Kernel (categorical)

Throughout, assume $\mathcal C$ is a category with a zero object (e.g. any additive category), so that for any objects $A,B$ there is a distinguished zero morphism $0_{A,B}:A\to B$ .

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

k:K\to A

such that:

$f\circ k = 0_{K,B}$ , and
(Universal property) for every morphism $t:T\to A$ with $f\circ t=0_{T,B}$ , there exists a unique morphism $u:T\to K$ with $k\circ u = t.$

It is the categorical version of the solution set of $f(x)=0$ , expressed by a universal property rather than by elements.

Equivalent characterizations

Equivalently, $k:K\to A$ is an equalizer of the parallel pair $f,0_{A,B}:A\rightrightarrows B$ .

Remarks

A kernel, if it exists, is unique up to unique isomorphism. In any category, kernels are monomorphisms (because equalizers are monic).

Examples

$\mathbf{Ab}$ . For a homomorphism $f:A\to B$ of abelian groups, $\ker(f)\subseteq A$ with inclusion $\ker(f)\hookrightarrow A$ is the categorical kernel.

$R$ $-$ $\mathbf{Mod}$ . For an $R$ -linear map $f:M\to N$ , the usual submodule $\{m\in M : f(m)=0\}$ with inclusion is the kernel.

$\mathbf{Grp}$ . For a group homomorphism $f:G\to H$ , the usual kernel $\{g\in G : f(g)=e\}$ with inclusion is the categorical kernel (here the zero morphism $G\to H$ is the constant map to the identity element).