category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

and its applications to

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

. A normal category is a category in which every monomorphism is normal. A conormal category is one in which every epimorphism is conormal.

Definition

A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism. A category C is binormal if it's both normal and conormal. But note that some authors will use the word "normal" only to indicate that C is binormal.

Examples

In the category of groups, a monomorphism ''f'' from ''H'' to ''G'' is normal if and only if its image is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

of ''G''. In particular, if ''H'' is a subgroup of ''G'', then the inclusion map ''i'' from ''H'' to ''G'' is a monomorphism, and 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 conormal (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.

References

*Section I.14 {{DEFAULTSORT:Normal Morphism Morphisms