Definition:Kernel (Category Theory)/Definition 2

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 a zero object $0$.

A kernel of $f$ is a morphism $\ker(f) \to A$, which is an equalizer of $f$ and the zero morphism $0: A \to B$.

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)

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