Vipul at 05:06, 26 December 2008

2008-12-26T05:06:05Z

← Older revision		Revision as of 05:06, 26 December 2008
Line 5:		Line 5:
	In a [[preadditive category]] (i.e., a category [[enriched category\|enriched]] over the [[monoidal category of Abelian groups]]), a '''normal epimorphism''' is an [[epimorphism]] that occurs as the [[cokernel]] of some [[monomorphism]]. In other words, an epimorphism <math>f:A \to B</math> in a preadditive category <math>\mathcal{C}</math> is termed '''normal''' if there exists a monomorphism <math>g:C \to A</math> for some object <math>C</math> such that <math>f</math> is a cokernel of <math>g</math>: in other words, <math>f</math> is a [[coequalizer]] of <math>g</math> and the zero morphism from <math>C</math> to <math>A</math>.		In a [[preadditive category]] (i.e., a category [[enriched category\|enriched]] over the [[monoidal category of Abelian groups]]), a '''normal epimorphism''' is an [[epimorphism]] that occurs as the [[cokernel]] of some [[monomorphism]]. In other words, an epimorphism <math>f:A \to B</math> in a preadditive category <math>\mathcal{C}</math> is termed '''normal''' if there exists a monomorphism <math>g:C \to A</math> for some object <math>C</math> such that <math>f</math> is a cokernel of <math>g</math>: in other words, <math>f</math> is a [[coequalizer]] of <math>g</math> and the zero morphism from <math>C</math> to <math>A</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: New page: ==Definition== ===In a preadditive category=== In a preadditive category (i.e., a category enriched over the monoidal category of Abelian groups), a '''norm...

2008-12-26T05:04:05Z

New page: ==Definition== ===In a preadditive category=== In a preadditive category (i.e., a category enriched over the monoidal category of Abelian groups), a '''norm...

New page

==Definition==

===In a preadditive category===

In a [[preadditive category]] (i.e., a category [[enriched category|enriched]] over the [[monoidal category of Abelian groups]]), a '''normal epimorphism''' is an [[epimorphism]] that occurs as the [[cokernel]] of some [[monomorphism]]. In other words, an epimorphism <math>f:A \to B</math> in a preadditive category <math>\mathcal{C}</math> is termed '''normal''' if there exists a monomorphism <math>g:C \to A</math> for some object <math>C</math> such that <math>f</math> is a cokernel of <math>g</math>: in other words, <math>f</math> is a [[coequalizer]] of <math>g</math> and the zero morphism from <math>C</math> to <math>A</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.

===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 epimorphism''' in <math>\mathcal{C}</math> is a [[monomorphism]] that occurs as the [[coequalizer]] of some monomorphism with the zero morphism. In symbols, an epimorphism <math>f:A \to B</math> in <math>\mathcal{C}</math> is termed '''normal''' if there exists a monomorphism <math>g: C \to A</math> for some object <math>C</math> such that <math>f</math> is the coequalizer of <math>g</math> and the zero morphism from <math>C</math> to <math>A</math>.

Normal epimorphism - Revision history

Vipul at 05:06, 26 December 2008

Vipul: New page: ==Definition== ===In a preadditive category=== In a preadditive category (i.e., a category enriched over the monoidal category of Abelian groups), a '''norm...