In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, an
equaliser, or
equalizer, is a set of arguments where two or more
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s have equal values.
An equaliser is the
solution setIn mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.For example, for a set of polynomials over a ring ,the solution set is the subset of on which the polynomials all vanish , formally...
of an
equationAn equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
.
In certain contexts, a
difference kernel is the equaliser of exactly two functions.
Let
X and
Y be sets.
Let
f and
g be
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s, both from
X to
Y.
Then the
equaliser of
f and
g is the set of elements
x of
X such that
f(
x) equals
g(
x) in
Y.
Symbolically:
The equaliser may be denoted Eq(
f,
g) or a variation on that theme (such as with lowercase letters "eq").
In informal contexts, the notation {
f =
g} is common.
The definition above used two functions
f and
g, but there is no need to restrict to only two functions, or even to only
finiteIn mathematics, finite set is a set that has a finite number of elements. For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set. A set that is not finite is called infinite...
ly many functions.
In general, if
F is a set of functions from
X to
Y, then the
equaliser of the members of
F is the set of elements
x of
X such that, given any two members
f and
g of
F,
f(
x) equals
g(
x) in
Y.
Symbolically:
This equaliser may be denoted Eq(
F), or Eq(
f,
g,
h,...) if
F is the set {
f,
g,
h,...}.
In the latter case, one may also find {
f =
g =
h = ···} in informal contexts.
As a degenerate case of the general definition, let
F be a singleton {
f}.
Since
f(
x) always equals itself, the equaliser must be the entire domain
X.
As an even more degenerate case, let
F be the
empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
{}.
Then the equaliser is again the entire domain
X, since the
universal quantificationIn predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
in the definition is vacuously true.
A binary equaliser (that is, an equaliser of just two functions) is also called a
difference kernel.
This may also be denoted DiffKer(
f,
g), Ker(
f,
g), or Ker(
f -
g).
The last notation shows where this terminology comes from, and why it is most common in the context of
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
:
The difference kernel of
f and
g is simply the
kernelIn 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...
of the difference
f -
g.
Conversely, the kernel of a single function
f can be reconstructed as the difference kernel Eq(
f,0), where 0 is the
constant functionIn mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4...
with value
zero0 is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems...
.
Of course, all of this presumes an algebraic context where the kernel of a function is its preimage under zero; that is not true in all situations.
However, the terminology "difference kernel" has no other meaning.
Equalisers can be defined by a
universal propertyIn 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...
, which allows the notion to be generalised from the
category of setsIn mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.-Properties of the category of sets:...
to arbitrary
categoriesIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
.
In the general context,
X and
Y are objects, while
f and
g are morphisms from
X to
Y.
These objects and morphisms form a
diagramIn mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition...
in the category in question, and the equaliser is simply the
limitIn category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of that diagram.
In more explicit terms, the equaliser consists of an object
E and a morphism
eq :
E →
X satisfying
f O eq =
g O eq,
and such that, given any other object
O and morphism
m :
O →
X, if
f O m =
g O m, then there exists a unique morphism
u :
O →
E such that
eq O u =
m.
In any
universal algebraUniversal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
ic category, including the categories where difference kernels are used, as well as the category of sets itself, the object
E can always be taken to be the ordinary notion of equaliser, and the morphism
eq can in that case be taken to be the inclusion function of
E as a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
of
X.
The generalisation of this to more than two morphisms is straightforward; simply use a larger diagram with more morphisms in it.
The degenerate case of only one morphism is also straightforward; then
eq can be any
isomorphismIn abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings....
from an object
E to
X.
The degenerate case of no morphisms at all might be confusing at first; it may seem that the diagram in question consists of only the objects
X and
Y, with no morphisms.
The limit of that diagram is the
productIn category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
of
X and
Y, which doesn't agree with the set-theoretic definition above.
However, the correct interpretation is that the diagram is based on
X and includes
Y only because
Y is the
codomainIn mathematics, the codomain, or target set, of a function is the set Y into which all of the output of the function is constrained to fall...
of a morphism in the diagram.
Then if there are no morphisms involved, then the diagram consists of
X alone, so the limit is again any isomorphism between
E and
X.
It can be proved that any equaliser in any category is a
monomorphismIn 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 ....
.
If the converse holds in a given category, then that category is said to be
regular (in the sense of monomorphisms).
More generally, a regular monomorphism in any category is any morphism
m that is an equaliser of some set of morphisms.
Some authorities require (more strictly) that
m be a
binary equaliser, that is an equaliser of exactly two morphisms.
However, if the category in question is
completeIn mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C...
, then both definitions agree.
The notion of difference kernel also makes sense in a category-theoretic context.
The terminology "difference kernel" is common throughout category theory for any binary equaliser.
In the case of a
preadditive categoryIn mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
(a category
enrichedIn category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner.-Definition:...
over the category of
Abelian groupAn 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 their order . Abelian groups generalize the arithmetic of addition of integers...
s), the term "difference kernel" may be interpreted literally, since subtraction of morphisms makes sense.
That is, Eq(
f,
g) = Ker(
f -
g), where Ker denotes the
category-theoretic kernelIn category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra...
.
Any category with fibre products (pull backs) and products has equalisers.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, an
equaliser, or
equalizer, is a set of arguments where two or more
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s have equal values.
