Coequalizer
In category theory, a coequalizer is a generalization of the quotient of a set by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.
Definition
A coequalizer is the colimit of a diagram consisting of two objects X and Y and two parallel morphisms.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 there exists a unique morphism such that. This information can be captured by the following commutative diagram:
[Image:Coequalizer-01.svg|x100px|class=skin-invert]
As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism.
It can be shown that a coequalizing arrow q is an epimorphism in any category.
Examples
- In the category of sets, the coequalizer of two functions is the quotient of Y by the smallest equivalence relation ~ such that for every, we have. In particular, if R is an equivalence relation on a set Y, and r1, r2 are the natural projections then the coequalizer of r1 and r2 is the quotient set.
- The coequalizer in the category of groups is very similar. Here if are group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set
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- For abelian groups the coequalizer is particularly simple. It is just the factor group..
- In the category of topological spaces, the circle object S1 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. In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference:
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 together with F is the coequalizer of F and F in the category D. Split coequalizers are examples of absolute coequalizers.