Editing Kernel (category theory) (section)

==Definition==
Let '''C''' be a [[category theory|category]].
In order to define a kernel in the general category-theoretical sense, '''C''' needs to have [[zero morphism]]s.
In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary [[morphism]] in '''C''', then a kernel of ''f'' is an [[Equaliser (mathematics)|equaliser]] of ''f'' and the zero morphism from ''X'' to ''Y''.
In symbols:
:ker(''f'') = eq(''f'', 0<sub>''XY''</sub>)

To be more explicit, the following [[universal property]] can be used. A kernel of ''f'' is an [[Object (category theory)|object]] ''K'' together with a morphism ''k'' : ''K'' → ''X'' such that:
* ''f''{{Hair space}}∘''k'' is the zero morphism from ''K'' to ''Y'';

<div style="text-align: center;">[[File:First_property_of_the_kernel.svg|100px|class=skin-invert]]</div>

* Given any morphism ''{{prime|k}}'' : ''{{prime|K}}'' → ''X'' such that ''f''{{Hair space}}∘''{{prime|k}}'' is the zero morphism, there is a unique morphism ''u'' : ''{{prime|K}}'' → ''K'' such that ''k''∘''u'' = ''{{prime|k}}''.

<div style="text-align: center;">[[File:Properties_of_a_kernel.svg|200px|class=skin-invert]]</div>

As for every universal property, there is a unique isomorphism between two kernels of the same morphism, and the morphism ''k'' is always a [[monomorphism]] (in the categorical sense). So, it is common to talk of ''the'' kernel of a morphism. In [[concrete categories]], one can thus take a [[subset]] of ''{{prime|K}}'' for ''K'', in which case, the morphism ''k'' is the [[inclusion map]]. This allows one to talk of ''K'' as the kernel, since ''k'' is implicitly defined by ''K''. There are non-concrete categories, where one can similarly define a "natural" kernel, such that ''K'' defines ''k'' implicitly.

Not every morphism needs to have a kernel, but if it does, then all its kernels are isomorphic in a strong sense: if ''k'' : ''K'' → ''X'' and {{math|''{{ell}}'' : ''L'' → ''X''}} are kernels of ''f'' : ''X'' → ''Y'', then there exists a unique [[isomorphism]] φ : ''K'' → ''L'' such that {{math|1=''{{ell}}''}}∘φ = ''k''.