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On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories
Ya. A. Kopylov
Yaroslav A. Kopylov, https://orcid.org/0000-0002-0343-4424, Sobolev Institute of Mathematics, 4 Ac. Koptyuga Ave., Novosibirsk 630090, Russia, yakop@math.nsc.ru
As is well known, many important additive categories in functional analysis and algebra are not abelian. Many classical diagram assertions valid in abelian categories fail in more general additive categories without additional assumptions concerning the properties of the morphisms of the diagrams under consideration. This in particular applies to the so-called Snake Lemma, or the Ker-Coker-sequence. We obtain a theorem about a diagram generalizing the classical situation of the Snake Lemma in the context of categories semi-abelian in the sense of Palamodov. It is also known that, already in P-semi-abelian categories, not all kernels (respectively, cokernels) are semi-stable, that is, stable under pushouts (respectively, pullbacks). We prove a proposition showing how non-semi-stable kernels and cokernels can arise in general preabelian categories.
Keywords: P-semi-abelian category, strict morphism, semi-stable kernels and cokernels, Snake Lemma, Ker-sequence, Coker-sequence.
Received: 15.12.2019 / Accepted: 23.03.2020 / Published: 30.11.2020
This is an open access article distributed under the terms of Creative Commons Attribution License (CC-BY 4.0)
DOI: https://doi.org/10.18500/1816-9791-2020-20-4-434-443
INTRODUCTION
As is well known, many classical diagram assertions valid in abelian categories fail in more general additive and nonadditive categories without additional assumptions concerning the properties of the morphisms of the diagrams under consideration. This in particular applies to the so-called Snake Lemma, or the Ker-Coker-sequence (see, for example, [1,2] or [3]). It is natural to expect that possible generalizations of the Snake Lemma in the non-abelian setting would also require additional conditions on the morphisms of the diagram under consideration.
In the present article, we consider a diagram of the form
Ao Bo Co
в
Y
(1)
A i -> Bi -> Ci,
Ф1
where = 0 and = 0, in P-semi-abelian categories, a class of additive categories with kernels and cokernels which appeared under different names in the 1960s in the works of Romanian mathematicians (see [4]) and were studied in more detail by D. A. Raikov (under the name of "preabelian") in [5] and V. P. Palamodov in [6].
In [7, Corollary A2], Nomura proved an assertion about the exactness of the Ker- and Coker-sequences corresponding to a diagram of the form (1) in a Puppe exact category, that is, informally speaking, in an "abelian category without additivity". In [2], we proved a version of Nomura's assertion for quasi-abelian categories. It turned out that
a
an analog of this assertion also holds for the larger class of P-semi-abelian categories. This is the main result of the present article.
The article is organized as follows.
In Section 1., we give the necessary definitions and recall some basic facts. In Section 2., we discuss one way for obtaining non-semi-stable kernels and cokernels in a pre-abelian category. In Section 4., we prove the above mentioned main result on the exactness of the Ker- and Coker-sequences (Theorem 1).
1. PREABELIAN AND P-SEMI-ABELIAN CATEGORIES
We consider preabelian categories, i.e., additive categories satisfying the following axiom.
Axiom 1. Each morphism has a kernel and a cokernel.
We denote an arbitrary kernel (cokernel) of a by ker a (coker a) and the corresponding object by Ker a (Coker a); the equality a = ker b (a = coker b) means that a is a kernel of b (a is a cokernel of b).
In a preabelian category, every morphism a admits a canonical decomposition a = (im a)a(coim a), where im a = ker coker a, coim a = coker ker a. A morphism a is called strict if a is an isomorphism.
We write a | p if a = ker p and p = coker a.
Lemma 1. [4,8-10] The following assertions hold in a preabelian category:
(i) a strict monomorphism is the same as a kernel; a strict epimorphism is the same as a cokernel;
(ii) a is a kernel ^^ a = im a, a is a cokernel ^^ a = coim a;
(iii) a morphism a is strict if and only if it is representable in the form a = aia0 with a0 a cokernel and a1 a kernel; in every such representation, a0 = coim a and
ai = im a;
(iv) the relations ker a = ker coim a and coker a = coker im a hold for every mor-phism a.
