# Normal (mono/epi)morphisms

## Idea

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.

Revised on September 11, 2011 00:52:02 by Mike Shulman (71.136.248.27)