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:

1. It admits finite biproducts (A biproduct is something that serves the role of both a product and a coproduct).
2. 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.