Kernel (categorical)

Kernels are a categorical version of “solutions of $f(x)=0$”, defined using universal properties.

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 $0_{A,B}:A\to B$.

Definition (Kernel)

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

such that:

1. $f\circ k = 0_{K,B}$, and
2. (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 $k\circ u = t.$

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

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

Examples

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

2. $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.

3. $\mathbf{Grp}$. 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).