Normal epimorphism

Definition

In a preadditive category

In a preadditive category (i.e., a category enriched over the monoidal category of Abelian groups), a normal epimorphism is an epimorphism that occurs as the cokernel of some monomorphism. In other words, an epimorphism $f:A\to B$ in a preadditive category $\mathcal{C}$ is termed normal if there exists a monomorphism $g:C\to A$ for some object $C$ such that $f$ is a cokernel of $g$: in other words, $f$ is a coequalizer of $g$ and the zero morphism from $C$ to $A$.

Every additive category, and more specifically, every Abelian category, is preadditive, so the notion of normal monomorphism makes sense for such a category. For an Abelian category, every monomorphism is normal.

In a category enriched over pointed sets

Suppose $\mathcal{C}$ is a category that is enriched over the monoidal category of pointed sets. In other words, the morphism sets of $\mathcal{C}$ have the additional structure of pointed sets and this structure is preserved by composition. This could happen, for instance, if $\mathcal{C}$ has a zero object. The distinguished point in each morphism set is termed the zero morphism.

A normal epimorphism in $\mathcal{C}$ is a monomorphism that occurs as the coequalizer of some monomorphism with the zero morphism. In symbols, an epimorphism $f:A\to B$ in $\mathcal{C}$ is termed normal if there exists a monomorphism $g:C\to A$ for some object $C$ such that $f$ is the coequalizer of $g$ and the zero morphism from $C$ to $A$.