Kernel

Remarks

Equivalently, $k$ is an equalizer of $f$ and $0 \colon X \to Y$. A kernel is unique up to a unique isomorphism, and is written $\ker f \to X$. Every kernel is a monomorphism, by the uniqueness in clause 2.

Example: in the category of modules over a ring, the inclusion $k \colon \set{x \in X : f(x) = 0} \to X$ of the kernel of the homomorphism is a kernel. If $f \circ u = 0$, then $u$ takes values in this submodule, so $u$ factors through $k$, and uniquely since $k$ is injective.

Non-example: for the reduction $f \colon \Z \to \Z/2$ of abelian groups, the inclusion $k \colon 4\Z \to \Z$ satisfies clause 1 but not clause 2. The inclusion $u \colon 2\Z \to \Z$ satisfies $f \circ u = 0$ but does not factor through $4\Z$, since $2 \notin 4\Z$.