Normal monomorphism: Difference between revisions

Revision as of 12:22, 25 December 2008

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 $C$ 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. For an Abelian category, every monomorphism is normal.

In a category enriched over pointed sets

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

A normal monomorphism in $C$ is a monomorphism that occurs as the equalizer of some epimorphism with the zero morphism. In symbols, a monomorphism $f : A \to B$ in $C$ 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

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.

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 ===In a preadditive category===
-In a [[preadditive category]] (i.e., a category [[enriched 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 <math>f:A \to B</math> in a preadditive category <math>\mathcal{C}</math> is termed '''normal''' if there exists an epimorphism <math>g:B \to C</math> for some object <math>C</math> such that <math>f</math> is a kernel of <math>g</math>: in other words, <math>f</math> is an [[equalizer]] of <math>g</math> and the zero morphism from <math>B</math> to <math>C</math>.
+In a [[preadditive category]] (i.e., a category [[enriched 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 <math>f:A \to B</math> in a preadditive category <math>\mathcal{C}</math> is termed '''normal''' if there exists an epimorphism <math>g:B \to C</math> for some object <math>C</math> such that <math>f</math> is a kernel of <math>g</math>: in other words, <math>f</math> is an [[equalizer]] of <math>g</math> and the zero morphism from <math>B</math> to <math>C</math>.
-Every [[additive category]], and more generally, every [[Abelian category]], is preadditive, so the notion of normal monomorphism makes sense for such a category.
+Every [[additive category]], and more generally, 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 <math>\mathcal{C}</math> is a [[category]] that is [[enriched category|enriched]] over the [[monoidal category of pointed sets]]. In other words, the morphism sets of <math>\mathcal{C}</math> have the additional structure of pointed sets and this structure is preserved by composition. This could happen, for instance, if <math>\mathcal{C}</math> has a [[zero object]]. The distinguished point in each morphism set is termed the ''zero morphism''.
+A '''normal monomorphism''' in <math>\mathcal{C}</math> is a [[monomorphism]] that occurs as the [[equalizer]] of some epimorphism with the zero morphism. In symbols, a monomorphism <math>f:A \to B</math> in <math>\mathcal{C}</math> is termed '''normal''' if there exists an epimorphism <math>g: B \to C</math> for some object <math>C</math> such that <math>f</math> is the equalizer of <math>g</math> and the zero morphism from <math>B</math> to <math>C</math>.
 ===In a protomodular category===
 {{fillin}}
+==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]].