Normal morphism

In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. A normal category is a category in which morphisms are normal, whenever reasonable.

Definition

A category C must have zero morphisms for the concept of normality to make complete sense. In that case, we say that a monomorphism is normal if it is the kernel of some morphism, and an epimorphism is normal (or conormal) if it is the cokernel of some morphism.

C itself is normal if every monomorphism is normal. C is conormal if every epimorphism is normal. Finally, C is binormal if it's both normal and conormal. But note that some authors will use only the word "normal" to indicate that C is actually binormal.

Examples

Suppose that G is a group and H is a subgroup of G. Then the inclusion map i from H to G is a monomorphism. i will be normal if and only if H is a normal subgroup of G. (In fact, this is the origin of the term "normal" for monomorphisms.) On the other hand, every epimorphism in the category of groups is normal (since it is the cokernel of its own kernel), so this category is conormal.

In an abelian category, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel. Thus, abelian categories are always binormal. The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.