A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an abelian group under addition, we require only that it forms an abelian monoid. A commutative rig is a rig in which multiplication induces an abelian monoid.
A distributive category is a small category with finite products and coproducts, which I'll denote * and + respectively, such that the canonical morphism X * Y + X * Z \rightarrow X * (Y+Z) is an isomorphism. Essentially, taking products distributes over taking coproducts.
There's an apparent formal similarity between the definitions, and in fact you can get a commutative rig out of a distributive category C by taking the objects of the rig to be isomorphism classes of objects in the category (equivalently, considering the skeletal category) and letting coproducts correspond to addition and products to multiplication. For instance, if you start with the category of finite sets, you can skeletonize to obtain a commutative rig isomorphic to the rig of natural numbers.
So the question is, does every commutative rig arise this way from some distributive category? I suspect the answer is "no," and I can even think of some likely counterexamples (the nonnegative real numbers, Z), but I don't see an obvious way to prove that they are counterexamples.
Supposing that my hunch is correct, is there a nice way to classify the rigs that do arise in this manner? Is there a way to classify the commutative rings that arise when you add negatives to these rigs (analogously to the Grothendieck group)?