TheInfoList

OR:

In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the ''null space'', is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.See and . For some types of structure, such as
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 com ...
s and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
for groups and two-sided ideals for
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
. Kernels allow defining quotient objects (also called quotient algebras in
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 groups as the object of stud ...
, and
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 ...
s in
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, cate ...
). For many types of algebraic structure, the
fundamental theorem on homomorphisms In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and ...
(or first isomorphism theorem) states that
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-dimension ...
of a homomorphism is isomorphic to the quotient by the kernel. The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...

# Survey of examples

## Linear maps

Let ''V'' and ''W'' be vector spaces over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
(or more generally, modules over a ring) and let ''T'' be a linear map from ''V'' to ''W''. If 0''W'' is the zero vector of ''W'', then the kernel of ''T'' is the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of the zero subspace ; that is, the subset of ''V'' consisting of all those elements of ''V'' that are mapped by ''T'' to the element 0''W''. The kernel is usually denoted as , or some variation thereof: :$\ker T = \ .$ Since a linear map preserves zero vectors, the zero vector 0''V'' of ''V'' must belong to the kernel. The transformation ''T'' is injective if and only if its kernel is reduced to the zero subspace. The kernel ker ''T'' is always a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals ...
of ''V''. Thus, it makes sense to speak of the quotient space ''V''/(ker ''T''). The first isomorphism theorem for vector spaces states that this quotient space is
naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
to 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-dimension ...
of ''T'' (which is a subspace of ''W''). As a consequence, the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of ''V'' equals the dimension of the kernel plus the dimension of the image. If ''V'' and ''W'' are finite-dimensional and bases have been chosen, then ''T'' can be described by a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''M'', and the kernel can be computed by solving the homogeneous
system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...
. In this case, the kernel of ''T'' may be identified to the kernel of the matrix ''M'', also called "null space" of ''M''. The dimension of the null space, called the nullity of ''M'', is given by the number of columns of ''M'' minus the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of ''M'', as a consequence of the rank–nullity theorem. Solving
homogeneous differential equation A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written :f(x,y) \, dy = g(x,y) \, dx, where and are homogeneous functions of the same degree of an ...
s often amounts to computing the kernel of certain
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retu ...
s. For instance, in order to find all twice-
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in ...
s ''f'' from the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
to itself such that :$x f\text{'}\text{'}\left(x\right) + 3 f\text{'}\left(x\right) = f\left(x\right),$ let ''V'' be the space of all twice differentiable functions, let ''W'' be the space of all functions, and define a linear operator ''T'' from ''V'' to ''W'' by :$\left(Tf\right)\left(x\right) = x f\text{'}\text{'}\left(x\right) + 3 f\text{'}\left(x\right) - f\left(x\right)$ for ''f'' in ''V'' and ''x'' an arbitrary real number. Then all solutions to the differential equation are in ker ''T''. One can define kernels for homomorphisms between modules over a ring in an analogous manner. This includes kernels for homomorphisms between
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 com ...
s as a special case. This example captures the essence of kernels in general abelian categories; see
Kernel (category theory) In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel o ...
.

## Group homomorphisms

Let ''G'' and ''H'' be
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 id ...
s and let ''f'' be 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) ...
from ''G'' to ''H''. If ''e''''H'' is the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
of ''H'', then the ''kernel'' of ''f'' is the preimage of the singleton set ; that is, the subset of ''G'' consisting of all those elements of ''G'' that are mapped by ''f'' to the element ''e''''H''. The kernel is usually denoted (or a variation). In symbols: : $\ker f = \ .$ Since a group homomorphism preserves identity elements, the identity element ''e''''G'' of ''G'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set . If ''f'' were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist $a, b \in G$ such that $a \neq b$ and $f\left(a\right) = f\left(b\right)$. Thus $f\left(a\right)f\left(b\right)^ = e_H$. ''f'' is a group homomorphism, so inverses and group operations are preserved, giving $f\left\left(ab^\right\right) = e_H$; in other words, $ab^ \in \ker f$, and ker ''f'' would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element $g \neq e_G \in \ker f$, then $f\left(g\right) = f\left(e_G\right) = e_H$, thus ''f'' would not be injective. is a
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 subgrou ...
of ''G'' and further it is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
. Thus, there is a corresponding
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exampl ...
. This is isomorphic to ''f''(''G''), the image of ''G'' under ''f'' (which is a subgroup of ''H'' also), by the first isomorphism theorem for groups. In the special case of
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 com ...
s, there is no deviation from the previous section.

### Example

Let ''G'' be the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
on 6 elements with modular addition, ''H'' be the cyclic on 2 elements with modular addition, and ''f'' the homomorphism that maps each element ''g'' in ''G'' to the element ''g'' modulo 2 in ''H''. Then , since all these elements are mapped to 0''H''. The quotient group has two elements: and . It is indeed isomorphic to ''H''.

