In
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 ...
, a coequalizer (or coequaliser) is a generalization of a
quotient by 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. A simpler example is equ ...
to objects in an arbitrary
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 ( ...
. It is the categorical construction
dual to the
equalizer.
Definition
A coequalizer is the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions s ...
of a diagram consisting of two objects ''X'' and ''Y'' and two parallel
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 ...
s .
More explicitly, a coequalizer of the parallel morphisms ''f'' and ''g'' can be defined as an object ''Q'' together with a morphism such that . Moreover, the pair must be
universal in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism such that . This information can be captured by the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
:
As with all
universal construction
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 ...
s, a coequalizer, if it exists, is unique
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
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 ...
(this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).
It can be shown that a coequalizing arrow ''q'' is an
epimorphism
In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f \implies g_1 = g_2.
Epimorphisms are categorical analo ...
in any category.
Examples
* In the
category of sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
, the coequalizer of two
functions is the
quotient of ''Y'' by the smallest
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. A simpler example is equ ...
~ such that for every , we have .
In particular, if ''R'' is an equivalence relation on a set ''Y'', and ''r''
1, ''r''
2 are the natural projections then the coequalizer of ''r''
1 and ''r''
2 is the quotient set . (See also:
quotient by an equivalence relation.)
* The coequalizer in the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
The ...
is very similar. Here if are
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, their coequalizer is the
quotient of ''Y'' by the
normal closure of the set
*:
* For
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 commu ...
s the coequalizer is particularly simple. It is just the
factor 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 exam ...
. (This is the
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 ...
of the morphism ; see the next section).
* In the
category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
, the circle object ''S''
1 can be viewed as the coequalizer of the two inclusion maps from the standard 0-simplex to the standard 1-simplex.
* Coequalizers can be large: There are exactly two
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s from the category 1 having one object and one identity arrow, to the category 2 with two objects and one non-identity arrow going between them. The coequalizer of these two functors is the
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 ...
of
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s under addition, considered as a one-object category. In particular, this shows that while every coequalizing arrow is
epic
Epic commonly refers to:
* Epic poetry, a long narrative poem celebrating heroic deeds and events significant to a culture or nation
* Epic film, a genre of film defined by the spectacular presentation of human drama on a grandiose scale
Epic(s) ...
, it is not necessarily
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
.
Properties
* Every coequalizer is an epimorphism.
* In a
topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...
, every
epimorphism
In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f \implies g_1 = g_2.
Epimorphisms are categorical analo ...
is the coequalizer of its kernel pair.
Special cases
In categories with
zero morphisms, one can define a ''
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 ...
'' of a morphism ''f'' as the coequalizer of ''f'' and the parallel zero morphism.
In
preadditive categories it makes sense to add and subtract morphisms (the
hom-set
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. Alt ...
s actually form
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 commu ...
s). In such categories, one can define the coequalizer of two morphisms ''f'' and ''g'' as the cokernel of their difference:
: coeq(''f'', ''g'') = coker(''g'' – ''f'').
A stronger notion is that of an absolute coequalizer, this is a coequalizer that is preserved under all functors.
Formally, an absolute coequalizer of a pair of parallel arrows in a category ''C'' is a coequalizer as defined above, but with the added property that given any functor , ''F''(''Q'') together with ''F''(''q'') is the coequalizer of ''F''(''f'') and ''F''(''g'') in the category ''D''.
Split coequalizers are examples of absolute coequalizers.
See also
*
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...
*
Pushout
Notes
References
*
** Coequalizers – page 65
** Absolute coequalizers – page 149
External links
Interactive Web page which generates examples of coequalizers in the category of finite sets. Written b
Jocelyn Paine
{{Category theory
Limits (category theory)