Recall that

• The biinitial monoidal category with 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 $(\mathsf{FinSets},*)$ consisting of the category of finite sets and morphisms between them 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. Do we know what, if it exists, is the biinitial semiring category with a (commutative) semiring object?

Recall that

• The biinitial monoidal category with a monoid is given by the augmented simplex category together with the monoid $([0],\sigma^{0}_{0},\delta^{0}_{0})$ there.
• The biinitial symmetric monoidal category with a commutative monoid is given by the pair $(\mathsf{FinSets},*)$ consisting of the category of finite sets and morphisms between them 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 semiring category. Do we know what, if it exists, is the biinitial semiring category with a (commutative) semiring object?

Recall that

• The biinitial monoidal category with 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. Do we know what, if it exists, is the biinitial semiring category with a (commutative) semiring object?

Recall that

• The biinitial monoidal category with 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 $(\mathsf{FinSets},*)$ consisting of the category of finite sets and morphisms between them 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. Do we know what, if it exists, is the biinitial semiring category with a (commutative) semiring object?