nLab
Ab

Ab is the category of abelian groups. So, it has abelian groups as objects and group homomorphisms between these as morphisms.

Ab can be made into a monoidal category in several ways, the most notable involving direct sum and tensor product of abelian groups. A monoid internal to (Ab,) is a ring, while a monoid in (Ab,) is just an abelian group (since is the coproduct in Ab, so every object has a unique monoid structure with respect to it).

Categories enriched over Ab are called pre-additive categories or sometimes just additive categories. If they satisfy an extra exactness condition they are called abelian categories.

category: category