Kernel of a morphism in a category

A concept generalizing that of the kernel of a linear transformation of vector spaces, the kernel of a homomorphism of groups, rings, etc. Let $\mathfrak{K}$ be a category with zero or null morphisms. A morphism $\mu : K \to A$ is called a kernel of a morphism $\alpha : A \to B$ if $\mu \, \alpha = 0$ and if every morphism $\phi$ for which $\phi \, \alpha = 0$ can be uniquely represented as $\phi = \psi \, \mu$. A kernel of a morphism $\alpha$ is denoted by $\ker \alpha$.

If $\mu$ and $\mu '$ are both kernels of $\alpha$, then $\mu ' = \xi \, \mu$ for a unique isomorphism $\xi$. Conversely, if $\mu = \ker \alpha$ and if $\xi$ is an isomorphism, then $\mu ' = \xi \, \mu$ is a kernel of $\alpha$. Thus, the kernels of a morphism $\alpha$ form a subobject of $A$, denoted by $\operatorname{Ker} \alpha$.

If $\mu = \ker \alpha$, then $\mu$ is a monomorphism. In general, the converse is not true; a monomorphism which occurs as a kernel is called a normal monomorphism. The kernel of the null morphism $0 : A \to B$ is the identity morphism $\mathbf{1}_A$. The kernel of $\mathbf{1}_A$ exists if and only if $\mathfrak{K}$ contains a null object (cf. Null object of a category).

Kernels do not always exist in a category with null morphisms. On the other hand, in a category $\mathfrak{K}$ with a null object a morphism $\alpha : A \to B$ has a kernel if and only if a pullback of $\alpha$ and $0 : 0 \to B$ exists in $\mathfrak{K}$.

The concept of the "kernel of a morphism" is the dual to that of the "cokernel of a morphism" .

Comments

The concept "kernel of a pair of morphisms" (not to be confused with "kernel pair of a morphism" ) is also frequently used. In English, the usual name for this concept is an equalizer. An equalizer of a parallel pair of morphism $\alpha, \beta : A \to B$ is a morphism $\mu : E \to A$ such that $\mu \, \alpha = \mu \, \beta$ and such that every $\phi$ satisfying $\phi \, \alpha = \phi \, \beta$ factors uniquely through $\mu$. Kernels are a special case of equalizers: $\mu$ is a kernel of $\alpha$ if and only if it is an equalizer of $\alpha$ and $0 : A \to B$. Conversely, in an additive category an equalizer of $\alpha$ and $\beta$ is the same thing as a kernel of $\alpha - \beta$; but in general the notion of equalizer is more widely applicable, since it does not require the existence of null morphisms. A monomorphism which occurs as an equalizer is called a regular monomorphism.

References

[a1]	B. Mitchell, "Theory of categories" , Acad. Press (1965)
[a2]	J. Adámek, "Theory of mathematical structures" , Reidel (1983)