Preabelian category: Difference between revisions
(New page: ==Definition== ===In terms of additive category=== A '''preabelian category''' (sometimes written '''preAbelian category''' or '''pre-Abelian category''') is an [[defining ingredient::ad...) |
|||
| Line 9: | Line 9: | ||
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]] | |||
Revision as of 06:48, 26 December 2008
Definition
In terms of additive category
A preabelian category (sometimes written preAbelian category or pre-Abelian category) is an 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 over the monoidal category of Abelian groups satisfying the following two conditions:
- It admits finite biproducts (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.
A category enriched over Abelian groups is termed a preadditive category, and a preadditive category satisfying condition (1) above is termed an additive category.