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| In [[category theory]], a '''coequalizer''' (or '''coequaliser''') is a generalization of a [[quotient set|quotient]] by an [[equivalence relation]] to objects in an arbitrary [[category (mathematics)|category]]. It is the categorical construction [[dual (category theory)|dual]] to the [[equaliser (mathematics)|equalizer]] (hence the name).
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| == Definition ==
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| A '''coequalizer''' is a [[colimit]] of the diagram consisting of two objects ''X'' and ''Y'' and two parallel [[morphism]]s ''f'', ''g'' : ''X'' → ''Y''.
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| More explicitly, a coequalizer can be defined as an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that ''q'' ∘ ''f'' = ''q'' ∘ ''g''. Moreover, the pair (''Q'', ''q'') must be [[universal property|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 [[commutative diagram|commutes]]:
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| <div style="text-align: center;">[[Image:Coequalizer-01.png]]</div>
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| As with all [[universal construction]]s, 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).
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| It can be shown that a coequalizer ''q'' is an [[epimorphism]] in any category.
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| == Examples ==
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| *In the [[category of sets]], the coequalizer of two [[function (mathematics)|function]]s ''f'', ''g'' : ''X'' → ''Y'' is the [[quotient set|quotient]] of ''Y'' by the smallest [[equivalence relation]] <math>~\sim</math> such that for every <math>x\in X</math>, we have <math>f(x)\sim g(x)</math>.<ref>{{cite book |last1=Barr |first1=Michael |authorlink1=Michael Barr (mathematician) |last2=Wells |first2=Charles |authorlink2=Charles Wells (mathematician) |year=1998 |title=Category theory for computing science |page=278 |url=http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf |accessdate=2013-07-25 |format=PDF}}</ref> In particular, if ''R'' is an equivalence relation on a set ''Y'', and ''r''<sub>1</sub>, ''r''<sub>2</sub> are the natural projections (''R'' ⊂ ''Y'' × ''Y'') → ''Y'' then the coequalizer of ''r''<sub>1</sub> and ''r''<sub>2</sub> is the quotient set ''Y''/''R''.
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| *The coequalizer in the [[category of groups]] is very similar. Here if ''f'', ''g'' : ''X'' → ''Y'' are [[group homomorphism]]s, their coequalizer is the [[quotient group|quotient]] of ''Y'' by the [[Normal closure (group theory)|normal closure]] of the set
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| :<math>S=\{f(x)g(x)^{-1}\ |\ x\in X\}</math>
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| *For [[abelian group]]s 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).
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| *In the [[category of topological spaces]], the circle object <math>S^1</math> can be viewed as the coequalizer of the two inclusion maps from the standard 0-simplex to the standard 1-simplex.
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| *Coequalisers 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 exactly one non-identity arrow going between them. The coequaliser 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 coequalising arrow is [[Epimorphism|epic]], it is not necessarily [[Surjective_function|surjective]].
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| == Properties ==
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| *In a [[topos]], every [[epimorphism]] is the coequalizer of its kernel pair.
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| *Every coequalizer is an epimorphism.
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| == Special cases ==
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| In categories with [[zero morphism]]s, one can define a ''[[cokernel]]'' of a morphism ''f'' as the coequalizer of ''f'' and the parallel zero morphism.
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| In [[preadditive category|preadditive categories]] it makes sense to add and subtract morphisms (the [[hom-set]]s actually form [[abelian group]]s). In such categories, one can define the coequalizer of two morphisms ''f'' and ''g'' as the cokernel of their difference:
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| :coeq(''f'', ''g'') = coker(''g'' – ''f'').
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| A stronger notion is that of an '''absolute coequalizer''', this is a coequalizer that is preserved under all [[functors]].
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| Formally, an absolute coequalizer of a pair <math>f,g: X \to Y</math> in a category C is a coequalizer as defined above but with the added property that given any functor <math>F: C \to D</math> 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.
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| == See also ==
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| *[[equalizer (mathematics)]]
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| *[[coproduct]]
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| *[[Pushout (category theory)|pushout]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *[[Saunders Mac Lane]]: [[Categories for the Working Mathematician]], Second Edition, 1998.
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| *Coequalizers - page 65
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| *Absolute coequalizers - page 149
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| == External links ==
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| *[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 [http://www.j-paine.org/ Jocelyn Paine].
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| [[Category:Limits (category theory)]]
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