Cokernel (categorical)

Cokernels are dual to kernels , capturing “quotienting out by the image of $f$” in contexts where that makes sense.

Throughout, assume $\mathcal C$ is a category with a zero object (e.g. any additive category ), hence zero morphisms $0_{A,B}:A\to B$.

Definition (Cokernel)

Given a morphism $f:A\to B$ in $\mathcal C$, a cokernel of $f$ is a morphism

such that:

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

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

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

Examples

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

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

3. $\mathbf{Grp}$. 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 $G\to H$.)