= 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.
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