# Day convolution for bimonoidal categories

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.

Regarding Q2: probably there is a way to avoid going deep into coherence conditions: instead of proving by hand the equivalence between promonoidal structures on $$C$$ and biclosed monoidal structures on $$\hat C$$, one can resort to a more conceptual pov.

What happens for pro/monoidal categories is that there is a pseudomonad $$S$$ on $$\sf Cat$$ with the property that $$S$$ lifts to a pseudomonad $$\hat S$$ on $$\sf Prof$$ (the Kleisli bicategory of $$P=\hat{(-)} = [(-)^{op},{\sf Set}]$$), and pseudo-$$S$$-algebra structures correspond to pseudo-$$\hat S$$-algebra structures (this is an equivalence of categories, in the appropriate sense; see here).

I believe a similar argument holds for every (almost every?) monad $$S$$ equipped with a distributive law over $$P$$ (the presheaf construction); this does not fall short from an equivalence $$\{S\text{-algebra structures on } PX\} \cong \{\hat S\text{-algebra structures on } X\}$$ where $$PX$$ is regarded as an object of $$\sf Cat$$, and $$X$$ as an object of $${\sf Kl}(P)$$.

Regarding Q1: have you tried to find the distributive and annullator morphisms for the putative 2-rig structure on $$\widehat{C}$$?

I was trying to find at least one distributive morphism, and I have no idea how to reduce $$F\hat{\otimes}(H\hat{\oplus} K)$$ to/from $$F\hat{\otimes} H \,\hat{\oplus}\, F\hat\otimes K$$, if $$F,H,K : \widehat{C}$$. If I'm not wrong (this is very back-of-the-envelope coend calculus), \begin{align*} F\hat\otimes H &= \int^{UA}FU\times HA\times [\_, U\otimes A]\\ F\hat\otimes K &= \int^{U'B}FU'\times KB\times [\_, U'\otimes B] \end{align*} whereas \begin{align*} F \hat\otimes \,(H\hat\oplus K) &= \int^{UV} FU \times (H\hat\oplus K)V \times [\_, U\otimes V] \\ &=\int^{UVAB} FU \times HA \times KB \times [V, A\oplus B] \times [\_, U\otimes V] \\ &=\int^{UAB} FU \times HA \times KB \times [\_, U\otimes (A\oplus B)] \\ &=\int^{UAB} FU \times HA \times KB \times [\_, U\otimes A \oplus U\otimes B] \\ \end{align*} ...and now we're stuck, unless we have either

• R1. a compatibility between $$\oplus$$ and $$\times$$, perhaps another distributive morphism;
• R2. a siftedness condition ensuring that $$\int^{UA}FU\times HA\times [\_, U\otimes A] \oplus \int^{U'B}FU'\times KB\times [\_, U'\otimes B]$$ can be reduced to an integral on just $$U$$.

Actually, you need both in order for the computation to proceed; but the conjunction of R1 and R2 is quite strong, as you can see.

Edit: the situation with annullators (for Laplaza, morphisms $${\bf 0}\otimes X \to {\bf 0}$$ and $$X\otimes {\bf 0} \to \bf 0$$) is even worse!

Let's open $$F \hat\otimes {\bf 0}$$ recalling that in this case $$\bf 0$$ is the representable $$y{\bf 0}$$ on the additive unit of $$C$$: \begin{align*} \int^{UV} FU \times [V,{\bf 0}] \times [\_,U\otimes V] &=\int^U FU \times [\_, U\otimes {\bf 0}] \\ &\overset{\rho_U}\to\int^U FU \times [\_, {\bf 0}]\\ &=\varinjlim F \times [\_, {\bf 0}] \end{align*} the cartesian structure on $$\sf Set$$ now entails that this is $$\bf 0$$ if and only if either factor is empty, but I see no way in which this can be or even map into $$y{\bf 0}$$ again, as it should.

• > have you tried to find the distributive and annullator morphisms for the putative 2-rig structure on $\hat{C}$? I hadn't. But yes, this seems totally hopeless in general! Thanks!
– Théo
Aug 27 at 20:16
• A bit off-topic, but I think things are as in this question: Day convolution works very well for duoidal categories, but not at all for bimonoidal ones. For the latter it totally fails, but this paper shows that it sends duoidal categories to duoidal categories (and there's also a correspondence involving "produoidal categories").
– Théo
Aug 27 at 20:18
• Additionally, since categories of presheaves have a pointwise product monoidal structure and Day convolution together with it is duoidal, in fact it (probably) defines a functor $$\mathsf{Day}\text{ }\mathsf{convolution} \colon \{ \text{n-fold monoidal categories} \} \to \{ \text{(n+1)-fold monoidal categories} \}!$$
– Théo
Aug 27 at 20:19
• (well, modulo size issues)
– Théo
Aug 27 at 20:26