The kernel of a function 
,
where 
is a set and
is a set with
a distinguished element denoted by 0, is defined by
 |
so the kernel gives the elements from the original set that are mapped to zero by the function.
is therefore a subset
of 
. For a linear
transformation, the kernel is also called the null
space.
The related image of a function is defined by
 |
is therefore a subset
of 
.
In a preadditive category, a kernel of a morphism 
is a morphism 
with 
having the following universal
property: for every morphism 
satisfying 
, there is a unique morphism 
such that 
.
Here 0 denotes the zero element of the relevant hom-set. This definition does not require a zero
object. The dual construction is the cokernel (The
Stacks Project Authors 2026).
