Preabelian category
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.