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Normal monomorphism
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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 
in a preadditive category 
is termed normal if there exists an epimorphism 
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 is a category that is enriched over the monoidal category of pointed sets. In other words, the morphism sets of have the additional structure of pointed sets and this structure is preserved by composition. This could happen, for instance, if has a zero object. The distinguished point in each morphism set is termed the zero morphism.
A normal monomorphism in is a monomorphism that occurs as the equalizer of some epimorphism with the zero morphism. In symbols, a monomorphism in is termed normal if there exists an epimorphism for some object C such that f is the equalizer of g and the zero morphism from B to C.
In a protomodular category
Fill this in later
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.
