# nLab normal monomorphism

Normal (mono/epi)morphisms

category theory

# Normal (mono/epi)morphisms

## Idea

The concept of normal monomorphism generalizes the concept of normal subgroup inclusions.

One often considers regular monomorphisms and regular epimorphisms. In the theory of abelian categories, one often equivalently uses normal monomorphisms and epimorphisms. In general, the concept makes sense in a category with zero morphisms, that is in a category enriched over pointed sets.

## Definition

A monomorphism $f\colon A \to B$ is normal if it is the kernel of some morphism $g\colon B \to C$, that is if it is the equalizer of $g$ and the zero morphisms $0_{B,C}\colon B \to C$.

An epimorphism $f\colon B \to A$ is normal if it is the cokernel of some morphism $g\colon C \to B$, that is if it is the coequalizer of $g$ and the zero morphism $0_{C,B}\colon C \to B$.

Note that a normal monomorphism in $C$ is the same as a normal epimorphism in the opposite category $C^{op}$, and vice versa.

A category is normal if every monomorphism is normal; it is conormal if every epimorphism is normal, and it is binormal if it is both normal and conormal. Note that $C$ is normal if and only if $C^{op}$ is conormal, and vice versa.

## Examples

The inclusion function of a subgroup into a group is normal if and only if the subgroup is normal; this is the origin of the terminology for normal monomorphisms.

Every normal monomorphism or epimorphism is clearly regular; the converse holds in any Ab-enriched category (the equalizer of $f$ and $g$ is the kernel of $f-g$).

Every abelian category must be binormal; this is one of the axioms in a common definition of abelian category.

Discussion of normalizer constructions for monomorphisms is in

• James Richard Andrew Gray, Normalizers, centralizers and action representability in semiabelian categories, Applied Categorical Structures 22(5-6), 981–1007, 2014.