Normal monomorphism: Difference between revisions
(New page: ==Definition== ===In a preadditive category=== In a preadditive category (i.e., a category enriched over the category of Abelian groups), a '''normal monomo...) |
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===In a preadditive category=== | ===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=== | ===In a protomodular category=== | ||
{{fillin}} | {{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]]. | |||
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 in a preadditive category is termed normal if there exists an epimorphism for some object such that is a kernel of : in other words, is an equalizer of and the zero morphism from to .
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 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 such that is the equalizer of and the zero morphism from to .
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.