# Normal monomorphism

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
A morphism having the characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let be a category with zero morphisms. A morphism is called a normal monomorphism if every morphism for which it always follows from , , that , can be uniquely represented in the form . 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 , 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 is an isomorphism of onto a normal subgroup of . However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.