Vipul: New page: {{preadditive category property}} ==Definition== An Abelian category is a category enriched over the monoidal category of Abelian groups satisfying the ...

2008-12-26T06:52:38Z

New page: {{preadditive category property}} ==Definition== An Abelian category is a category enriched over the monoidal category of Abelian groups satisfying the ...

New page

{{preadditive category property}}

==Definition==

An [[Abelian category]] is a [[category]] [[enriched category|enriched]] over the [[monoidal category of Abelian groups]] satisfying the following additional conditions:

# It admits finite [[biproduct]]s and has a zero object.
# Every morphism has a [[kernel]] and a [[cokernel]].
# Every [[monomorphism]] is a [[normal monomorphism]] and every [[epimorphism]] is a [[normal epimorphism]]. Equivalently, every monomorphism equals the kernel of its cokernel and every epimorphism equals the cokernel of its kernel.

A category enriched over Abelian groups is termed a [[preadditive category]]. A preadditive category satisfying condition (1) is termed an [[additive category]]. A preadditive category satisfying conditions (1) and (2) is termed a [[preabelian category]].

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Exact category]]
* [[Stronger than::Preabelian category]]
* [[Stronger than::Additive category]]
* [[Stronger than::Normal category]]
* [[Stronger than::Conormal category]]

Abelian category - Revision history

Vipul: New page: {{preadditive category property}} ==Definition== An Abelian category is a category enriched over the monoidal category of Abelian groups satisfying the ...