Editing Coequalizer (section)

== 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]].