In category theory, a normal morphism (resp. Conormal morphism ) is a morphism that is the core (resp. Cokadrom ) of a certain morphism. A normal category is a category in which every monomorphism is normal. Accordingly, in the conormal category, every epimorphism is conormal. A category is called binormal if it is normal and conormal at the same time.
Examples
In the category of groups, the monomorphism f from H to G is normal if and only if its image is a normal subgroup of G. This is the reason for the origin of the term “normal morphism”.
On the other hand, every epimorphism in the category of groups is conormal (since it is the co-nucleus of its core), therefore this category is conormal.
In an arbitrary abelian category, each monomorphism is the core of its cokernel and each epimorphism is the co-nucleus of its core. Consequently, the abelian categories are binormal. The category of abelian groups is the most important example of an abelian category and, in particular, each subgroup of an abelian group is normal.
Notes
- Mitchell, Barry (1965), Theory of categories, - Pure and applied mathematics 17, Academic Press, - Section I.14 - ISBN 978-0-124-99250-4 .