# Is there a universal way to construct a bimonoidal category structure on $\mathsf{PSh}(\mathcal{C})$ from such a structure on $\mathcal{C}$?

The Day convolution monoidal category structure $$(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$$ on the category of presheaves of a monoidal category $$(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$$ satisfies the following universal properties:

• In A universal property of the convolution monoidal structure, Im–Kelly prove that it is the free monoidal cocompletion of $$\mathcal{C}$$. Here, a monoidal category is monoidally cocomplete if it is cocomplete and the functors $$A\otimes-$$ and $$-\otimes B$$ preserve colimits for all $$A,B\in\mathrm{Obj}(\mathcal{C})$$. Im–Kelly then prove:

The monoidal category $$(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$$ is the universal monoidally cocomplete category on $$\mathcal{C}$$ in that, given any monoidally cocomplete monoidal category $$(\mathcal{D},\otimes,\mathbf{1}_{\mathcal{D}})$$, precomposition with $$よ\colon\mathcal{C}\hookrightarrow\mathsf{PSh}(\mathcal{C})$$ defines an equivalence of categories $$よ^*\colon\mathsf{Fun}^{\otimes,\mathsf{strong}}_{\mathsf{cocont.}}(\mathsf{PSh}(\mathcal{C}),\mathcal{D})\longrightarrow\mathsf{Fun}^{\otimes,\mathsf{strong}}(\mathcal{C},\mathcal{D}).$$ That is, $$(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$$ is uniquely determined by the following requirements:

1. The Yoneda embedding $$よ\colon\mathcal{C}\hookrightarrow\mathsf{PSh}(\mathcal{C})$$ is strong monoidal.
2. $$\circledast$$ is cocontinuous in each variable.

The analogue of the first of these for bimonoidal categories, however, doesn't work.

Question. Given a bimonoidal category $$(\mathcal{C},\otimes,\oplus,\mathbf{1}_{\mathcal{C}},\mathbf{0}_{\mathcal{C}})$$, is there a universal way to put a bimonoidal category structure on $$\mathsf{PSh}(\mathcal{C})$$?

In particular, is there a bimonoidal category structure on $$\mathsf{PSh}(\mathcal{C})$$ such that, given another bimonoidal category $$(\mathcal{D},\otimes_{\mathcal{D}},\oplus_{\mathcal{D}},\mathbf{1}_{\mathcal{D}},\mathbf{0}_{\mathcal{D}})$$, we have an equivalence of categories $$\left\{ \begin{gathered} \text{symmetric strong }\color{red}{\text{bi}}\text{monoidal}\\ \text{functors \mathcal{C}\times\mathcal{D}\to\mathsf{Sets}} \end{gathered} \right\} \cong \left\{ \begin{gathered} \text{symmetric strong }\color{red}{\text{bi}}\text{monoidal}\\ \text{functors \mathcal{D}\to\mathsf{PSh}(\mathcal{C})} \end{gathered} \right\},$$ natural in $$\mathcal{D}$$?

$$\def\C{\mathcal{C}}\def\Set{\mathsf{Set}}$$ (What follows comes from a private chat with Todd Trimble)

Notation. If $$(\C,\otimes,\oplus)$$ is a bimonoidal category, I will call $$\otimes$$ the multiplicative structure and $$\oplus$$ the additive structure; if $$\oplus$$ is the cocartesian monoidal structure, I will call $$\C$$ a 2-rig, following https://arxiv.org/abs/2103.00938.

I am convinced that in general (=for a general bimonoidal category) little can be said, because one needs some compatibility between the multiplicative structure and co/products. Even when $$\C$$ is a 2-rig the most I can formulate until now is a

Conjecture. When $$\C$$ is a 2-rig, the category $$[\C^o,\Set]_\times$$ of functors $$\C^o \to \Set$$ that are product-preserving (=sending coproducts in $$\C$$ to products in $$\Set$$) is the free 2-rig on $$\C$$.

