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 $2\leq k\leq\infty$?
Best Answer
$$Fun^{\otimes}(\mathbb Z/2, C) \simeq map_{E_1}(\mathbb Z/2, C^\simeq) \simeq map_{E_k}(\mathrm{Ind}_{E_1}^{E_k}\mathbb Z/2, C^\simeq)$$ 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.