Equaliser
Encyclopedia
In mathematics
, an equaliser, or equalizer, is a set of arguments where two or more function
s 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.
Let f and g be function
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 finitely 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 written as Eq(f,g,h,...) if
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 set
{}.
Then the equaliser is again the entire domain X, since the universal quantification
in the definition is vacuously true.
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 algebra
:
The difference kernel of f and g is simply the kernel
of the difference f − g.
Furthermore, the kernel of a single function f can be reconstructed as the difference kernel Eq(f,0), where 0 is the constant function
with value zero
.
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.
, which allows the notion to be generalised from the category of sets
to arbitrary categories
.
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 diagram
in the category in question, and the equaliser is simply the limit
of that diagram.
In more explicit terms, the equaliser consists of an object E and a morphism eq : E → X satisfying
,
and such that, given any object O and morphism m : O → X, if
, then there exists a unique morphism u : O → E such that 
.
In any universal algebra
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 subset
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 isomorphism
from an object E to X.
The correct diagram for the degenerate case with no morphisms is slightly subtle: one might initially draw the diagram as consisting of the objects X and Y and no morphisms. This is incorrect, however, since the limit of such a diagram is the product
of X and Y, rather than the equalizer. (And indeed products and equalizers are different concepts: the set-theoretic definition of product doesn't agree with the set-theoretic definition of the equalizer mentioned above, hence they are actually different.) Instead, the appropriate insight is that every equalizer diagram is fundamentally concerned with X, including Y only because Y is the codomain
of morphisms which appear in the diagram. With this view, we see that if there are no morphisms involved, Y does not make an appearance and the equalizer diagram consists of X alone. The limit of this diagram is then any isomorphism between E and X.
It can be proved that any equaliser in any category is a monomorphism
.
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 complete
, 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 category
(a category enriched
over the category of Abelian group
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 kernel
.
Any category with fibre products (pull backs) and products has equalisers.
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...
, an equaliser, or equalizer, is a set of arguments where two or more function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s have equal values.
An equaliser is the solution set of an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
.
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 function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
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 finitely 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 written as Eq(f,g,h,...) if
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 set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality 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 quantification
Universal quantification
In 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 algebra
Abstract algebra
Abstract 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 kernel
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...
of the difference f − g.
Furthermore, the kernel of a single function f can be reconstructed as the difference kernel Eq(f,0), where 0 is the constant function
Constant function
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...
with value zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 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 propertyUniversal 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...
, which allows the notion to be generalised from the category of sets
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
to arbitrary categories
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...
.
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 diagram
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
in the category in question, and the equaliser is simply the limit
Limit (category theory)
In 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
,and such that, given any object O and morphism m : O → X, if
, then there exists a unique morphism u : O → E such that .In any universal algebra
Universal algebra
Universal 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 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.
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 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...
from an object E to X.
The correct diagram for the degenerate case with no morphisms is slightly subtle: one might initially draw the diagram as consisting of the objects X and Y and no morphisms. This is incorrect, however, since the limit of such a diagram is the product
Product (category theory)
In 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, rather than the equalizer. (And indeed products and equalizers are different concepts: the set-theoretic definition of product doesn't agree with the set-theoretic definition of the equalizer mentioned above, hence they are actually different.) Instead, the appropriate insight is that every equalizer diagram is fundamentally concerned with X, including Y only because Y is the codomain
Codomain
In mathematics, the codomain or target set 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...
of morphisms which appear in the diagram. With this view, we see that if there are no morphisms involved, Y does not make an appearance and the equalizer diagram consists of X alone. The limit of this diagram is then any isomorphism between E and X.
It can be proved that any equaliser in any category 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....
.
If the converse
Converse (logic)
In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S is P, the converse is All P is S. In neither case does the converse necessarily follow from...
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 complete
Complete category
In 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. Dually, a cocomplete category is one in which all small colimits exist...
, 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 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...
(a category enriched
Enriched category
In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
over the category of Abelian group
Abelian group
In abstract algebra, 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 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 kernel
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...
.
Any category with fibre products (pull backs) and products has equalisers.
External links
- Interactive Web page which generates examples of equalizers in the category of finite sets. Written by Jocelyn Paine.
See also
- Coequaliser, the dualDual (category theory)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 spaceTopological spaceTopological 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. - PullbackPullback (category theory)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 X \rightarrow Z \leftarrow Y...
, a special limitLimit (category theory)In 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....
that can be constructed from equalisers and products.