In
mathematics, specifically in
category theory, a pre-abelian category is an
additive category that has all
kernels
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
and
cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
# C is
preadditive, that is
enriched over the
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left a ...
of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s (equivalently, all
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s in C are
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s and composition of
morphisms is
bilinear);
# C has all
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
products
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
(equivalently, all finite
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
s); note that because C is also preadditive, finite products are the same as finite coproducts, making them
biproduct
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide fo ...
s;
# given any morphism ''f'': ''A'' → ''B'' in C, the
equaliser of ''f'' and the
zero morphism In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
Definitions
Suppose C is a category, and ''f'' : ''X'' → ''Y'' is a morphism in C. The ...
from ''A'' to ''B'' exists (this is by definition the kernel of ''f''), as does the
coequaliser (this is by definition the cokernel of ''f'').
Note that the zero morphism in item 3 can be identified as the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
Hom(''A'',''B''), which is an abelian group by item 1; or as the unique morphism ''A'' → 0 → ''B'', where 0 is a
zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
, guaranteed to exist by item 2.
Examples
The original example of an additive category is the category Ab of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s.
Ab is preadditive because it is a
closed monoidal category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.
A classic exa ...
, the biproduct in Ab is the finite
direct sum, the kernel is inclusion of the
ordinary kernel from group theory and the cokernel is the quotient map onto the
ordinary cokernel from group theory.
Other common examples:
* The category of (left)
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'', in particular:
** the category of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K''.
* The category of (
Hausdorff)
abelian topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s.
* The category of
Banach spaces.
* The category of
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s.
* The category of (Hausdorff)
bornological spaces.
These will give you an idea of what to think of; for more examples, see
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
(every abelian category is pre-abelian).
Elementary properties
Every pre-abelian category is of course an
additive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that hold specifically because of the existence of kernels and cokernels.
Although kernels and cokernels are special kinds of
equalisers and
coequalisers, a pre-abelian category actually has ''all'' equalisers and coequalisers.
We simply construct the equaliser of two morphisms ''f'' and ''g'' as the kernel of their difference ''g'' − ''f''; similarly, their coequaliser is the cokernel of their difference.
(The alternative term "difference kernel" for binary equalisers derives from this fact.)
Since pre-abelian categories have all finite
products
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
and
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
s (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of
category theory, they have all finite
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
and
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
s.
That is, pre-abelian categories are
finitely complete.
The existence of both kernels and cokernels gives a notion of
image and
coimage In algebra, the coimage of a homomorphism
:f : A \rightarrow B
is the quotient
:\text f = A/\ker(f)
of the domain by the kernel.
The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies.
...
.
We can define these as
:im ''f'' := ker coker ''f'';
:coim ''f'' := coker ker ''f''.
That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel.
Note that this notion of image may not correspond to the usual notion of image, or
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
, of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, even assuming that the morphisms in the category ''are'' functions.
For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the ''
closure'' of the range of the function.
For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary set-theoretic concept.
In many common situations, such as the category of
sets, where images and coimages exist, their objects are
isomorphic.
Put more precisely, we have a factorisation of ''f'': ''A'' → ''B'' as
:''A'' → ''C'' → ''I'' → ''B'',
where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle (called the ''parallel'' of ''f'') is an isomorphism.
In a pre-abelian category, ''this is not necessarily true''.
The factorisation shown above does always exist, but the parallel might not be an isomorphism.
In fact, the parallel of ''f'' is an isomorphism for every morphism ''f''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
the pre-abelian category is an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
.
An example of a non-abelian, pre-abelian category is, once again, the category of topological abelian groups.
As remarked, the image is the inclusion of the ''closure'' of the range; however, the coimage is a quotient map onto the range itself.
Thus, the parallel is the inclusion of the range into its closure, which is not an isomorphism unless the range was already
closed.
Exact functors
Recall that all finite
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
and
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
s exist in a pre-abelian category.
In general
category theory, a functor is called ''
left exact'' if it preserves all finite limits and ''
right exact'' if it preserves all finite colimits. (A functor is simply ''exact'' if it's both left exact and right exact.)
In a pre-abelian category, exact functors can be described in particularly simple terms.
First, recall that an
additive functor
In mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.
That is, an Ab-category C is a category such that
every hom- ...
is a functor ''F'': C → D between
preadditive categories
In mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.
That is, an Ab-category C is a category such that
every h ...
that acts as a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
on each
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
.
Then it turns out that a functor between pre-abelian categories is left exact
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels.
Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages.
Exact functors are most useful in the study of
abelian categories
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
, where they can be applied to
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s.
Maximal exact structure
On every pre-abelian category
there exists an
exact structure that is maximal in the sense that it contains every other exact structure. The exact structure
consists of precisely those kernel-cokernel pairs
where
is a semi-stable kernel and
is a semi-stable cokernel. Here,
is a semi-stable kernel if it is a kernel and for each morphism
in the
pushout
A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is consider ...
diagram
the morphism
is again a kernel.
is a semi-stable cokernel if it is a cokernel and for every morphism
in the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
diagram
the morphism
is again a cokernel.
A pre-abelian category
is
quasi-abelian if and only if all kernel-cokernel pairs form an exact structure. An example for which this is not the case is the category of (Hausdorff) bornological spaces.
The result is also valid for additive categories that are not pre-abelian but
Karoubian.
[Crivei, 2012, p. 445.]
Special cases
* An ''
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
'' is a pre-abelian category such that every
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
and
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
.
* A ''
quasi-abelian category'' is a pre-abelian category in which kernels are stable under pushouts and cokernels are stable under pullbacks.
* A ''
semi-abelian category'' is a pre-abelian category in which for each morphism
the induced morphism
is always a monomorphism and an epimorphism.
The pre-abelian categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category. Pre-abelian categories that are not abelian appear for instance in functional analysis.
Citations
References
*
Nicolae Popescu
Nicolae Popescu (; 22 September 1937 – 29 July 2010) was a Romanian mathematician and professor at the University of Bucharest. He also held a research position at the Institute of Mathematics of the Romanian Academy, and was elected corresp ...
; 1973;
Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print
* Dennis Sieg and Sven-Ake Wegner, Maximal exact structures on additive categories, Math. Nachr. 284 (2011), 2093–2100.
* Septimu Crivei, Maximal exact structures on additive categories revisited, Math. Nachr. 285 (2012), 440–446.
Additive categories