nLab cokernel

Idea

Definition

In a category with a terminal object $1$, the cokernel of a morphism $f : A \to B$ is the pushout

$$coker(f) \;\coloneqq\; 1 \underset{A}{\sqcup} B \phantom{AAAAAA} \array{ A &\stackrel{\phantom{A}f\phantom{A}}{\longrightarrow}& B \\ \big\downarrow &{}^{(po)}& \big\downarrow \\ 1 &\longrightarrow& coker(f) }$$

Remark

In the case when the terminal object is in fact zero object, one can, more explicitly, characterize the object $coker(f)$ as the object (unique up to unique isomorphism) that satisfies the following universal property:

for every object $C$ and every morphism $h \colon B \longrightarrow C$ such that $h \circ f = 0$ is the zero morphism, there is a unique morphism $\phi \colon coker(f) \to C$ such that $h = \phi \circ i$.

Remark

The notion of cokernel is dual to that of kernel. A cokernel in a category $\mathcal{C}$ is a kernel in the opposite category $\mathcal{C}^{op}$.

Properties

Exactness properties

• Taking cokernels is a right exact functor on arrow-categories.

Examples

Example

In the category Ab of abelian groups the cokernel of a morphism $f : A \to B$ is the quotient of $B$ by the image (of the underlying morphism of sets) of $f$.

Example

More generally, for $R$ any ring, this is true in the category $R$Mod of modules: the cokernel of a morphism is the quotient by its set-theoretic image.

Example

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.

Example

In an abelian category the coimage of any morphism $f$ is the cokernel of its kernel

$$coim(f) = coker(ker(f)) \,.$$

Related concepts

• kernel

• cocone

• cofiber, homotopy cofiber