category with duals (list of them)
dualizable object (what they have)
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.
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.
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).
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