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Kernel (category theory)
Encyclopedia
In category theory
and its applications to other branches of mathematics
, kernels are a generalization of the kernels of group homomorphism
s, the kernels of module homomorphisms and certain other kernels from algebra
. Intuitively, the kernel of the morphism
f : X → Y is the "most general" morphism k : K → X that yields zero when composed with (followed by) f.
Note that kernel pairs and difference kernels (aka binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article.
.
In order to define a kernel in the general category-theoretical sense, C needs to have zero morphism
s.
In that case, if f : X → Y is an arbitrary morphism
in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y.
In symbols:
To be more explicit, the following universal property
can be used. A kernel of f is any morphism k : K → X such that:
Note that in many concrete
contexts, one would refer to the object K as the "kernel", rather than the morphism k.
In those situations, K would be a subset
of X, and that would be sufficient to reconstruct k as an inclusion map
; in the nonconcrete case, in contrast, we need the morphism k to describe how K is to be interpreted as a subobject
of X. In any case, one can show that k is always a monomorphism
(in the categorical sense of the word). One may prefer to think of the kernel as the pair (K,k) rather than as simply K or k alone.
Not every morphism needs to have a kernel, but if it does, then all its kernels are isomorphic in a strong sense: if k : K → X and l : L → X are kernels of f : X → Y, then there exists a unique isomorphism
φ : K → L such that l o φ = k.
, such as the category of groups or the category of (left) modules
over a fixed ring
(including vector space
s over a fixed field
).
To be explicit, if f : X → Y is a homomorphism
in one of these categories, and K is its kernel in the usual algebraic sense
, then K is a subalgebra
of X and the inclusion homomorphism from K to X is a kernel in the categorical sense.
Note that in the category of monoid
s, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes.
Therefore, the notion of kernel studied in monoid theory is slightly different.
Conversely, in the category of rings
, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms.
Nevertheless, there is still a notion of kernel studied in ring theory.
See Relationship to algebraic kernels below for the resolution of this paradox.
In the category of pointed topological spaces
, if f :X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.
.
That is, the kernel of a morphism is its cokernel in the opposite category
, and vice versa.
As mentioned above, a kernel is a type of binary equaliser, or difference kernel.
Conversely, in a preadditive category
, every binary equaliser can be constructed as a kernel.
To be specific, the equaliser of the morphisms f and g is the kernel of the difference
g − f.
In symbols:
It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphisms cannot be subtracted.
Every kernel, like any other equaliser, is a monomorphism
.
Conversely, a monomorphism is called normal
if it is the kernel of some morphism.
A category is called normal if every monomorphism is normal.
Abelian categories, in particular, are always normal.
In this situation, the kernel of the cokernel
of any morphism (which always exists in an abelian category) turns out to be the image
of that morphism; in symbols:
When m is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel.
In symbols:
defines a notion of kernel for homomorphisms between two algebraic structure
s of the same kind.
This concept of kernel measures how far the given homomorphism is from being injective.
There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above.
In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair.
In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic terms.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
and its applications to other branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, kernels are a generalization of the kernels of group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s, the kernels of module homomorphisms and certain other kernels from algebra
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
. Intuitively, the kernel of the morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
f : X → Y is the "most general" morphism k : K → X that yields zero when composed with (followed by) f.
Note that kernel pairs and difference kernels (aka binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article.
Definition
Let C be a categoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
.
In order to define a kernel in the general category-theoretical sense, C needs to have zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...
s.
In that case, if f : X → Y is an arbitrary morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y.
In symbols:
- ker(f) = eq(f, 0XY)
To be more explicit, the following universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
can be used. A kernel of f is any morphism k : K → X such that:
- f o k is the zero morphism from K to Y;
- Given any morphism k′ : K′ → X such that f o k′ is the zero morphism, there is a unique morphism u : K′ → K such that k o u = k.
Note that in many concrete
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
contexts, one would refer to the object K as the "kernel", rather than the morphism k.
In those situations, K would be a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of X, and that would be sufficient to reconstruct k as an inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
; in the nonconcrete case, in contrast, we need the morphism k to describe how K is to be interpreted as a subobject
Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...
of X. In any case, one can show that k is always a monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
(in the categorical sense of the word). One may prefer to think of the kernel as the pair (K,k) rather than as simply K or k alone.
Not every morphism needs to have a kernel, but if it does, then all its kernels are isomorphic in a strong sense: if k : K → X and l : L → X are kernels of f : X → Y, then there exists a unique isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
φ : K → L such that l o φ = k.
Examples
Kernels are familiar in many categories from abstract algebraAbstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, such as the category of groups or the category of (left) modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a fixed ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
(including vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over a fixed field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
).
To be explicit, if f : X → Y is a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
in one of these categories, and K is its kernel in the usual algebraic sense
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
, then K is a subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of X and the inclusion homomorphism from K to X is a kernel in the categorical sense.
Note that in the category of monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
s, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes.
Therefore, the notion of kernel studied in monoid theory is slightly different.
Conversely, in the category of rings
Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms...
, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms.
Nevertheless, there is still a notion of kernel studied in ring theory.
See Relationship to algebraic kernels below for the resolution of this paradox.
In the category of pointed topological spaces
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
, if f :X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.
Relation to other categorical concepts
The dual concept to that of kernel is that of cokernelCokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
.
That is, the kernel of a morphism is its cokernel in the opposite category
Opposite category
In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite...
, and vice versa.
As mentioned above, a kernel is a type of binary equaliser, or difference kernel.
Conversely, in a preadditive category
Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
, every binary equaliser can be constructed as a kernel.
To be specific, the equaliser of the morphisms f and g is the kernel of the difference
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
g − f.
In symbols:
- eq (f,g) = ker (g − f).
It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphisms cannot be subtracted.
Every kernel, like any other equaliser, is a monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
.
Conversely, a monomorphism is called normal
Normal morphism
In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which every monomorphism is normal...
if it is the kernel of some morphism.
A category is called normal if every monomorphism is normal.
Abelian categories, in particular, are always normal.
In this situation, the kernel of the cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
of any morphism (which always exists in an abelian category) turns out to be the image
Image (category theory)
Given a category C and a morphismf\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:#There exists a morphism g\colon X\to I such that f = hg....
of that morphism; in symbols:
- im f = ker coker f (in an abelian category)
When m is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel.
In symbols:
- m = ker (coker m) (for monomorphisms in an abelian category)
Relationship to algebraic kernels
Universal algebraUniversal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
defines a notion of kernel for homomorphisms between two algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s of the same kind.
This concept of kernel measures how far the given homomorphism is from being injective.
There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above.
In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair.
In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic terms.