Kernel of a morphism in a category
A concept generalizing that of the kernel of a linear transformation of vector spaces, the kernel of a homomorphism of groups, rings, etc. Let be a category with null (or zero) morphisms. A morphism
is called a kernel of a morphism
if
and if every morphism
for which
can be uniquely represented as
. A kernel of a morphism
is denoted by
.
If and
are both kernels of
, then
for a unique isomorphism
. Conversely, if
and if
is an isomorphism, then
is a kernel of
. Thus, the kernels of a morphism
form a subobject of
, denoted by
.
If , then
is a monomorphism. In general, the converse is not true; a monomorphism which occurs as a kernel is called a normal monomorphism. The kernel of the null morphism
is the identity morphism
. The kernel of
exists if and only if
contains a null object (cf. Null object of a category).
Kernels do not always exist in a category with null morphisms. On the other hand, in a category with a null object a morphism
has a kernel if and only if a pullback of
and
exists in
.
The concept of the "kernel of a morphism" is the dual to that of the "cokernel of a morphism" .
Comments
The concept "kernel of a pair of morphismskernel of a pair of morphisms" (not to be confused with "kernel pair of a morphism" ) is also frequently used. In English, the usual name for this concept is an equalizer. An equalizer of a parallel pair of morphism is a morphism
such that
and such that every
satisfying
factors uniquely through
. Kernels are a special case of equalizers:
is a kernel of
if and only if it is an equalizer of
and
. Conversely, in an additive category an equalizer of
and
is the same thing as a kernel of
; but in general the notion of equalizer is more widely applicable, since it does not require the existence of null morphisms. A monomorphism which occurs as an equalizer is called a regular monomorphism.
References
[a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |
[a2] | J. Adámek, "Theory of mathematical structures" , Reidel (1983) |
Kernel of a morphism in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_morphism_in_a_category&oldid=14382