Normal monomorphism - Revision history

Vipul at 05:05, 26 December 2008

2008-12-26T05:05:59Z

← Older revision		Revision as of 05:05, 26 December 2008
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	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>.		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. For an Abelian category, every monomorphism is normal.		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===		===In a category enriched over pointed sets===

Vipul at 12:22, 25 December 2008

2008-12-25T12:22:18Z

← Older revision		Revision as of 12:22, 25 December 2008
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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]].

Vipul: 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...

2008-12-25T12:16:30Z

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...

New page

==Definition==

===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>.

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===

{{fillin}}