Editing Coequalizer

{{Short description|Generalization of a quotient by an equivalence relation to objects in an arbitrary category}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}

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]].

== Definition ==
A '''coequalizer''' is the [[colimit]] of a diagram consisting of two objects ''X'' and ''Y'' and two parallel [[morphism]]s {{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 property|universal]] in the sense that given any other such pair (''Q''&prime;, ''q''&prime;) there exists a unique morphism {{nowrap|''u'' : ''Q'' → ''Q''&prime;}} such that {{nowrap|1=''u'' ∘ ''q'' = ''q''&prime;}}. This information can be captured by the following [[commutative diagram]]:
<div style="text-align: center;">[[Image:Coequalizer-01.svg|x100px|class=skin-invert]]</div>


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).

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 [[function (mathematics)|function]]s {{nowrap|''f'', ''g'' : ''X'' → ''Y''}} is the [[quotient set|quotient]] of ''Y'' by the smallest [[equivalence relation]] ~ such that for every {{nowrap|''x'' &isin; ''X''}}, we have {{nowrap|''f''(''x'') ~ ''g''(''x'')}}.<ref>{{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)}}</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 {{nowrap|(''R'' ⊂ ''Y'' &times; ''Y'') → ''Y''}} then the coequalizer of ''r''<sub>1</sub> and ''r''<sub>2</sub> 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 homomorphism]]s, their coequalizer is the [[quotient group|quotient]] of ''Y'' by the [[Normal closure (group theory)|normal closure]] of the set
*: <math>S=\{f(x)g(x)^{-1}\mid x\in X\}</math>
* For [[abelian group]]s 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 ''S''<sup>1</sup> 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 [[functor]]s 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 number]]s under addition, considered as a one-object category. In particular, this shows that while every coequalizing arrow is [[Epimorphism|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 morphism]]s, one can define a ''[[cokernel]]'' of a morphism ''f'' as the coequalizer of ''f'' and the parallel zero morphism.

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:
: 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 coequalizer]]s are examples of absolute coequalizers.

== See also ==
* [[Coproduct]]
* [[Pushout (category theory)|Pushout]]

== Notes ==
{{reflist}}

== References ==
* {{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

== External links ==
* [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)]]