Editing Cokernel (section)

== Formal definition ==
One can define the cokernel in the general framework of [[category theory]]. In order for the definition to make sense the category in question must have [[zero morphism]]s. The '''cokernel''' of a [[morphism]] {{math|''f'' : ''X'' → ''Y''}} is defined as the [[coequalizer]] of {{mvar|f}} and the zero morphism {{math|0<sub>''XY''</sub> : ''X'' → ''Y''}}.

Explicitly, this means the following. The cokernel of {{math|''f'' : ''X'' → ''Y''}} is an object {{mvar|Q}} together with a morphism {{math|''q'' : ''Y'' → ''Q''}} such that the diagram

<div style="text-align: center;">[[Image:Cokernel-01.svg|class=skin-invert]]</div>

[[commutative diagram|commutes]]. Moreover, the morphism {{mvar|q}} must be [[universal property|universal]] for this diagram, i.e. any other such {{math|''q''′ : ''Y'' → ''Q''′}} can be obtained by composing {{mvar|q}} with a unique morphism {{math|''u'' : ''Q'' → ''Q''′}}:

<div style="text-align: center;">[[Image:Cokernel-02.png|class=skin-invert]]</div>

As with all universal constructions the cokernel, if it exists, is unique [[up to]] a unique [[isomorphism]], or more precisely: if {{math|''q'' : ''Y'' → ''Q''}} and {{math|''q''′ : ''Y'' → ''Q''′}} are two cokernels of {{math|''f'' : ''X'' → ''Y''}}, then there exists a unique isomorphism {{math|''u'' : ''Q'' → ''Q''′}} with {{math|1=''q''' = ''u'' ''q''}}.

Like all coequalizers, the cokernel {{math|''q'' : ''Y'' → ''Q''}} is necessarily an [[epimorphism]]. Conversely an epimorphism is called ''[[normal morphism|normal]]'' (or ''conormal'') if it is the cokernel of some morphism. A category is called ''conormal'' if every epimorphism is normal (e.g. the [[category of groups]] is conormal).

=== Examples ===
In the [[category of groups]], the cokernel of a [[group homomorphism]] {{math|''f'' : ''G'' → ''H''}} is the [[quotient group|quotient]] of {{mvar|H}} by the [[Normal closure (group theory)|normal closure]] of the image of {{mvar|f}}. In the case of [[abelian group]]s, since every [[subgroup]] is normal, the cokernel is just {{mvar|H}} [[Ideal (ring theory)|modulo]] the image of {{mvar|f}}:
:<math>\operatorname{coker}(f) =  H / \operatorname{im}(f).</math>

=== Special cases ===
In a [[preadditive category]], it makes sense to add and subtract morphisms. In such a category, the [[coequalizer]] of two morphisms {{mvar|f}} and {{mvar|g}} (if it exists) is just the cokernel of their difference:

: <math>\operatorname{coeq}(f, g) = \operatorname{coker}(g - f).</math>

In an [[abelian category]] (a special kind of preadditive category) the [[image (category theory)|image]] and [[coimage]] of a morphism {{mvar|f}} are given by
:<math>\begin{align}
    \operatorname{im}(f) &= \ker(\operatorname{coker} f), \\
  \operatorname{coim}(f) &= \operatorname{coker}(\ker f).
\end{align}</math>

In particular, every abelian category is normal (and conormal as well). That is, every [[monomorphism]] {{mvar|m}} can be written as the kernel of some morphism. Specifically, {{mvar|m}} is the kernel of its own cokernel:
:<math>m = \ker(\operatorname{coker}(m))</math>