimorphism of 𝒞 by Lemma 2.4.

Theorem 3.7. Let 𝒞 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 that lim𝑓 is a strong epimorphism. Suppose that the following diagram

is 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 if 𝑓∶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. Let 𝑋 be an object of pro-𝒞. Then the following conditions on 𝑋 are equivalent.(i)There is a strong monomorphism 𝑓∶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 is 𝛼∈𝐼(𝑋) such that 𝑝(𝑋)_𝛽 is a strong monomorphism of pro-𝒞 for all 𝛽≥𝛼.

Proof. (i)⇒(ii) Let 𝑔∶𝑋_𝛼→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)^𝛽_𝛼∘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 Proposition 4.2 in [1].

Proposition 3.10. Let 𝒞 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 epimorphism 𝑓∶𝑃→𝑋, where 𝑃 is an object of 𝒞.(ii)𝑋 is uniformly movable.

Theorem 3.11. Suppose that 𝑓={𝑓_𝛼∶𝑋_𝛼→𝑌_𝛼}_{𝛼∈𝐼(𝑋)} is a level morphism of pro-𝒞 where 𝒞 is a category with direct sums such that each 𝑝(𝑋)^𝛽_𝛼 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 𝑓_𝛽 is isomorphism of 𝒞 for all 𝛽≥𝛼.

Proof. Assume that 𝑓 is an isomorphism of pro-𝒞. By Corollary 3.4, 𝑓_𝛼 is a strong monomorphism of 𝒞 for some 𝛼∈𝐼(𝑋). By Lemma 3.3, 𝑓_𝛼 is a strong epimorphism of 𝒞 for each 𝛼∈𝐼(𝑋). Since 𝑓_𝛼 is a strong monomorphism of 𝒞 for some 𝛼∈𝐼(𝑋), we have that 𝑓_𝛼 is a monomorphism by Lemma 2.3. Thus, there is 𝛼∈𝐼(𝑋) such that 𝑓_𝛽 is a monomorphism of 𝒞 for all 𝛽≥𝛼 by Corollary 2.9 in [3]. Therefore, 𝑓_𝛽 is a strong epimorphism and a monomorphism. Hence, 𝑓_𝛽 is isomorphism of 𝒞 for each 𝛽≥𝛼 by Theorem 2.6.