Definition:Kernel (Category Theory)/Definition 1

Definition
Let $\mathbf C$ be a category.

Let $A$ and $B$ be objects of $\mathbf C$.

Let $f : A \to B $ be a morphism in $\mathbf C$. 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 $A$ together with the morphism $K \to A$ coming from the definition of pullback.

Also see

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