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).
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 speak 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 r1,2 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=\\ |\ 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.
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).
See also
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