 Semi-additive category

A semi-additive category has a commutative-monoid enrichment and finite products in a category and coproducts in a category. Composition preserves addition and zero in each variable. Every finite product in a category is canonically a coproduct in a category and conversely, producing finite biproducts. Some authors use "semi-additive structure" for the commutative-monoid enrichment alone; that weaker convention also permits one-object categories without finite products in a category.

A preadditive category instead has abelian-group-valued hom sets. Semi-additivity alone does not require additive inverses.