category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

and its applications to other branches of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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) whe ...

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, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

''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 (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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

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

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

As for every universal property, there is a unique isomorphism between two kernels of the same morphism, and the morphism ''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). So, it is common to talk of ''the'' kernel of a morphism. In concrete categories, one can thus take a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...

of ' for ''K'', in which case, the morphism ''k'' is the inclusion map. This allows one to talk of ''K'' as the kernel, since ''k'' is implicitly defined by ''K''. There are non-concrete categories, where one can similarly define a "natural" kernel, such that ''K'' defines ''k'' implicitly. 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 or morphism 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 the ...

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, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, such as the category of groups 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

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

in one of these categories, and ''K'' is its kernel in the usual algebraic sense, then ''K'' 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 ''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 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 . Monoids are semigroups with identity ...

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 nam ...

. 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^ of a given Category (mathematics), category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal ...

, 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: :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 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 ...

, 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 or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

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 in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of stud ...

defines a notion of kernel for homomorphisms between two

algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...

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 of its codomain; that is, implies (equivalently by contraposition, impl ...

. 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