category with duals (list of them)
dualizable object (what they have)
A bipermuatative category is a semistrict rig category. More concretely, it is a permutative category with a second symmetric monoidal category structure that distributes over , with, again, some of the coherence laws required to hold strictly.
For a plain ring, regarded as a discrete rig category, it is a bipermutative category. The corresponding K-theory of a bipermutative category is ordinary cohomology with coefficients in , given by the Eilenberg-MacLane spectrum .
with being the symmetric group of permutations of elements. The two monoidal structures ar given by addition and multiplication of natural numbers. This is a bipermutative version of the , the core of the category FinSet of finite sets.