Preabelian category: Difference between revisions
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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]]. | ||
== | ==Relation with other properties== | ||
===Stronger | ===Stronger properties=== | ||
* [[Weaker than::Abelian category]] | * [[Weaker than::Abelian category]] | ||
===Weaker | ===Weaker properties=== | ||
* [[Stronger than::Additive category]] | * [[Stronger than::Additive category]] | ||
* [[Stronger than::Preadditive category]] | * [[Stronger than::Preadditive category]] | ||
Latest revision as of 06:54, 26 December 2008
This article defines a preadditive category property: a property that can be evaluated to true/false given a preadditive category.
View a complete list of preadditive category properties|Get preadditive category property lookup help |Get exploration suggestions
VIEW RELATED: Preadditive category property implications | Preadditive category property non-implications | Preadditive category metaproperty satisfactions | Preadditive category metaproperty dissatisfactions | Preadditive category property satisfactions |Preadditive category property dissatisfactions
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.