Definition:Kernel (Category Theory)/Definition 1

Definition
Let $\mathbf C$ have an initial object $0$.

A kernel of $f$ is a morphism $\map \ker f \to A$ which is a pullback of the unique morphism $0 \to B$ via $f$ to $A$.

Also see

 * Equivalence of Definitions of Kernel of Morphism
 * Definition:Cokernel (Category Theory)
 * Definition:Image (Category Theory)
 * Definition:Coimage (Category Theory)