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Coequalizer - Definition | | Coequalizer - Definition: Encyclopedia II - Coequalizer - Definition | | The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y.
More explicity, the coequalizer can be defined as an object Q and 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′ ...
See also:Coequalizer, Coequalizer - Definition, Coequalizer - Examples, Coequalizer - Special cases | | | Coequalizer, Coequalizer - Definition, Coequalizer - Examples, Coequalizer - Special cases, equalizer, coproduct, pushout – a special colimit construction making use a coequalizer and coproduct | | |
| | | Coequalizer: Encyclopedia II - Coequalizer - Definition
Coequalizer - Definition
The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y.
More explicity, the coequalizer can be defined as an object Q and 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, the coequalizer, if it exists, is unique up to a unique isomorphism.
It can be shown that the coequalizer q is an epimorphism in any category.
Other related archivesCategory theory, abelian groups, category, category of groups, category of sets, category theory, cokernel, colimit, commutes, coproduct, dual, epimorphism, equalizer, equivalence relation, factor group, functions, group homomorphisms, hom-sets, isomorphism, mathematics, morphisms, normal closure, preadditive categories, pushout, quotient, universal, universal constructions, up to, zero morphisms
 Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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