A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite coproducts (this includes an initial object $0$) with the property that $\otimes$ preserves finite coproducts in each variable. In other words, we have a distributive law $$X \otimes (\bigoplus_{i \in I} Y_i) \cong \bigoplus_{i \in I} (X \otimes Y_i).$$ This is also known as a finitary distributive symmetric monoidal category. Notice that to each symmetric monoidal category with finite coproducts we may associate a (possibly large) commutative semiring: The elements are the isomorphism classes of objects in $\mathcal{C}$. The addition is $[X] + [Y] := [X \oplus Y]$, the multiplication is $[X] \cdot [Y] := [X \otimes Y]$. The zero element is $[0]$, the multiplicative unit is $[1]$.

A familar example is the the symmetric monoidal category of vector bundles on a space $X$, which produces the semiring of isomorphism classes of vector bundles on $X$. If $X$ is a point, we see that the underlying semiring of $\mathsf{FinSet}$ is $\mathbb{N}$. As a consequence, we may think of symmetric monoidal categories with finite coproducts as a possible categorification of semirings.

Note, however, that not every commutative semiring arises from the construction above. Basically this is because we have agreed that the addition is not induced by some symmetric monoidal structure which distributes over $\otimes$, but rather from the coproduct. The coproduct satisfies $X \oplus Y \cong 0 \Rightarrow X \cong Y \cong 0$. It follows that in the underlying semiring only $0$ has an additive inverse.

Question. How can we classify those commutative semirings which are induced by symmetric monoidal categories with finite coproducts?

Every commutative monoid with the property $a+b=0 \Rightarrow a=b=0$ arises from a category with finite coproducts, see SE/834869. Does every commutative semiring with the same property arise from a symmetric monoidal category with finite coproducts?


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