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The cokernel of a
linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
vector spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
is the quotient space of the
codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

codomain
of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to the kernels of category theory, hence the name: the kernel is a
subobject In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
of the domain (it maps to the domain), while the cokernel is a
quotient objectIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
of the codomain (it maps from the codomain). Intuitively, given an equation that one is seeking to solve, the cokernel measures the ''constraints'' that must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the ''degrees of freedom'' in a solution, if one exists. This is elaborated in
intuition Intuition is the ability to acquire knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is ...
, below. More generally, the cokernel of a
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
in some
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
(e.g. a
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

homomorphism
between
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
s or a
bounded linear operator In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
between
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s) is an object and a morphism such that the composition is the
zero morphismIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
of the category, and furthermore is
universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a te ...
with respect to this property. Often the map is understood, and itself is called the cokernel of . In many situations in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, such as for
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s,
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s or
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
s, the cokernel of the
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
is the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of by the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of . In
topological In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topological
settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.


Formal definition

One can define the cokernel in the general framework of
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. In order for the definition to make sense the category in question must have
zero morphismIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
s. The cokernel of a
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
is defined as the
coequalizer In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled ...
of and the zero morphism . Explicitly, this means the following. The cokernel of is an object together with a morphism such that the diagram
commutes
commutes
. Moreover, the morphism must be
universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a te ...
for this diagram, i.e. any other such can be obtained by composing with a unique morphism :
As with all universal constructions the cokernel, if it exists, is unique
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
a unique
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
, or more precisely: if and are two cokernels of , then there exists a unique isomorphism with . Like all coequalizers, the cokernel is necessarily an
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
. Conversely an epimorphism is called '' normal'' (or ''conormal'') if it is the cokernel of some morphism. A category is called ''conormal'' if every epimorphism is normal (e.g. the
category of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is conormal).


Examples

In the
category of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, the cokernel of a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

group homomorphism
is the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of by the normal closure of the image of . In the case of
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, since every
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
is normal, the cokernel is just modulo the image of : :\operatorname(f) = H / \operatorname(f).


Special cases

In a
preadditive category In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab. That is, an Ab-catego ...
, it makes sense to add and subtract morphisms. In such a category, the
coequalizer In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled ...
of two morphisms and (if it exists) is just the cokernel of their difference: : \operatorname(f, g) = \operatorname(g - f). In an
abelian category In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...
(a special kind of preadditive category) the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
and
coimageIn Abstract algebra, algebra, the coimage of a homomorphism :f : A \rightarrow B is the quotient group, quotient :\text f = A/\ker(f) of the domain of a function, domain by the kernel (algebra), kernel. The coimage is natural isomorphism, canoni ...
of a morphism are given by :\begin \operatorname(f) &= \ker(\operatorname f), \\ \operatorname(f) &= \operatorname(\ker f). \end In particular, every abelian category is normal (and conormal as well). That is, every
monomorphism In the context of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...
can be written as the kernel of some morphism. Specifically, is the kernel of its own cokernel: :m = \ker(\operatorname(m))


Intuition

The cokernel can be thought of as the space of ''constraints'' that an equation must satisfy, as the space of ''obstructions'', just as the
kernel 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 ...
is the space of ''solutions.'' Formally, one may connect the kernel and the cokernel of a map by the
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
:0 \to \ker T \to V \overset T \longrightarrow W \to \operatorname T \to 0. These can be interpreted thus: given a linear equation to solve, * the kernel is the space of ''solutions'' to the ''homogeneous'' equation , and its dimension is the number of ''degrees of freedom'' in solutions to , if they exist; * the cokernel is the space of ''constraints'' on ''w'' that must be satisfied if the equation is to have a solution, and its dimension is the number of independent constraints that must be satisfied for the equation to have a solution. The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space is simply the dimension of the space ''minus'' the dimension of the image. As a simple example, consider the map , given by . Then for an equation to have a solution, we must have (one constraint), and in that case the solution space is , or equivalently, , (one degree of freedom). The kernel may be expressed as the subspace : the value of is the freedom in a solution. The cokernel may be expressed via the real valued map : given a vector , the value of is the ''obstruction'' to there being a solution. Additionally, the cokernel can be thought of as something that "detects"
surjection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

surjection
s in the same way that the kernel "detects"
injection Injection or injected may refer to: Science and technology * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...
s. A map is injective if and only if its kernel is trivial, and a map is surjective if and only if its cokernel is trivial, or in other words, if .


References

*
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ...
: ''
Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
'', Second Edition, 1978, p. 64 *
Emily Riehl Emily Riehl is an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...

Emily Riehl

Category Theory in Context
2014, p. 82, p. 139 footnote 8. {{Category theory Abstract algebra Category theory Isomorphism theorems de:Kern (Algebra)#Kokern