In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an equaliser is a set of arguments where two or more
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 ...
s have
equal
Equal(s) may refer to:
Mathematics
* Equality (mathematics).
* Equals sign (=), a mathematical symbol used to indicate equality.
Arts and entertainment
* ''Equals'' (film), a 2015 American science fiction film
* ''Equals'' (game), a board game
...
values.
An equaliser is the
solution set
In 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 (evaluate ...
of an
equation
In mathematics, an equation is a formula that expresses the equality (mathematics), equality of two Expression (mathematics), expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may h ...
.
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 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 ...
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 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
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
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 written as Eq(''f'', ''g'', ''h'', ...) if
is the set .
In the latter case, one may also find in informal contexts.
As a
degenerate case of the general definition, let F be a
singleton .
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
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
. Then the equaliser is again the entire domain ''X'', since the
universal quantification
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In othe ...
in the definition is
vacuously true
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
.
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
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 ...
: The difference kernel of ''f'' and ''g'' is simply the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
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
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properties ...
with value
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...
.
Of course, all of this presumes an algebraic context where the kernel of a function 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 zero under that function; 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 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 fr ...
, which allows the notion to be generalised from the
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 the total functions from ''A'' to ''B'', and the composition of ...
to arbitrary
categories
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 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
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a thr ...
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
.
A morphism
is said to equalise
and
if
.
In any
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 study ...
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
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 ...
of ''E'' as 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''.
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
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 ...
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 equaliser. (And indeed products and equalisers are different concepts: the set-theoretic definition of product doesn't agree with the set-theoretic definition of the equaliser mentioned above, hence they are actually different.) Instead, the appropriate insight is that every equaliser diagram is fundamentally concerned with ''X'', including ''Y'' only because ''Y'' is 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 the ...
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 equaliser 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
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 monomorphi ...
.
If the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical ...
holds in a given category, then that category is said to be ''regular'' (in the sense of monomorphisms).
More generally, a
regular 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 monomorphism ...
in any category is any morphism ''m'' that is an equaliser of some set of morphisms.
Some authors 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
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 hom- ...
(a category
enriched over the category 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 comm ...
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 (pullbacks) and products has equalisers.
See also
*
Coequaliser, the
dual notion, obtained by reversing the arrows in the equaliser definition.
*
Coincidence theory, a topological approach to equaliser sets in
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s.
*
Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
, a special
limit that can be constructed from equalisers and products.
Notes
References
*
External links
Interactive Web page which generates examples of equalisers in the category of finite sets. Written b
Jocelyn Paine
{{Category theory
Set theory
Limits (category theory)