Coequalizer

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Generalization of a quotient by an equivalence relation to objects in an arbitrary category

title: "Coequalizer" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["limits-(category-theory)"] description: "Generalization of a quotient by an equivalence relation to objects in an arbitrary category" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Coequalizer" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Generalization of a quotient by an equivalence relation to objects in an arbitrary category ::

In category theory, a coequalizer (or coequaliser) 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 f, g : X → Y.

More explicitly, a coequalizer of the parallel morphisms f and g can be defined as an object Q together with a morphism q : Y → Q such that . 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 . 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 (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).

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 f, g : X → Y is the quotient of Y by the smallest equivalence relation ~ such that for every x ∈ X, we have f(x) ~ g(x). In particular, if R is an equivalence relation on a set Y, and r1, r2 are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r1 and r2 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={f(x)g(x)^{-1}\mid 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 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 (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.

Notes

References

    • Coequalizers – page 65
    • Absolute coequalizers – page 149

References

  1. (1998). "Category theory for computing science". [[Prentice Hall International Series in Computer Science]].

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limits-(category-theory)