Cokernel
of a morphism in a category
The concept dual to the concept of the kernel of a morphism in a category. In categories of vector spaces, groups, rings, etc. it describes a largest quotient object of an object that annihilates the image of a homomorphism
.
Let be a category with null morphisms. A morphism
is called a cokernel of a morphism
if
and if any morphism
such that
can be expressed in unique way as
. A cokernel of a morphism
is denoted by
.
If and
then
for a unique isomorphism
.
Conversely, if and
is an isomorphism, then
is a cokernel of
. Thus, all cokernels of a morphism
form a quotient object of
, which is denoted by
. If
, then
is a normal epimorphism. The converse need not be true. The cokernel of the zero morphism
is
. The cokernel of the unit morphism
exists if and only if
contains a zero object.
In a category with a zero object, a morphism
has a cokernel if and only if
contains a co-Cartesian square with respect to the morphisms
and
. This condition is satisfied, in particular, for any morphism of a right locally small category with a zero object and products.
Comments
The co-Cartesian square, or fibred sum or pushout, of two morphisms ,
is (if it exists) a commutative diagram
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such that for any two morphisms ,
such that
there exists a unique morphism
for which
,
.
Cokernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cokernel&oldid=15734