For ordinary categories, the assignment $\mathcal{C}\mapsto\mathsf{Mon}(\mathcal{C})$ defines a functor $\mathsf{Mon}\colon\mathsf{Alg}_{\mathbb{E}_{k}}(\mathsf{Cats})\to\mathsf{Alg}_{\mathbb{E}_{k-1}}(\mathsf{Cats})$, 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 $\mathsf{Alg}_{\mathbb{E}_{2}}(\mathcal{C})\cong\mathsf{Alg}_{\mathbb{E}_{3}}(\mathcal{C})\cong\cdots$), 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 $\mathcal{C}\mapsto\mathsf{CMon}(\mathcal{C})$ gives a functor $\mathsf{CMon}\colon\mathsf{Alg}_{\mathbb{E}_{k}}(\mathsf{Cats})\to\mathsf{Alg}_{\mathbb{E}_{k-2}}(\mathsf{Cats})$.
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 $1\leq k\leq\infty$, 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 $1\leq k,m\leq n-1$?
- 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.
