Abelian category
This article defines a preadditive category property: a property that can be evaluated to true/false given a preadditive category.
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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
An Abelian category is a category enriched over the monoidal category of Abelian groups satisfying the following additional conditions:
- It admits finite biproducts 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.