In order for this freeness property to be legitimate, the least we can ask is that

1. $$F\C=[\C^o,\Set]_\times$$ is a 2-rig if the multiplicative structure is Day convolution;
2. The Yoneda embedding $$y : \C \to F\C$$ is a morphism of 2-rigs.

Unfortunately, I am still unable to prove that the Day convolution restricts to "models" of the "theory" $$\C$$ (it is a fruitful intuition to think of $$\C$$ like it was a Lawvere theory even if it's not, were it only because it's easier to query google with questions ;-) )

Update: I couldn't because it's not true, but it falls very short from being true, in the sense that the conjecture is "true up to reflecting the monoidal structure": observe that $$[\C^o,\Set]_\times$$ has many desirable properties for the free cocomplete 2-rig on $$\C$$:

1. $$[\C^o,\Set]_\times$$ is a cocomplete[¹], reflective subcategory of the entire $$[\C^o,\Set]$$.
2. the yoneda embedding $$y : \C \to [\C^o,\Set]$$ clearly factors through $$[\C^o,\Set]_\times$$.
3. $$F\C=[\C^o,\Set]_\times$$ has the following universal property, if $$\mathcal D$$ is cocomplete: $$\{\text{cocontinuous } g : F\C \to\mathcal D\}\cong \{\text{coproduct preserving } h : \C \to \mathcal D\}$$
4. Another useful universal property for $$[\C^o,\Set]_\times$$ is that it is the cocompletion of $$\C$$ under sifted colimit.

All these facts turn out to be useful to establish that $$[\C^o,\Set]_\times$$ is the free 2-rig over $$\C$$: the coend that expresses the usual Day convolution formula is to be interpreted in a way that we first take the coend in the usual category of presheaves, but then to that apply the reflection functor $$r$$ that is left adjoint to the full inclusion. So, it's not true that the ordinary Day convolution takes a pair of product-preserving functors to a product-preserving functor; you have to sheafify. After you do that, everything falls into place.

Look how neat everything becomes!

Suppose we have a functor $$F$$ in $$[\C^o,\Set]_\times$$; by 4 above, it's a sifted colimit of representables $$\C(-, c)$$. Since we assume $$\C$$ is a 2-rig, we have a composite of (finite) coproduct-preserving functors

$$Y : \C \stackrel{C \otimes -}{\to} \C \stackrel{y}{\to} [\C^o,\Set]_\times$$

under the equivalence of point 3, this becomes a colimit preserving functor $$[\C^o,\Set]_\times \to [\C^o,\Set]_\times$$, exactly the reflected convolution $$\C(-,C)\ast \_$$.

But since the convolution we want to build must be cocontinuous, it is uniquely determined by this construction!

The missing detail is some result ensuring that $$\ast$$ is a monoidal structure. All in all I expect this to be a consequence of a theorem about transport of monoidal structures $$\otimes$$ (Day convolution of presheaves) into $$\ast$$ ("reflected" Day convolution) given some lax monoidality assumptions on $$r$$, so I won't enter the details.

Let me just add a final neat detail: the coend formula expressing the Day convolution product is a reflexive coequalizer; a reflexive coequalizer is a sifted colimit, so it can be interpreted as the usual, pointwise colimit in Set. Moreover, the sums = coproducts involved in the coend formula are filtered colimits of finite coproducts; again, filtered colimits are sifted colimits. So the only "re-interpretation" involved with the reflector resides in the way you compute finite coproducts in models, i.e.: you don't compute finite coproducts set-wise, but at the level of models, and every other colimit as usual.

[¹] But colimits in $$[\C^o,\Set]_\times$$ are not computed as colimits in the presheaf category (thikn again to the case of Lawvere theories and coproducts of monoids...), and this will be crucial, because for example, in $$[\C^o,\Set]_\times$$ the object $$\C(-,A+B)$$ has the universal property of the coproduct $$\C(-,A)+\C(-,B)$$.