# $\mathbb{E}_\infty$-refinements of the graded tensor product of $\mathbb{Z}$-graded spectra

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:

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:

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:

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?