Definition:Kernel (Category Theory)

Definition
Let $C$ be a category.

Let $f : A \to B $ be a morphism.

Definition 1: for categories with initial objects
Let $C$ have an inital object $0$.

A kernel of $f$ is a morphism $\operatorname{ker}(f) \to A$ which is a pullback of the unique morphism $0 \to B$ via $f$ to $A$.

Definition 2: for categories with zero objects
Let $C$ have an zero object $0$.

A kernel of $f$ is the equalizer of $f$ and the zero morphism $0 : A \to B$.

Also see

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