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The category $$\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathsf{Mod}_R)$$ of $\mathbb{Z}$-graded $R$-modules has a natural monoidal category structure obtained via Day convolution, whose monoidal product is given by

$$(A\otimes_RB)_n\overset{\mathrm{def}}{=}\bigoplus_{p+q=n}A_p\otimes_RB_q.$$

A natural question then arises: what kinds of braided or symmetric monoidal structures can we put on $\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R$ lifting the above monoidal structure?

The answer to this problem goes back to Joyal–Street and Eilenberg–Kelly, and can be rephrased as follows:

First, define the $2$-category $\mathcal{M}_{\mathbb{E}_{2},\mathbb{E}_{1}}(\mathcal{C})$ of braided monoidal refinements of a monoidal category $(\mathcal{C},\otimes_\mathcal{C},\mathbf{1}_{\mathcal{C}})$ to be the following (strict) fibre product in the $2$-category of categories:

enter image description here

Similarly, define the $2$-category $\mathcal{M}_{\mathbb{E}_{\infty},\mathbb{E}_{1}}(\mathcal{C})$ of symmetric monoidal refinements of $(\mathcal{C},\otimes_\mathcal{C},\mathbf{1}_{\mathcal{C}})$ as follows:

enter image description here

Then, Joyal–Street and Eilenberg–Kelly's result can be stated as the following pair of isomorphisms of discrete $2$-categories: \begin{align*} \mathcal{M}_{\mathbb{E}_{2},\mathbb{E}_{1}}(\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R) &\cong R^\times,\\ \mathcal{M}_{\mathbb{E}_{\infty},\mathbb{E}_{1}}(\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R) &\cong \mathrm{Inv}(R). \end{align*} (Here $\mathrm{Inv}(R)=\{a\in R\ |\ a^2=1\}$.)

For example, when $R=\mathbb{Z}$, we have $\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_\mathbb{Z}=\mathsf{Gr}_{\mathbb{Z}}\mathsf{Ab}$, which admits only two symmetries:

  • The one corresponding to $u=1$, given by $\beta(x\otimes y)=y\otimes x$;
  • The one corresponding to $u=-1$, given by the Koszul rule $\beta(x\otimes y)=(-1)^{|x||y|}y\otimes x$.

Meanwhile, for $R=\mathbb{Z}/24$, we have five symmetries, corresponding to $u=1$, $5$, $7$, $11$, or $13$.


Now, passing to $\infty$-categories, let $\mathcal{C}$ be an $\mathbb{E}_{k}$-monoidal category.

For $k\geq1$, define the $(\infty,2)$-category of $\mathbb{E}_{k}$-refinements of an $\mathbb{E}_{1}$-monoidal category $\mathcal{C}$ to be the following pullback of simplicial sets:

enter image description here

Finally, define $\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R$ for $R$ a ring spectrum by

$$\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_{\mathsf{disc}},\mathsf{Sp}),$$ and equip it with the Day convolution $\mathbb{E}_{1}$-monoidal structure.

Question. What do we know about the $(\infty,2)$-categories $\mathcal{M}_{\mathbb{E}_k,\mathbb{E}_{1}}(\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R)$ parametrising $\mathbb{E}_k$-refinements of the Day convolution $\mathbb{E}_1$-monoidal structure on $\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R$, for $2\leq k\leq\infty$?

Can we describe them in terms of properties of $R$, as in the classical case of $\mathbb{Z}$-graded $R$-modules?

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