Coequalizer

In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.

Definition

A coequalizer is a colimit of the diagram consisting of two objects ''X'' and ''Y'' and two parallel morphisms ''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 in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism ''u'' : ''Q'' → ''Q''′ such that ''u'' ∘ ''q'' = ''q''′. This information can be captured by the following commutative diagram:

As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (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 in any category.

Examples

*In the category of sets, the coequalizer of two functions ''f'', ''g'' : ''X'' → ''Y'' is the quotient of ''Y'' by the smallest equivalence relation $\sim$ such that for every $x\in X$, we have $f(x)\sim g(x)$. In particular, if ''R'' is an equivalence relation on a set ''Y'', and ''r''₁, ''r''₂ are the natural projections (''R'' ⊂ ''Y'' × ''Y'') → ''Y'' then the coequalizer of ''r''₁ and ''r''₂ is the quotient set ''Y''/''R''. (See also: quotient by an equivalence relation.)
*The coequalizer in the category of groups is very similar. Here if ''f'', ''g'' : ''X'' → ''Y'' are group homomorphisms, their coequalizer is the quotient of ''Y'' by the normal closure of the set
:$S=\$

*For abelian groups the coequalizer is particularly simple. It is just the factor group ''Y'' / im(''f'' – ''g''). (This is the cokernel of the morphism ''f'' – ''g''; see the next section).
*In the category of topological spaces, 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 functors 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 of natural numbers under addition, considered as a one-object category. In particular, this shows that while every coequalizing arrow is epic, it is not necessarily surjective.

Properties

*Every coequalizer is an epimorphism.
*In a topos, every epimorphism is the coequalizer of its kernel pair.

Special cases

In categories with zero morphisms, one can define a ''cokernel'' 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-sets actually form abelian groups). 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 ''f'', ''g'' : ''X'' → ''Y'' in a category ''C'' is a coequalizer as defined above, but with the added property that given any functor ''F'': ''C'' → ''D'', ''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
*Pushout

Notes

References

*Saunders Mac Lane: Categories for the Working Mathematician, Second Edition, 1998.
*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)