Cokernel (categorical)

Let $\mathcal C$ be a category with a zero object, and hence with zero morphisms $0_{A,B}:A\to B$ . Given a morphism $f:A\to B$ , a cokernel of $f$ is a morphism

q:B\to Q

such that:

$q\circ f = 0_{A,Q}$ , and
for every morphism $s:B\to T$ with $s\circ f=0_{A,T}$ , there exists a unique morphism $v:Q\to T$ with $v\circ q = s.$

This construction is dual to a kernel.

Equivalent characterizations

Equivalently, $q:B\to Q$ is a coequalizer of the parallel pair $f,0_{A,B}:A\rightrightarrows B$ .

Remarks

A cokernel, if it exists, is unique up to unique isomorphism. Cokernels are epimorphisms (because coequalizers are epic).

Examples

$\mathbf{Ab}$ . For $f:A\to B$ a homomorphism, $\operatorname{coker}(f)\cong B/\operatorname{im}(f)$ , and the cokernel map is the quotient $B\twoheadrightarrow B/\operatorname{im}(f)$ .

$R$ - $\mathbf{Mod}$ . For $f:M\to N$ an $R$ -linear map, $\operatorname{coker}(f)\cong N/\operatorname{im}(f)$ .

$\mathbf{Grp}$ . For a group homomorphism $f:G\to H$ , the cokernel is the quotient $H \twoheadrightarrow H/\langle\!\langle f(G)\rangle\!\rangle$ where $\langle\!\langle f(G)\rangle\!\rangle$ is the normal closure of the subgroup $f(G)$ in $H$ . (This is the coequalizer of $f$ and the trivial map $G\to H$ .)