(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
In a category with a terminal object , the cokernel of a morphism is the pushout
In the case when the terminal object is in fact zero object, one can, more explicitly, characterize the object as the object (unique up to unique isomorphism) that satisfies the following universal property:
for every object and every morphism such that is the zero morphism, there is a unique morphism such that .
The notion of cokernel is dual to that of kernel. A cokernel in a category is a kernel in the opposite category .
In the category Ab of abelian groups the cokernel of a morphism is the quotient of by the image (of the underlying morphism of sets) of .
More generally, for any ring, this is true in the category Mod of modules: the cokernel of a morphism is the quotient by its set-theoretic image.
In the category Grp of general (not necessarily abelian) groups, the cokernel is instead the quotient group by the normal closure of the image.
The following example is by the very definition of abelian category.
In an abelian category the coimage of any morphism is the cokernel of its kernel
Last revised on July 8, 2023 at 13:46:45. See the history of this page for a list of all contributions to it.