Preabelian category - Revision history

Vipul at 06:54, 26 December 2008

2008-12-26T06:54:04Z

← Older revision		Revision as of 06:54, 26 December 2008
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			{{preadditive category property}}

	==Definition==		==Definition==

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	A category enriched over Abelian groups is termed a [[defining ingredient::preadditive category]], and a preadditive category satisfying condition (1) above is termed an [[additive category]].		A category enriched over Abelian groups is termed a [[defining ingredient::preadditive category]], and a preadditive category satisfying condition (1) above is termed an [[additive category]].

	==~~Related notions~~==		==Relation with other properties==

	===Stronger ~~notions~~===		===Stronger properties===

	* [[Weaker than::Abelian category]]		* [[Weaker than::Abelian category]]

	===Weaker ~~notions~~===		===Weaker properties===

	* [[Stronger than::Additive category]]		* [[Stronger than::Additive category]]
	* [[Stronger than::Preadditive category]]		* [[Stronger than::Preadditive category]]

Vipul: /* In terms of preadditive category */

2008-12-26T06:48:17Z

In terms of preadditive category

← Older revision		Revision as of 06:48, 26 December 2008
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	More explicitly, it is a [[category]] [[enriched category\|enriched]] over the [[monoidal category of Abelian groups]] satisfying the following two conditions:		More explicitly, it is a [[category]] [[enriched category\|enriched]] over the [[monoidal category of Abelian groups]] satisfying the following two conditions:

	# It admits finite [[biproduct]]s (A biproduct is something that serves the role of both a product and a coproduct).		# It admits finite [[biproduct]]s (A biproduct is something that serves the role of both a product and a coproduct) and has a [[zero object]].
	# Every morphism has a kernel and a cokernel.		# Every morphism has a kernel and a cokernel.

	A category enriched over Abelian groups is termed a [[defining ingredient::preadditive category]], and a preadditive category satisfying condition (1) above is termed an [[additive category]].		A category enriched over Abelian groups is termed a [[defining ingredient::preadditive category]], and a preadditive category satisfying condition (1) above is termed an [[additive category]].

			==Related notions==

			===Stronger notions===

			* [[Weaker than::Abelian category]]

			===Weaker notions===

			* [[Stronger than::Additive category]]
			* [[Stronger than::Preadditive category]]

Vipul: New page: ==Definition== ===In terms of additive category=== A '''preabelian category''' (sometimes written '''preAbelian category''' or '''pre-Abelian category''') is an [[defining ingredient::ad...

2008-12-26T06:43:03Z

New page: ==Definition== ===In terms of additive category=== A '''preabelian category''' (sometimes written '''preAbelian category''' or '''pre-Abelian category''') is an [[defining ingredient::ad...

New page

==Definition==

===In terms of additive category===

A '''preabelian category''' (sometimes written '''preAbelian category''' or '''pre-Abelian category''') is an [[defining ingredient::additive category]] satisfying the additional condition that every morphism has a [[kernel]] and a [[cokernel]].

===In terms of preadditive category===

More explicitly, it is a [[category]] [[enriched category|enriched]] over the [[monoidal category of Abelian groups]] satisfying the following two conditions:

# It admits finite [[biproduct]]s (A biproduct is something that serves the role of both a product and a coproduct).
# Every morphism has a kernel and a cokernel.

A category enriched over Abelian groups is termed a [[defining ingredient::preadditive category]], and a preadditive category satisfying condition (1) above is termed an [[additive category]].