Editing Normal morphism

{{Short description|Type of morphism}}
In [[category theory]] and its applications to [[mathematics]], a '''normal monomorphism''' or '''conormal epimorphism''' is a particularly well-behaved type of [[morphism]].
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 (category theory)|kernel]] of some morphism, and an epimorphism is '''conormal''' if it is the [[cokernel (category theory)|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.{{Citation needed|date=January 2010}}

==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]] 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.{{Citation needed|date=January 2010}}

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 group]]s is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.

==References==
*Section I.14 {{Mitchell TOC}}

{{DEFAULTSORT:Normal Morphism}}
[[Category:Morphisms]]