category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A bimonoidal category is a category equipped with two monoidal category structure (in the role of the tensor product) and (in the role of the direct sum) such that distributes over up to coherent natural isomorphism.
This may be thought of as a natural categorification of the notion of rig in the way that the notion of monoidal category is the categorification of the notion of monoid.
A bimonoidal category is a category with two structures and of a monoidal category, which satisfy a distributivity law up to natural isomorphism among each, paralleling that for addition and multiplication in a rig.
The notion of a bipermutative category is a strictification of the notion of symmetric bimonidal categories.
Every symmetric bimonoidal category is equivalent to a bipermutative category (May, prop. VI 3.5).
Every bimonoidal category is equivalent to a strict bimonoidal category (Guillou, theorem 1.2).
Set is bimonoidal under disjoint union and cartesian product.
Ab, Mod, Vect are bimonoidal under direct sum and tensor product of abelian groups/tensor product of modules.
The coherence for the distributivity law in bimonoidal categories has been given in
M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72.
G. Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin, 1974, pp. 281-375.
where these categories are called ring categories. Discussion with an eye towers the K-theory of a bipermutative category is in