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 $A$.

Uniqueness

Since the kernel is defined by a universal property it is only unique up to unique isomorphism.

While for example in group theory the kernel of a group homomorphism $f : G \to H$ is a subset of $G$, not all categorical kernels of $f$ in the category of groups are subsets of $G$.

Also see

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