nLab
Ab

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

$\mathrm{Ab}$ 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 $(\mathrm{Ab},\otimes )$ is a ring, while a monoid in $(\mathrm{Ab},\oplus )$ is just an abelian group (since $\oplus$ is the coproduct in $\mathrm{Ab}$, so every object has a unique monoid structure with respect to it).

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

category: category