A preabelian category is abelian if and only if a is an isomorphism for every a, that is, if and only if every morphism is strict.
We call a sequence ... A B A ... in an additive category semi-exact at the term
B if ba = 0. A sequence ... A B A ... in a preabelian category is said to be exact at the term B if im a = ker b. Lemma 1 (iv), which is Lemma 1 of [10], implies that the sequence is exact at the term B if and only if coker a = coim b.
A preabelian category is called P-semi-abelian or semi-abelian (in the sense of Palamodov) [6,11] if it satisfies
Axiom 2. For every morphism a, a is a bimorphism, that is, a monomorphism and an epimorphism.
If the morphism a is a monomorphism (an epimorphism) for every a then, following Rump [11, p. 167], we call the preabelian category left semi-abelian (right semi-abelian).
Note that a preabelian category is right (left) semi-abelian if and only if the composition of any two kernels (respectively, cokernels) in it is again a kernel (respectively, a cokernel), which is equivalent to the statement that if gf is a kernel then f is a kernel (if gf is a cokernel then g is a cokernel). For a detailed characterization of P-semi-abelianity, the reader is referred to [12].
A preabelian category A is called left quasi-abelian (or left almost abelian, see [11]) if it satisfies
Axiom 3. If a commutative square
C
A
D
f B
(2)
is a pullback then f is a cokernel g is a cokernel.
Dually, a preabelian category A is called right quasi-abelian (or right almost abelian [11]) if it satisfies
Axiom 3*. If (2) is a pushout then g is a kernel f is a kernel.
A left and right quasi-abelian category is referred to as quasi-abelian [13] (semi-abelian in the sense of Raikov [5], or almost abelian [11]).
As is well-known [5,9,11,13], every quasi-abelian category is P-semi-abelian. Kuz'minov and Cherevikin [9, Theorem 2] and later Rump [11, Proposition 3] noticed that a P-semi-abelian category is quasi-abelian if and only if it is left or right quasi-abelian. In 2006, Bonet and Dierolf [14] constructed an example of a pullback violating Axiom 3 in the category Bor of bornological locally convex spaces, thus proving that it is not quasi-abelian. Later Rump [15] gave an algebraic example of a P-semi-abelian but not quasi-abelian category. In [16], he carried out a thorough study of P-semi-abelian subcategories of quasi-abelian categories and proved that Bor and the category Bar of barreled locally convex spaces are P-semi-abelian but not quasi-abelian. Later in [17] Wengenroth explained that the non-semi-stability of cokernels in Bor is not rare.
2. SEMI-STABLE KERNELS AND COKERNELS IN A PREABELIAN CATEGORY
If, for a cokernel f in a preabelian category, in every pullback (2), g is a cokernel (for a kernel g in a preabelian category, in every pushout (2), f is a kernel) then f is called a semi-stable cokernel (g is called a semi-stable kernel).
We recall some basic properties of semi-stable kernels and cokernels (following from [18, Propositions 5.11 and 5.12]).
Lemma 2. The following hold in a preabelian category:
(i) if gf is a semi-stable kernel then so is f, if gf is a semi-stable cokernel then so is g;
(ii) if f and g are semi-stable kernels and gf is defined then gf is a semi-stable kernel; if f and g are semi-stable cokernels and gf is defined then gf is a semi-stable cokernel;
(iii) a pushout of a semi-stable kernel is a semi-stable kernel; a pullback of a semi-stable cokernel is a semi-stable cokernel.
The following Lemma is due to Kuz'minov and Cherevikin [9, Lemma 2].
Lemma 3. Let
E ^^ C —^ A
E
g
B
Ф D
g
be a commutative diagram in a preabelian category. Assume that fa = 0, f is an epimorphism, and p = coker p. Then the pg = ^f is a pushout. The dual assertion also holds.
