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 $\mathbb{S}$.




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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 $2\leq k\leq\infty$?



Corepresentability of involutory objects in monoidal $\infty$-categories
$$Fun^{\otimes}(\mathbb Z/2, C) \simeq map\_{E\_1}(\mathbb Z/2, C^\simeq) \simeq map\_{E\_k}(\mathrm{Ind}\_{E\_1}^{E\_k}\mathbb Z/2, C^\simeq)$$
where $\mathrm{Ind}\_{E\_1}^{E\_k}$ denotes the left adjoint to the forgetful functor.


So $Inv$ is representable, and the natural $E\_{k-1}$-structure (see my comments for why I wrote $E\_{k-1}$ and not $E\_k$ - it is possible that in this special case too we could get $E\_k$, but I don't see a reason why, and what I wrote works for any $E\_1$-space $X$) on this space gives $\mathrm{Ind}\_{E\_1}^{E\_k}\mathbb Z/2$ a natural co-$E\_{k-1}$-structure (in $E\_k$-spaces - with the coproduct monoidal structure).


Now does the space $\mathrm{Ind}\_{E\_1}^{E\_k}\mathbb Z/2$ have a reasonably concrete description ? I think it's something like a bar construction $Bar(E\_1,E\_k, \mathbb Z/2)$ so you can get an explicit description involving the space of little $k$-disks, but I'm not entirely sure you can get much better. I'd love to hear about a better description.


