coequalizer

A coequalizer is the colimit of a diagram consisting of two objects X and Y and two parallel morphisms {{nowrap|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 {{nowrap|q : Y → Q}} such that {{nowrap|1=q ∘ f = q ∘ g}}. Moreover, the pair {{nowrap|(Q, q)}} must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism {{nowrap|u : Q → Q′}} such that {{nowrap|1=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 coequalizing arrow q is an epimorphism in any category.

• In the category of sets, the coequalizer of two functions {{nowrap|f, g : X → Y}} is the quotient of Y by the smallest equivalence relation ~ such that for every {{nowrap|x ∈ X}}, we have {{nowrap|f(x) ~ g(x)}}.{{cite book|last1=Barr|first1=Michael|url=http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf|title=Category theory for computing science|last2=Wells|first2=Charles|publisher=Prentice Hall International Series in Computer Science|year=1998|page=278|format=PDF|authorlink1=Michael Barr (mathematician)|authorlink2=Charles Wells (mathematician)}} In particular, if R is an equivalence relation on a set Y, and r₁, r₂ are the natural projections {{nowrap|(R ⊂ Y × Y) → Y}} then the coequalizer of r₁ and r₂ is the quotient set {{nowrap|Y / R}}. (See also: quotient by an equivalence relation.)
• The coequalizer in the category of groups is very similar. Here if {{nowrap|f, g : X → Y}} are group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set
• : $S=\backslash \{f(x)g(x)^\{-1\}\backslash mid\; x\backslash in\; X\backslash \}$
• For abelian groups the coequalizer is particularly simple. It is just the factor group {{nowrap|Y / im(f – g)}}. (This is the cokernel of the morphism {{nowrap|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.

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

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 {{nowrap|f, g : X → Y}} in a category C is a coequalizer as defined above, but with the added property that given any functor {{nowrap|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.

• Coproduct
• Pushout

{{reflist}}

• {{cite book | last = Mac Lane | first = Saunders | author-link = Saunders Mac Lane | year = 1998 | title = Categories for the Working Mathematician | series = Graduate Texts in Mathematics | volume = 5 | edition = 2nd | publisher = Springer-Verlag | isbn = 0-387-98403-8 | zbl=0906.18001 }}
• Coequalizers – page 65
• Absolute coequalizers – page 149

• [https://web.archive.org/web/20080916162345/http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page], which generates examples of coequalizers in the category of finite sets. Written by [https://web.archive.org/web/20081223001815/http://www.j-paine.org/ Jocelyn Paine].

{{Category theory}}

Category:Limits (category theory)