The idea of the following assertion, allowing to construct examples of non-semi-stable kernels and cokernels, comes from the proof of [9, Theorem 1(3)].
Proposition 1. Let a be a morphism in a preabelian category for which a is not an epimorphism. Then im a is a non-semi-stable kernel.
By duality, if a morphism a is such that a is not a monomorphism then coim a is a non-semi-stable cokernel.
Proof. Let a : A 4 B be a morphism such that a is not epic and p is the morphism of the cokernels of the rows in the commutative square
A a coim a c
A -y B
a
Then the commutative diagram
4 a coim a ^ coker a ^ ■■ _
A -> C -> Coker a
A -> B -> Coker a
a coker a
satisfies the hypothesis of Lemma 3. Thus, p coker a = (coker a) im a(= 0) is a pushout. Since coker a is epic, the equality p coker a = 0 implies that p = 0. In particular, since coker a = 0, we infer that p is not a monomorphism and, thus, im a is a non-semi-stable kernel.
The second assertion of the lemma is obtained from the first by duality. □
3. THE LEFT AND RIGHT HOMOLOGY OBJECTS
Suppose first that the ambient category is preabelian. Given a sequence of the form
A 4 B 4 C (3)
such that ^p = 0, there are a natural morphism a : A 4 Ker ^ such that p = (ker and a natural morphism t : Coker p 4 C such that ^ = t coker p.
Definition 1. Call H- (B) = H- (B,p» = Coker a and H+ (B) = H+ (B,p>) = = Ker t the left and right homology objects of (3) at the term B.
It is classical that these two notions coincide for abelian categories (see, for example, [19]). This remains valid for quasi-abelian categories [20] and even in the nonadditive setting of homological categories in the sense of Grandis [3].
If the ambient category is P-semi-abelian then there is an equivalent description of the left and right homology objects. Consider the natural morphisms r : Im p 4 Ker ^
im a
P
and r' : Coker p ^ Coim ф. Then coker r = coker a and ker r' = ker т, and hence H- (B, p, ф) = Coker r and H+ (В,р,ф) = Ker r'.
As was shown in [20], in a preabelian category, there is a unique morphism m : H-(B) ^ H+ (B) such that
(ker т)m coker a = (coker p) (ker ф). (4)
The following assertion holds ([21, Lemma 7], [22, Proposition 1]).
Lemma 4. (i) Let the ambient category be P-semi-abelian. The morphism m : H- (B) ^ H+ (B) is a bimorphism. If ker ф is a semi-stable kernel or coker p is a semi-stable cokernel then m is an isomorphism.
(ii) Let the ambient category be preabelian. If ker ф is a semi-stable kernel then m is a semi-stable kernel and if coker p is a semi-stable cokernel then m is a semi-stable cokernel. Thus, if both conditions are fulfilled then m is an isomorphism.
Examples of situations when the left and right homology objects do not coincide can be obtained from the following observation [22, Lemma 4]: Let '
P ^^ F
E -► G
u
be a pullback in a P-semi-abelian category such that v is a kernel, u is a cokernel, and u' is not a cokernel. Let H-(E) and H+ (E) be the left and right homology objects of the sequence
ker u t—r coker v' T
K —> E —> L
at the term E. Then the canonical morphism m : H- (E) ^ H+ (E) is not an isomorphism.
As was shown by Wengenroth (see [17]), such pullbacks are not unusual, for example, in the P-semi-abelian category of bornological locally convex spaces and arise when non-a-regular inductive limits in the sense of Makarov [23] are considered.
4. A GENERALIZATION OF THE SNAKE LEMMA
Consider a diagram of the form (1) in a P-semi-abelian category. As in the case of the classical diagram of the Snake Lemma, diagram (1) gives rise to a Ker-sequence
Ker a -4 Ker p Ker 7 (5)
with (e = 0 and a Coker-sequence
Coker a -— Coker p Coker 7 (6)
with Or = 0.
