Coequalizador

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

En la teoría de categorías , un coequalizador (o coecualizador ) es una generalización de un cociente por una relación de equivalencia con objetos en una categoría arbitraria . Es la construcción categórica dual al ecualizador .

Definición

A coequalizer es un colimit del diagrama que consta de dos objetos X y Y y dos paralelas morfismos f , g  : X → Y .

Más explícitamente, un coecualizador se puede definir como un objeto Q junto con un morfismo q  : Y → Q tal que q ∘ f = q ∘ g . Además, el par ( Q , q ) debe ser universal en el sentido de que dado cualquier otro par ( Q ′, q ′) existe un morfismo único u  : Q → Q ′ tal que u ∘ q = q ′. Esta información puede ser capturada por los siguientesdiagrama conmutativo :

Como ocurre con todas las construcciones universales , un coequalizador, si existe, es único hasta un isomorfismo único (por eso, por abuso del lenguaje, a veces se habla del coequalizador de dos flechas paralelas).

Se puede demostrar que un coequalizador q es un epimorfismo en cualquier categoría.

Ejemplos

• En la categoría de conjuntos , el coecualizador de dos funciones f , g  : X → Y es el cociente de Y por la relación de equivalencia más pequeña tal que para cada , tenemos . ^[1] En particular, si R es una relación de equivalencia en un conjunto Y , y r ₁ , r ₂ son las proyecciones naturales ( R ⊂ Y × Y ) → Y entonces el coecualizador de r ₁${\displaystyle \sim }$${\displaystyle x\in X}$${\displaystyle f(x)\sim g(x)}$ 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

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

• 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 ${\displaystyle S^{1}}$ 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.

See also

• Coproduct
• Pushout

Notes

1. Barr, Michael; Wells, Charles (1998). Category theory for computing science (PDF). p. 278. Archived from the original (PDF) on March 4, 2016. Retrieved July 25, 2013.

References

• Saunders Mac Lane: Categories for the Working Mathematician, Second Edition, 1998.
• Coequalizers - page 65
• Absolute coequalizers - page 149

External links

• Interactive Web page which generates examples of coequalizers in the category of finite sets. Written by Jocelyn Paine.