normal monomorphism

Normal (mono/epi)morphisms


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.


A monomorphism f:ABf\colon A \to B is normal if it is the kernel of some morphism g:BCg\colon B \to C, that is if it is the equalizer of gg and the zero morphisms 0 B,C:BC0_{B,C}\colon B \to C.

An epimorphism f:BAf\colon B \to A is normal if it is the cokernel of some morphism g:CBg\colon C \to B, that is if it is the coequalizer of gg and the zero morphism 0 C,B:CB0_{C,B}\colon C \to B.

Note that a normal monomorphism in CC is the same as a normal epimorphism in the opposite category C opC^{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 CC is normal if and only if C opC^{op} is conormal, and vice versa.


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 ff and gg is the kernel of fgf-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 (