For diagram (1), we have a commutative diagram of natural morphisms
Ao Ker
в
Ai -> Ker фх.
pi
(7)
a
Here / : Ker 4 Ker ^ is the natural morphism of the kernels of the rows of the square
Bo
Bi
■> Co
Vl
Y
Ci,
and p0 and p1 are uniquely defined by = (ker )p0 and = (ker )p1. We thus have a natural morphism x- : H-(B0) 4 H-(B1) of the cokernels of the rows in (7) such that
X- coker p0 = (coker p1 )/•
In the dual manner, we have a commutative diagram of natural morphisms
Coker ———4 C0
P Y
Coker (p1 -> C1,
ni
where 3 is the morphism of the cokernels of the rows of the square
A ——4 B0
(8)
A1
—1
P
B1
and n0 and n1 are uniquely defined by = n0 coker and = n1 coker This gives a natural morphism x+ : H+ (B0) 4 H+ (B1) of the kernels of the rows in (8) such that
3 ker n0 = (ker )x+ • In [2], we proved the following assertion (Lemma 10):
Suppose in (1) that = ker = coker ). Then e = ker Z (respectively,
9 = coker t ).
We will prove the following generalization of this assertion, which is a P-semi-abelian version of [7, Corollary A2] and [2, Theorem 4].
Theorem 1. The following hold:
(1) if, in a diagram of the form (1) in a P-semi-abelian category, the morphism
is strict and and x- : H- (B0) 4 H- (B1) are monomorphisms then sequence (5) is exact at the term Ker
(2) if, in a diagram of the form (1) in a P-semi-abelian category, the morphism
is strict and and x+ : H+ (B0) 4 H+(B1) are epimorphisms then sequence (6) is exact at the term Coker /.
Proof. 1. Take a morphism x : X 4 Ker / with Zx = 0- Show that x = (im e)x for some unique X. We may assume without loss of generality that x = im x.
P
Since 0 = (ker 7)(x = ^0(ker p)x, there is a morphism z : X ^ Ker such that
(ker p)x = (ker ^0)z. Further, (ker ^i)pz = p(ker )z = p(ker p)x = 0 and ker fa is a mo-
nomophism; therefore, pz = 0. Let r0 be the natural morphism Im p0 ^ Ker such that
im p0 = (ker )r0. Since x(coker p0)z = (coker pi)pz = 0 and x is a monomorphism, we get (coker p0)z = 0. Note that r0 = ker(coker p0) = im p0. Hence, z = r0p for some p.
Let ri be the natural morphism Im pi ^ Ker fa such that im pi = (ker fa)ri. As was observed in Section 3., coker p0 = coker r0 and coker pi = coker ri. Let s : Im p0 ^ Im pi be the natural morphism of the kernels of the rows of the square
Bf
Bi
coker ^q
coker
> Coker p0
в
> Coker pi.
We have the commutative diagram
Im po
Im pi
ro
Ker ф0
coker ro
r1
Ker ф1
coker r1
> H- (Bo) = Coker ro
> H- (Bi ) = Coker ri.
We infer ^ ^
ri sp = pr0 p = pp = 0.
Since ri is a monomorphism, this gives sp = 0.
Represent p0 in the form p0 = (im p0)p'0. Since p0 is strict, p0 is a cokernel. Consider the pullback
Y
У2
y 1
Ao
X
Im p0.
^0
Then y2 is an epimorphism. Observe that pim p0 = (im pi)s. We infer
pi ayi = pp0 yi = p (im p0)p0) yi = p (im P0 )py2 = (im pi )spy2 = 0.
But pi is a monomorphism; therefore, ayi = 0. Hence, there exists a morphism y : Y ^ Ker a with the property yi = (ker a)y. Then
(ker p )xy2 = (ker ^0 )zy2 = (ker ^0 )r0 py2 = = (im p0 )py2 = (im p0)p0 yi = p0 yi = p0 (ker a)y = (ker p )ey.
