In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of

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) ...

s, the kernels of

module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ' ...

s and certain other kernels from algebra. Intuitively, the kernel of the

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, morphis ...

''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 kernel In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Definition ...

s (also known as 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

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) ...

. In order to define a kernel in the general category-theoretical sense, C needs to have

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. T ...

s. In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary

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'', 0_''XY'') To be more explicit, the following

universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...

can be used. A kernel of ''f'' is an object ''K'' together with a morphism ''k'' : ''K'' → ''X'' such that: * ''f''∘''k'' is the zero morphism from ''K'' to ''Y'';

* Given any morphism ''k''′ : ''K''′ → ''X'' such that ''f''∘''k''′ is the zero morphism, there is a unique morphism ''u'' : ''K''′ → ''K'' such that ''k''∘''u'' = ''k′''.

Note that in many

concrete Concrete is a composite material composed of fine and coarse aggregate bonded together with a fluid cement (cement paste) that hardens (cures) over time. Concrete is the second-most-used substance in the world after water, and is the most ...

contexts, one would refer to the object ''K'' as the "kernel", rather than the morphism ''k''. In those situations, ''K'' would be a

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of ''X'', and that would be sufficient to reconstruct ''k'' as an

inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...

; in the nonconcrete case, in contrast, we need the morphism ''k'' to describe ''how'' ''K'' is to be interpreted as 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 theor ...

of ''X''. In any case, one can show that ''k'' is always a

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 monomorphis ...

(in the categorical sense). 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 are kernels of ''f'' : ''X'' → ''Y'', then there exists 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 ...

φ : ''K'' → ''L'' such that ∘φ = ''k''.

Examples

Kernels are familiar in many categories from

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 te ...

, such as the category of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

s or the category of (left) modules over a fixed ring (including

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 ...

s over a fixed field). To be explicit, if ''f'' : ''X'' → ''Y'' is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

in one of these categories, and ''K'' is its kernel in the usual algebraic sense, then ''K'' is a

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear oper ...

of ''X'' and the inclusion homomorphism from ''K'' to ''X'' is a kernel in the categorical sense. Note that in the category of

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

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 (see #Relationship to algebraic kernels below). In the category of unital 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 that corresponds to kernels in the category of non-unital rings. 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''.

Relation to other categorical concepts

The dual concept to that of kernel is that of

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

. That is, the kernel of a morphism is its cokernel in the

opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op 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 yield ...

, and vice versa. As mentioned above, a kernel is a type of binary equaliser, or

. Conversely, 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 ho ...

, 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: :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

. 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

of any morphism (which always exists in an abelian category) turns out to be 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 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 algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular Group (mathematics), groups as ...

defines a notion of kernel for homomorphisms between two

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

s of the same kind. This concept of kernel measures how far the given homomorphism is from being

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

. 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.

Sources

* * {{DEFAULTSORT:Kernel (Category Theory) Category theory