For ordinary categories, the assignment $mathcal{C}mapstomathsf{Mon}(mathcal{C})$ defines a functor $mathsf{Mon}colonmathsf{Alg}_{mathbb{E}_{k}}(mathsf{Cats})tomathsf{Alg}_{mathbb{E}_{k-1}}(mathsf{Cats})$, which is to say that:

- If $mathcal{C}$ is monoidal ($mathbb{E}_{1}$), then $mathsf{Mon}(mathcal{C})$ exists;
- If $mathcal{C}$ is braided ($mathbb{E}_{2}$), then $mathsf{Mon}(mathcal{C})$ is a monoidal category ($mathbb{E}_{1}$) in two different ways, related by replacing $beta_{A,B}$ by $beta^{-1}_{B,A}$;¹
- If $mathcal{C}$ is symmetric ($mathbb{E}_{3}=mathbb{E}_{4}=cdots$), then $mathsf{Mon}(mathcal{C})$ is braided ($mathbb{E}_{2}$), and also symmetric ($mathbb{E}_{3}$), since $mathbb{E}_{3}=mathbb{E}_{4}=cdots$.

Morevoer, if one replaces $mathsf{Mon}(mathcal{C})$ by $mathsf{CMon}(mathcal{C})$ (i.e. $mathsf{Alg}_{mathbb{E}_{1}}(mathcal{C})$ by $mathsf{Alg}_{mathbb{E}_{2}}(mathcal{C})congmathsf{Alg}_{mathbb{E}_{3}}(mathcal{C})congcdots$), then $mathsf{CMon}(mathcal{C})$ is still monoidal ($mathbb{E}_{1}$… and braided ($mathbb{E}_{2}$) and symmetric ($mathbb{E}_{3}$)) when $mathcal{C}$ is symmetric ($mathbb{E}_{3}$… as again $mathbb{E}_{3}=mathbb{E}_{4}=cdots$), but having $mathcal{C}$ be braided fails to endow $mathsf{CMon}(mathcal{C})$ with a monoidal structure. So now the assignment $mathcal{C}mapstomathsf{CMon}(mathcal{C})$ gives a functor $mathsf{CMon}colonmathsf{Alg}_{mathbb{E}_{k}}(mathsf{Cats})tomathsf{Alg}_{mathbb{E}_{k-2}}(mathsf{Cats})$.

This whole situation made me wonder what happens in the $infty$-setting, where now we have not only monoidal, braided, and symmetric structures, but the whole array of $mathbb{E}_{k}$-monoidal structures for $1leq kleqinfty$, starting with $mathbb{E}_{1}$ (i.e. monoidal $infty$-categories) all the way up to $mathbb{E}_{infty}$ (i.e. symmetric monoidal $infty$-categories):

- Given an $mathbb{E}_{n}$-monoidal $infty$-category $mathcal{C}$, is there a sensible way to “count” how many natural induced $mathbb{E}_{n-k}$-structures are there on $mathsf{Alg}_{mathbb{E}_{n-m}}(mathcal{C})$, where $1leq k,mleq n-1$?
- Does the “space” of these induced structures have some kind of symmetry, such as in the case mentioned above where having a braided monoidal structure on $mathcal{C}$ gave $mathsf{Mon}(mathcal{C})$ two different monoidal structures related by exchanging $beta_{A,B}$ with $beta^{-1}_{B,A}$?

¹This was pointed out by Amar Hadzihasanovic on Zulip in reply to a question of David Roberts.