The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$).

A similar story is true for $\mathbb{Z}$ and invertible elements, but let's instead tell it in the $\infty$-setting: namely, the $\infty$-category of $\mathbb{E}_1$-monoidal functors $\mathbb{Z}_\mathsf{disc}\to\mathcal{C}$ is just $\mathsf{Pic}(\mathcal{C})$, and thus $\mathbb{Z}_\mathsf{disc}$ corepresents the functor $$\mathsf{Pic}\colon\mathsf{Mon}_{\mathbb{E}_1}(\mathsf{Cats}_{\infty})\to\mathcal{S}$$

However, replacing

- $\mathsf{Mon}_{\mathbb{E}_1}(\mathsf{Cats}_\infty)$ by $\mathsf{Mon}_{\mathbb{E}_\infty}(\mathsf{Cats}_\infty)$, the $\infty$-category of symmetric monoidal $\infty$-categories;
- $\mathcal{S}$ by $\mathsf{Grp}_{\mathbb{E}_\infty}(\mathcal{S})$;

changes the corepresenting object from $\mathbb{Z}_{\mathsf{disc}}$ to the sphere spectrum $\mathbb{S}$. Similarly, if we pass to $\mathbb{E}_k$ rather than $\mathbb{E}_{\infty}$, we get $\Omega^kS^k$ instead of $\mathbb{S}$.

Now, define an **involutory object** of a monoidal $\infty$-category $\mathcal{C}$ to be a strong monoidal functor $(\mathbb{Z}/2)_{\mathsf{disc}}\to\mathcal{C}$. By definition, $(\mathbb{Z}/2)_{\mathsf{disc}}$ corepresents the functor
$$\mathsf{Inv}\colon\mathsf{Mon}_{\mathbb{E}_1}(\mathsf{Cats}_{\infty})\to\mathcal{S}$$
sending $\mathcal{C}$ to $\mathsf{Inv}(\mathcal{C})\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes((\mathbb{Z}/2)_{\mathsf{disc}},\mathcal{C})$.

**Question.** Is the functor
$$\mathsf{Inv}\colon\mathsf{Mon}_{\mathbb{E}_{k}}(\mathsf{Cats}_\infty)\to\mathsf{Grp}_{\mathbb{E}_{k-1}}(\mathcal{S})$$
corepresentable by an $\mathbb{E}_{k}$-monoidal category for $2\leq k\leq\infty$?

co-abelian group in $CMon$ - which is not a surprise, everyone is a co-(commutative monoid) in $CMon$, and the corresponding shear map is just the shear map $\endgroup$