In category theory

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...

, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

 to objects in an arbitrary category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

. It is the categorical construction dual

Dual (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...

 to the equalizer (hence the name).

Definition

A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s f, g : X → Y.

More explicitly, a coequalizer can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g. Moreover, the pair (Q, q) must be universal

Universal 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...

 in the sense that given any other such pair (Q′, q′) there exists a unique morphism u : Q → Q′ for which the following diagram commutes

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...

:
As with all universal constructions, a coequalizer, if it exists, is unique up to

Up to

In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

 a unique 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...

 (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).

It can be shown that a coequalizer q is an epimorphism

Epimorphism

In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

 in any category.

Examples

• In 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...
  
  , the coequalizer of two 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 f, g : X → Y is the quotient of Y by the smallest equivalence relation
  Equivalence relation
  In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
  
    such that for every , we have . In particular, if R is an equivalence relation on a set Y, and r_1,2 are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r₁ and r₂ is the quotient set Y/R.

• The coequalizer in the category of groups
  Category of groups
  In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
  
   is very similar. Here if f, g : X → Y are group homomorphism
  Group homomorphism
  In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
  
  s, their coequalizer is the quotient
  Quotient group
  In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
  
   of Y by the normal closure
  Normal closure
  The term normal closure is used in two senses in mathematics:* In group theory, the normal closure of a subset of a group is the smallest normal subgroup that contains the subset; see conjugate closure....
  
   of the set

• For 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 coequalizer is particularly simple. It is just the factor group Y / im(f – g). (This is the cokernel
  Cokernel
  In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
  
   of the morphism f – g; see the next section).

• In the category of topological spaces
  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 continuous...
  
  , the circle object can be viewed as the coequalizer of the two inclusion maps from the standard 0-simplex to the standard 1-simplex.

• Coequalisers can be large: There are exactly two functors from the category 1 having one object and one identity arrow, to the category 2 with two objects and exactly one non-identity arrow going between them. The coequaliser of these two functors is the monoid of natural numbers under addition, considered as a one-object category. In particular, this shows that while every coequalising arrow is epic
  Epimorphism
  In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
  
  , it is not necessarily surjective
  Surjective function
  In mathematics, a function f from a set X to a set Y is surjective , or a surjection, if every element y in Y has a corresponding element x in X so that f = y...
  
  .

Properties

• In a topos
  Topos
  In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
  
  , every epimorphism
  Epimorphism
  In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
  
   is the coequalizer of its kernel pair.
• Every coequalizer is an epimorphism.

Special cases

In categories with zero morphism

Zero morphism

In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

s, one can define a cokernel

Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

of a morphism f as the coequalizer of f and the parallel zero morphism.

In preadditive categories

Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

 it makes sense to add and subtract morphisms (the hom-sets actually form 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). 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 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.

External links

• Interactive Web page which generates examples of coequalizers in the category of finite sets. Written by Jocelyn Paine.