**Semiring categories**, also called **rig categories** or **bimonoidal categories**, are pseudomonoids in the symmetric monoidal bicategory $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$¹. These are a categorification of semirings, the monoids in $(\mathsf{CMon},\otimes_{\mathbb{N}},\mathbb{N})$, and carry two monoidal structures, one additive and one multiplicative, with the multiplicative one being coherently bilinear over the additive monoidal structure. A great introduction for these is Johnson–Yau's *Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory*.

Similarly, **ring categories** are pseudomonoids in $(\mathsf{2Ab},\otimes_{\mathbb{S}},\mathbb{S})$. They are a categorification of rings, the monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, and are more simply those semiring categories having (weak) additive inverses (i.e. for each object $A$ in a ring category, there exists an object $-A$ such that $A\oplus(-A)\cong\mathbf{0}_{\mathcal{C}}$ via a coherent isomorphism).

It is well-known that given a monoidal category $(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$, we can use Day convolution to get a monoidal structure on presheaves on $\mathcal{C}$, obtaining a monoidal category $(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$.

**Question 1.** Given a semi/ring category $(\mathcal{C},\oplus,\otimes,\mathbf{1}_{\mathcal{C}},\mathbf{0}_{\mathcal{C}},\ldots)$, is the tuple $(\mathcal{C},\circledast^{\oplus},\circledast^{\otimes},\mathsf{h}_{\mathbf{1}_{C}},\mathsf{h}_{\mathbf{0}_{C}},\ldots)$ obtained by applying Day convolution to both monoidal structures also a semi/ring category?

**Question 2.** Day convolution gives a bijection
$$
\{\text{promonoidal structures on $\mathcal{C}$}\}
\cong
\{\text{biclosed monoidal structures on $\mathsf{PSh}(\mathcal{C}$})\}.
$$
Assuming the statement in question 1 holds, is there an analogue of this bijection for semi/ring categories?

¹More or less―for nonsymmetric bimonoidal categories, 19 of the 22 nonsymmetric bimonoidal category axioms of Johnson–Yau, Definition 2.1.2 hold. The exceptions are 2.1.13, 2.1.15, and 2.1.16.