# Definition:Kernel (Category Theory)

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This page is about Kernel in the context of Category Theory. For other uses, see Kernel.

## 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 $\mathbf C$.

### Definition 1: for categories with initial objects

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 $A$.

### Definition 2: for categories with zero objects

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 $G$.