Recall that

- The biinitial [monoidal category with a monoid](https://mathoverflow.net/questions/115517/what-is-the-free-monoidal-category-generated-by-a-monoid) is given by the augmented simplex category together with the monoid $([0],\sigma^{0}_{0},\delta^{0}_{0})$ there.
- The biinitial monoidal category with a commutative monoid is given by the pair $(\mathbb{F},*)$ consisting of the groupoid of finite sets and permutations $\mathbb{F}$ equipped with the coproduct as the monoidal structure, and the triple $(*,*\coprod*\to*,\emptyset\to*)$ with $*$ the punctual set as the commutative monoid.

There's a natural notion of a [semiring object in a bimonoidal category](https://mathoverflow.net/questions/403014). Do we know what, if it exists, is the biinitial semiring category with a (commutative) semiring object?