$\newcommand{\lax}{\mathsf{lax}}\newcommand{\oplax}{\mathsf{oplax}}\newcommand{\str}{\mathsf{str}}\renewcommand{\S}{\mathbb{S}}\newcommand{\F}{\mathbb{F}}\newcommand{\Hom}{\mathrm{Hom}}$We have tensor products $\otimes_{\mathbb{N}}$ and $\otimes_{\mathbb{Z}}$ of commutative monoids and abelian groups, determined by requiring natural bijections of the form \begin{align*} \Hom_{\mathsf{CMon}}(A\otimes_\mathbb{N}B,C) &\cong\Hom_{\mathsf{CMon}}(A,\mathbf{Hom}_{\mathsf{CMon}}(B,C)),\\ \Hom_{\mathsf{Ab}}(A\otimes_\mathbb{Z}B,C) &\cong\Hom_{\mathsf{Ab}}(A,\mathbf{Hom}_{\mathsf{Ab}}(B,C)) \end{align*} to hold. Similarly, it has been shown that there is a tensor product $\otimes_{\mathbb{F}}$ in the $2$-category of symmetric monoidal categories, determined by requiring a natural isomorphism of categories of the form \begin{equation} \mathsf{SymMonFun}^{\mathsf{strong}}(\mathcal{C}\otimes_\mathbb{F}\mathcal{D},\mathcal{E}) \cong \mathsf{SymMonFun}^{\mathsf{strong}}(\mathcal{C},\mathsf{SymMonFun}^{\mathsf{strong}}(\mathcal{D},\mathcal{E})),\tag{1} \end{equation} or, equivalently, by asking for a natural isomorphism of categories $$ \mathsf{SymMonFun}^{\mathsf{strong}}(\mathcal{C}\otimes_\mathbb{F}\mathcal{D},\mathcal{E}) \cong \mathsf{SymMonFun}^{\mathsf{strong},\mathsf{bil}}(\mathcal{C}\times\mathcal{D},\mathcal{E}), $$ where the objects of the category on the RHS are “bilinear symmetric monoidal functors $F\colon\mathcal{C}\times\mathcal{D}\to\mathcal{E}$”, coming equipped with natural families of isomorphisms \begin{align*} F(A,B\otimes_{\mathcal{D}}B') &\overset{\sim}{\dashrightarrow} F(A,B)\otimes_{\mathcal{E}}F(A,B'),\\ F(A\otimes_{\mathcal{C}}A',B) &\overset{\sim}{\dashrightarrow} F(A,B)\otimes_{\mathcal{E}}F(A',B),\\ F(\mathbf{1}_{\mathcal{C}},B) &\overset{\sim}{\dashrightarrow} \mathbf{1}_{\mathcal{E}},\\ F(A,\mathbf{1}_{\mathcal{D}}) &\overset{\sim}{\dashrightarrow} \mathbf{1}_{\mathcal{E}}. \end{align*} However, both proofs that I'm aware of―the first given by Hyland–Powell using abstract $2$-categorical machinery and the second by Gepner–Groth–Nikolaus via $\infty$-categories–are non-constructive. There's also Schmitt's work, in which they construct a tensor product $\otimes_{\mathbb{F}}'$ characterised by a natural isomorphism of the form $$ \mathsf{SymMonFun}^{\color{blue}{\mathsf{strict}}}(\mathcal{C}\otimes'_\mathbb{F}\mathcal{D},\mathcal{E}) \cong \mathsf{SymMonFun}^{\mathsf{strong}}(\mathcal{C},\mathsf{SymMonFun}^{\mathsf{strong}}(\mathcal{D},\mathcal{E})). $$
Question 1. What is an explicit description of the tensor product $\otimes_{\mathbb{F}}$ making $(1)$ above hold?
Question 2. I believe there should be further variants of the tensor product $\otimes_{\mathbb{F}}$, going off two orthogonal directions:
- Replacing $\mathsf{SymMonCats}$ by $\mathsf{2Ab}$, the $2$-category of abelian $2$-groups (also known as Gr-categories or Picard groupoids), there should be a tensor product $\otimes_{\mathbb{S}}$ whose unit is now given by the $1$-truncation of the sphere spectrum, rather than the groupoid of sets and permutations. The $2$-category $\mathsf{2Ab}$ is to $\mathsf{SymMonCats}$ as $\mathsf{Ab}$ is to $\mathsf{CMon}$.
- Replacing strong monoidal functors (on both sides of $(1)$) by strict, lax, or oplax ones should give tensor products $\otimes^\str_\F$, $\otimes^\lax_\F$, and $\otimes^\oplax_\F$.
Do the tensor products $\otimes_\S$, $\otimes^\str_\S$, $\otimes^\lax_\S$, $\otimes^\oplax_\S$, $\otimes^\str_\F$, $\otimes^\lax_\F$, and $\otimes^\oplax_\F$ indeed exist? If so, what is an explicit description of $\otimes_\S$ and $\otimes^\lax_\F$?
(Why should one care? Here are two reasons:
While semirings and rings are monoids in $(\mathsf{CMon},\otimes_{\mathbb{N}},\mathbb{N})$ and $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, bimonoidal categories and $2$-rings are pseudomonoids in $(\mathsf{SymMonCats},\otimes_{\F},\F)$ and $(\mathsf{2Ab},\otimes_\S,\S)$. (Well, almost: the data is very slightly different (isos instead of monos or just any morphisms) and only $19$ of Laplaza's $22$ axioms hold in the non-symmetric case).
Just like
- The abelian group completion of a commutative monoid is given by $\mathbb{Z}\otimes_\mathbb{N}-$;
- The $\mathbb{E}_{\infty}$-group completion of an $\mathbb{E}_{\infty}$-space is given by $QS^0\otimes_{\mathbb{F}}-$;
- The ring completion of a semiring is given by $\mathbb{Z}\otimes_\mathbb{N}-$;
I suspect the $2$-ring completion functor of Baas–Dundas–Richter–Rognes might be given by $\S\otimes_\F-$, leading to an alternative description of their celebrated construction.)
