Normal monomorphism

Definition

In a preadditive category

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

In a protomodular category

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Examples

• The category of groups has a zero object (The trivial group) and can hence be viewed as a category enriched over pointed sets. A normal monomorphism in this category is an injective homomorphism whose image is a normal subgroup.