## Ring homomorphisms

Let ''R'' and ''S'' be rings (assumed unital) and let ''f'' be a ring homomorphism from ''R'' to ''S''. If 0''S'' is the zero element of ''S'', then the ''kernel'' of ''f'' is its kernel as linear map over the integers, or, equivalently, as additive groups. It is the preimage of the zero ideal , which is, the subset of ''R'' consisting of all those elements of ''R'' that are mapped by ''f'' to the element 0''S''. The kernel is usually denoted (or a variation). In symbols: : $\operatorname f = \\mbox$ Since a ring homomorphism preserves zero elements, the zero element 0''R'' of ''R'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set . This is always the case if ''R'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a gras ...
, and ''S'' is not the zero ring. Since ker ''f'' contains the multiplicative identity only when ''S'' is the zero ring, it turns out that the kernel is generally not a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For thos ...
of ''R.'' The kernel is a sub rng, and, more precisely, a two-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of ''R''. Thus, it makes sense to speak of the quotient ring ''R''/(ker ''f''). The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of ''f'' (which is a subring of ''S''). (Note that rings need not be unital for the kernel definition). To some extent, this can be thought of as a special case of the situation for modules, since these are all
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in ...
s over a ring ''R'': * ''R'' itself; * any two-sided ideal of ''R'' (such as ker ''f''); * any quotient ring of ''R'' (such as ''R''/(ker ''f'')); and * 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 t ...
of any ring homomorphism whose domain is ''R'' (such as ''S'', the codomain of ''f''). However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not. This example captures the essence of kernels in general Mal'cev algebras.

## Monoid homomorphisms

Let ''M'' and ''N'' be monoids and let ''f'' be a monoid homomorphism from ''M'' to ''N''. Then the ''kernel'' of ''f'' is the subset of the direct product consisting of all those
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
s of elements of ''M'' whose components are both mapped by ''f'' to the same element in ''N''. The kernel is usually denoted (or a variation thereof). In symbols: : $\operatorname f = \left\.$ Since ''f'' is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
, the elements of the form must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the diagonal set . It turns out that is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
on ''M'', and in fact a
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
. Thus, it makes sense to speak of the
quotient monoid In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
. The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of ''f'' (which is a
submonoid 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 ...
of ''N''; for the congruence relation). This is very different in flavour from the above examples. In particular, the preimage of the identity element of ''N'' is ''not'' enough to determine the kernel of ''f''.

# Universal algebra

All the above cases may be unified and generalized in
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 groups as the object of stud ...
.

## General case

Let ''A'' and ''B'' be
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 a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. Then the ''kernel'' of ''f'' is the subset of the direct product ''A'' × ''A'' consisting of all those
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
s of elements of ''A'' whose components are both mapped by ''f'' to the same element in ''B''. The kernel is usually denoted (or a variation). In symbols: : $\operatorname f = \left\\mbox$ Since ''f'' is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
, the elements of the form (''a'', ''a'') must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is exactly the diagonal set . It is easy to see that ker ''f'' is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
on ''A'', and in fact a
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
. Thus, it makes sense to speak of the quotient algebra ''A''/(ker ''f''). The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which 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 ope ...
of ''B''). Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept. For more on this general concept, outside of abstract algebra, see kernel of a function.

## Malcev algebras

In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special
neutral element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
(the zero vector in the case of vector spaces, the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
in the case of commutative groups, and the zero element in the case of rings or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker ''f'' from the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of the neutral element. To be specific, let ''A'' and ''B'' be Malcev algebraic structures of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. If ''e''''B'' is the neutral element of ''B'', then the ''kernel'' of ''f'' is the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of the singleton set ; that is, the subset of ''A'' consisting of all those elements of ''A'' that are mapped by ''f'' to the element ''e''''B''. The kernel is usually denoted (or a variation). In symbols: : $\operatorname f = \\mbox$ Since a Malcev algebra homomorphism preserves neutral elements, the identity element ''e''''A'' of ''A'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set . The notion of
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
generalises to any Malcev algebra (as
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals ...
in the case of vector spaces,
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
in the case of groups, two-sided ideals in the case of rings, and
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the ...
in the case of modules). It turns out that ker ''f'' is not 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 ope ...
of ''A'', but it is an ideal. Then it makes sense to speak of the quotient algebra ''G''/(ker ''f''). The first isomorphism theorem for Malcev algebras states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a subalgebra of ''B''). The connection between this and the congruence relation for more general types of algebras is as follows. First, the kernel-as-an-ideal is the equivalence class of the neutral element ''e''''A'' under the kernel-as-a-congruence. For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication * Division algorithm, a method for computing the result of mathematical division Military * Division (military), a formation typically consistin ...
on either side for groups and
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
for vector spaces, modules, and rings). Using this, elements ''a'' and ''b'' of ''A'' are equivalent under the kernel-as-a-congruence if and only if their quotient ''a''/''b'' is an element of the kernel-as-an-ideal.

# Algebras with nonalgebraic structure

Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
s or
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is a ...
s, which are equipped with a topology. In this case, we would expect the homomorphism ''f'' to preserve this additional structure; in the topological examples, we would want ''f'' to be a continuous map. The process may run into a snag with the quotient algebras, which may not be well-behaved. In the topological examples, we can avoid problems by requiring that topological algebraic structures be Hausdorff (as is usually done); then the kernel (however it is constructed) will be a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
and the quotient space will work fine (and also be Hausdorff).

# Kernels in category theory

The notion of ''kernel'' in
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, cate ...
is a generalisation of the kernels of abelian algebras; see
Kernel (category theory) In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel o ...
. The categorical generalisation of the kernel as a congruence relation is the '' kernel pair''. (There is also the notion of difference kernel, or binary equaliser.)