Coequalizer

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In mathematics, 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 (hence the name).

1 Definition
2 Examples
3 Special cases
4 References
5 See also

[edit] 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 O f = q O 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′ for which the following diagram commutes:

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.

[edit] 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_1,2 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 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

$S=\{f(x)g(x)^{-1}\ |\ x\in X\}$

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.

[edit] 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 $f,g: X \to Y$ in a category C is a coequalizer as defined above but with the added property that given any functor $F: C \to D$ 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.