An equaliser is the
solution setIn mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.For example, for a set of polynomials over a ring ,the solution set is the subset of on which the polynomials all vanish , formally...
of an
equationAn equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
.
In certain contexts, a
difference kernel is the equaliser of exactly two functions.
Definitions
Let
X and
Y be sets.
Let
f and
g be
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s, both from
X to
Y.
Then the
equaliser of
f and
g is the set of elements
x of
X such that
f(
x) equals
g(
x) in
Y.
Symbolically:
The equaliser may be denoted Eq(
f,
g) or a variation on that theme (such as with lowercase letters "eq").
In informal contexts, the notation {
f =
g} is common.
The definition above used two functions
f and
g, but there is no need to restrict to only two functions, or even to only
finiteIn mathematics, finite set is a set that has a finite number of elements. For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set. A set that is not finite is called infinite...
ly many functions.
In general, if
F is a set of functions from
X to
Y, then the
equaliser of the members of
F is the set of elements
x of
X such that, given any two members
f and
g of
F,
f(
x) equals
g(
x) in
Y.
Symbolically:
This equaliser may be denoted Eq(
F), or Eq(
f,
g,
h,...) if
F is the set {
f,
g,
h,...}.
In the latter case, one may also find {
f =
g =
h = ···} in informal contexts.
As a degenerate case of the general definition, let
F be a singleton {
f}.
Since
f(
x) always equals itself, the equaliser must be the entire domain
X.
As an even more degenerate case, let
F be the
empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
{}.
Then the equaliser is again the entire domain
X, since the
universal quantificationIn predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
in the definition is vacuously true.
Difference kernels
A binary equaliser (that is, an equaliser of just two functions) is also called a
difference kernel.
This may also be denoted DiffKer(
f,
g), Ker(
f,
g), or Ker(
f -
g).
The last notation shows where this terminology comes from, and why it is most common in the context of
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
:
The difference kernel of
f and
g is simply the
kernelIn 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...
of the difference
f -
g.
Conversely, the kernel of a single function
f can be reconstructed as the difference kernel Eq(
f,0), where 0 is the
constant functionIn mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4...
with value
zero0 is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems...
.
Of course, all of this presumes an algebraic context where the kernel of a function is its preimage under zero; that is not true in all situations.
However, the terminology "difference kernel" has no other meaning.
In category theory
Equalisers can be defined by a
universal propertyIn 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...
, which allows the notion to be generalised from the
category of setsIn mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.-Properties of the category of sets:...
to arbitrary
categoriesIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
.
In the general context,
X and
Y are objects, while
f and
g are morphisms from
X to
Y.
These objects and morphisms form a
diagramIn mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition...
in the category in question, and the equaliser is simply the
limitIn category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of that diagram.
In more explicit terms, the equaliser consists of an object
E and a morphism
eq :
E →
X satisfying
f O eq =
g O eq,
and such that, given any other object
O and morphism
m :
O →
X, if
f O m =
g O m, then there exists a unique morphism
u :
O →
E such that
eq O u =
m.
In any
universal algebraUniversal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
ic category, including the categories where difference kernels are used, as well as the category of sets itself, the object
E can always be taken to be the ordinary notion of equaliser, and the morphism
eq can in that case be taken to be the inclusion function of
E as a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
of
X.
The generalisation of this to more than two morphisms is straightforward; simply use a larger diagram with more morphisms in it.
The degenerate case of only one morphism is also straightforward; then
eq can be any
isomorphismIn abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings....
from an object
E to
X.
The degenerate case of no morphisms at all might be confusing at first; it may seem that the diagram in question consists of only the objects
X and
Y, with no morphisms.
The limit of that diagram is the
productIn category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
of
X and
Y, which doesn't agree with the set-theoretic definition above.
However, the correct interpretation is that the diagram is based on
X and includes
Y only because
Y is the
codomainIn mathematics, the codomain, or target set, of a function is the set Y into which all of the output of the function is constrained to fall...
of a morphism in the diagram.
Then if there are no morphisms involved, then the diagram consists of
X alone, so the limit is again any isomorphism between
E and
X.
It can be proved that any equaliser in any category is a
monomorphismIn 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 ....
.
If the converse holds in a given category, then that category is said to be
regular (in the sense of monomorphisms).
More generally, a regular monomorphism in any category is any morphism
m that is an equaliser of some set of morphisms.
Some authorities require (more strictly) that
m be a
binary equaliser, that is an equaliser of exactly two morphisms.
However, if the category in question is
completeIn mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C...
, then both definitions agree.
The notion of difference kernel also makes sense in a category-theoretic context.
The terminology "difference kernel" is common throughout category theory for any binary equaliser.
In the case of a
preadditive categoryIn mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
(a category
enrichedIn category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner.-Definition:...
over the category of
Abelian groupAn 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 their order . Abelian groups generalize the arithmetic of addition of integers...
s), the term "difference kernel" may be interpreted literally, since subtraction of morphisms makes sense.
That is, Eq(
f,
g) = Ker(
f -
g), where Ker denotes the
category-theoretic kernelIn category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra...
.
Any category with fibre products (pull backs) and products has equalisers.
External links
See also
- Coequaliser, the dual
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
notion, obtained by reversing the arrows in the equaliser definition.
- Coincidence theory, a topological approach to equalizer sets in topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s.
- Pullback
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan...
, a special limit, that can be constructed from equalisers and products.