Since ker p is a monomorphism, this yields
xy2 = sy.
(9)
Let e = (ime)e'. In (9), x is a kernel, y2 is an epimorphism; therefore, x = im(xy2) = = (im e)(im(e'y)). We can take x = im(e'y). The condition x = (im e)x defines x uniquely because im e is a monomorphism.
в
x
>
u
Assertion 1 of the theorem is proved. Assertion 2 results from it by duality.
The theorem is proved. □
Let us formulate explicitly what Theorem 1 means for the categories Bor and Bar of bornological and barreled locally convex spaces respectively.
We say that a sequence A 4 B 4 C in either of these categories such that ^ = 0 is approximately exact at the term B whenever the closure of the range of the operator ^ coincides with the kernel of ^. It is not hard to see that our categorical exactness is in fact approximate exactness in this sense. Moreover, a continuous linear operator between bornological or barreled spaces is strict if and only if it has closed range and is an open mapping onto its range.
Corollary. Consider a commutative diagram of the form (1) constituted by bornological or barreled locally convex spaces and continuous linear operators. The following hold:
(1) if in (1) the operator has closed range and is open onto its range and x- : H-(B0) 4 H-(B1) are infective then the corresponding left sequence (5) is approximately exact at the term Ker /;
(2) if in (1) the operator has closed range and is open onto its range and and x+ : H+(B0) 4 H+ (B1) have dense range then the corresponding sequence (6) is approximately exact at the term Coker /.
Acknowledgements: The author is indebted to the referee for valuable remarks which substantially improved the exposition. The work was carried out in the framework of the State Contract of the Sobolev Institute of Mathematics (project No. 0314-20190006).
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Cite this article as:
Kopylov Ya. A. On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 4, pp. 434-443. DOI: https://doi.org/10.18500/1816-9791-2020-20-4-434-443
УДК 512.66:517.982.2
О некоторых диаграммных утверждениях в предабелевых и Р-полуабелевых категориях
Я. А. Копылов
Копылов Ярослав Анатольевич, кандидат физико-математических наук, Институт математики им. С. Л. Соболева СО РАН, Россия, 630090, г. Новосибирск, просп. Ак. Коптюга, д. 4, yakop@math.nsc.ru
Как известно, многие важные аддитивные категории функционального анализа и алгебры неабелевы. Многие классические диаграммные утверждения, справедливые в абелевых категориях, оказываются неверны в более общих аддитивных категориях без дополнительных предположений о свойствах морфизмов рассматриваемых диаграмм. Это, в частности, относится к так называемой лемме о змее, или Ker-Coker-последовательности.
В статье получена теорема о диаграмме, обобщающей классическую ситуацию леммы о змее в контексте категорий, полуабелевых в смысле Паламодова. Известно также, что уже в P-полуабелевых категориях не все ядра (соответственно, коядра) полустабильны, т. е. стабильны относительно универсальных (соответственно, коуниверсальных) квадратов. Мы доказываем предложение, показывающее, как неполустабильные ядра и коядра могут возникнуть в общих предабелевых категориях.
Ключевые слова: P-полуабелева категория, строгий морфизм, полустабильные ядра и коядра, лемма о змее, Ker-последовательность, Coker-последовательность.
Поступила в редакцию: 15.12.2019 / Принята: 23.03.2020 / Опубликована: 30.11.2020
Статья опубликована на условиях лицензии Creative Commons Attribution License (CC-BY4.0)
Благодарности. Автор благодарит рецензента за ценные замечания, которые помогли существенно улучшить работу. Работа выполнена в рамках государственного задания ИМ СО РАН (проект № 0314-2019-0006).
Образец для цитирования:
Kopylov Ya. A. On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories [Копылов Я. А. О некоторых диаграммных утверждениях в предабелевых и P-полуабелевых категориях] // Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2020. Т. 20, вып. 4. С. 434-443. DOI: https://doi.org/10.18500/1816-9791-2020-20-4-434-443
