The cokernel of a linear mapping of vector spaces is the quotient space of the

codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...

of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the name: the kernel is a

subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...

of the domain (it maps to the domain), while the cokernel is a

quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...

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 without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...

, below. More generally, the cokernel of a

morphism 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 ...

in some

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

(e.g. a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

between groups or a

bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...

between

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

s) is an object and a morphism such that the composition is the zero morphism 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 t ...

with respect to this property. Often the map is understood, and itself is called the cokernel of . In many situations in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

, such as for

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 ...

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 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 * Modul ...

s, the cokernel of the

is the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of by the

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...

of . In 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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

. In order for the definition to make sense the category in question must have zero morphisms. The cokernel of a

is defined as the coequalizer 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. Moreover, the morphism must be

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 objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...

a unique

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

, 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 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 ...

. 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 is conormal).

Examples

In the category of groups, the cokernel of 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 ...

is the

of by the normal closure of the image of . In the case of

s, since every

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

is normal, the cokernel is just

modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...

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 that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom- ...

, it makes sense to add and subtract morphisms. In such a category, the coequalizer 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 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 ...

(a special kind of preadditive category) the

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. ...

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 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 ...

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 is the space of ''solutions.'' Formally, one may connect the kernel and the cokernel of a map by the exact sequence :

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, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

s in the same way that the kernel "detects" injections. 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 who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...

: ''

Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on ...

'', Second Edition, 1978, p. 64 *

Emily Riehl Emily Riehl is an American mathematician who has contributed to higher category theory and homotopy theory. Much of her work, including her PhD thesis, concerns model structures and more recently the foundations of infinity-categories. She is ...

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