Note
Commutative Rings as Split Grothendieck Rings
Abstract
Every commutative unital ring is the split Grothendieck ring of a small additive rigid symmetric monoidal category in which every object is self-dual. This answers Question 15 of Levy. We first construct categories of polynomial objects and finite-pattern maps whose coproduct and product relations are prescribed by nonnegative polynomials. A unification argument supplies finite limits, while a development invariant determines the object isomorphisms. Passing to additive spans produces a rigid category; categorical dimension identifies its split Grothendieck ring with the original Burnside ring. An object representing the negative of the unit then allows us to extract any prescribed ring from the resulting polynomial extension. We also describe the construction for , where a square-zero endomorphism has trace one.