Editing Kernel (category theory) (section)

==Relation to other categorical concepts==
The dual concept to that of kernel is that of [[cokernel]].
That is, the kernel of a morphism is its cokernel in the [[opposite category]], and vice versa.

As mentioned above, a kernel is a type of binary equaliser, or [[difference kernel]].
Conversely, in a [[preadditive category]], every binary equaliser can be constructed as a kernel.
To be specific, the equaliser of the morphisms ''f'' and ''g'' is the kernel of the [[subtraction|difference]] ''g'' &minus; ''f''.
In symbols:
:eq (''f'', ''g'') = ker (''g'' &minus; ''f'').
It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphisms cannot be subtracted.

Every kernel, like any other equaliser, is a [[monomorphism]].
Conversely, a monomorphism is called ''[[normal morphism|normal]]'' if it is the kernel of some morphism.
A category is called ''normal'' if every monomorphism is normal.

[[Abelian categories]], in particular, are always normal.
In this situation, the kernel of the [[cokernel]] of any morphism (which always exists in an abelian category) turns out to be the [[image (category theory)|image]] of that morphism; in symbols:
:im ''f'' = ker coker ''f'' (in an abelian category)
When ''m'' is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know ''which'' morphism the monomorphism is a kernel of, to wit, its cokernel.
In symbols:
:''m'' = ker (coker ''m'') (for monomorphisms in an abelian category)