# Normal monomorphism

A morphism having the characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let $\mathfrak{K}$ be a category with zero morphisms. A morphism $\mu : U \to A$ is called a normal monomorphism if every morphism $\phi : X \to A$ for which it always follows from $\mu \, \alpha = 0$, $\alpha : A \to Y$, that $\phi \, \alpha = 0$, can be uniquely represented in the form $\phi = \phi ' \mu$. The kernel of any morphism (cf. Kernel of a morphism in a category) is a normal monomorphism. The converse is not true, in general; however, if cokernels (cf. Cokernel) of morphisms exist in $\mathfrak{K}$, then every normal monomorphism turns out to be the kernel of its cokernel. In an Abelian category every monomorphism is normal. The concept of a normal monomorphism is dual to that of a normal epimorphism.
The above definition is not entirely standard: many authors would define a normal monomorphism to be a morphism which occurs as a kernel. (The term normal subobject is also in use, for an isomorphism class of normal monomorphisms.) In categories without zero morphisms, a substitute for normal monomorphisms is provided by regular monomorphisms, which are those morphisms which occur as equalizers (cf. Kernel of a morphism in a category). Every normal monomorphism is regular, but not conversely: in the category of all groups every injective homomorphism is a regular monomorphism, but a normal monomorphism $G \to H$ is an isomorphism of $G$ onto a normal subgroup of $H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.