Normal monomorphism

Definition

In a preadditive category

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

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

In a protomodular category

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