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 $a^2=1_A$). Moreover, this refines to a corepresentation of the functor $$\mathrm{Inv} \colon \mathsf{CMon} \to \mathsf{Ab},$$ making $\mathbb{Z}/2$ into an abelian group.

A similar story is true for $\mathbb{Z}$ and invertible elements, but let's 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 $\mathbb{E}_{\infty}$, then we get $\Omega^kS^k$ instead of $\mathbb{S}$.

Finally, 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 $$\mathsf{Inv}\colon\mathsf{Mon}_{\mathbb{E}_{k}}(\mathsf{Cats}_\infty)\to\mathsf{Grp}_{\mathbb{E}_{k}}(\mathcal{S})$$ corepresentable by an $\mathbb{E}_{k}$-space for $2\leq k\leq\infty$?

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 $2\leq k\leq\infty$?

Inverses in monoids.

The monoid of integers $\mathbb{Z}$ is the "universal monoid with an invertible element": for any other monoid $A$, we have a bijection of sets $$\mathrm{Hom}_{\mathsf{Mon}}(\mathbb{Z},A)\cong A^\times,$$ i.e. a morphism of monoids $\mathbb{Z}\to A$ is the same as specifying an invertible element $a$ of $A$ (if $A$ has any) via $1\mapsto a$.

  Moreover, $\mathbb{Z}$ is also the "universal commutative monoid with an invertible element": again we have a bijection $$\mathrm{Hom}_{\mathsf{CMon}}(\mathbb{Z},A)\cong A^\times$$ for any commutative monoid $A$.

Inverses in monoidal categories.

When passing from sets to categories, one sees a number of changes, all coming from homotopy coherence:

• We replace monoids by monoidal categories;
• We replace morphisms of monoids by strong monoidal functors;
• There are now more notions of commutativity: instead of "nothing" and "commutative", we have "nothing", "braided", and "symmetric";
• A morphism between braided or symmetric monoidal categories now has to additionally respect the commutative structure, so we speak also of braided strong monoidal functors.

Then, something very interesting happens: the extra homotopy coherence creates more diversity when considering "universal objects with inverses". So now:

• The "universal monoidal category with an invertible element" is still going to be $\mathbb{Z}$ (or more precisely $\mathbb{Z}_\mathsf{disc}$). It is characterised by an isomorphism of categories $$\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathcal{C})\cong\mathsf{Pic}(\mathcal{C});$$
• The "universal braided monoidal category with an invertible element" is $\tau_{\leq1}\Omega^2S^2$, the fundamental groupoid of the $2$-fold loop space of $S^2$, and it is characterised by $$\mathsf{Fun}^{\otimes,\color{red}{\textbf{br}}}(\tau_{\leq1}\Omega^2S^2,\mathcal{C})\cong\mathsf{Pic}(\mathcal{C});$$
• The "universal symmetric monoidal category with an invertible element" is $\tau_{\leq1}\mathbb{S}$, the $1$-truncation of the sphere spectrum, characterised by $$\mathsf{Fun}^{\otimes,\color{red}{\textbf{sym}}}(\tau_{\leq1}\mathbb{S},\mathcal{C})\cong\mathsf{Pic}(\mathcal{C}).$$

Inverses in monoidal $\infty$-categories.

For $\infty$-categories, we speak of $\mathbb{E}_k$-commutativity. Here the "universal $\mathbb{E}_k$-monoidal $\infty$-category with an invertible element", meaning the free $\mathbb{E}_k$-group on a point, is $\Omega^k S^k$, the $k$-fold loop space of the $k$-sphere. Similarly, when $k=\infty$, we get the sphere spectrum $\mathbb{S}$.

Question. Is there an analogue of this story for idempotent and involutory objects in $\mathbb{E}_k$-monoidal $\infty$-categories?

The answer for the "categorical level" 0 version of this question is clear:

• An idempotent element in a monoid $A$ is an element $a\in A$ with $a^2=a$, which is the same as a morphism of monoids from the "Boolean monoid" $\mathbb{B}=(\{0,1\},\vee,1)$;
• An involutory element in a monoid $A$ is an element $a\in A$ with $a^2=1_A$. Again, this is the same as a morphism of monoids from $\mathbb{Z}/2$.

However, things get more complicated in the "categorical level $1$" setting:

• An "idempotent object" $A$ of a monoidal category $\mathcal{C}$ should include an isomorphism $A\otimes A\cong A$ as part of the data, and it should satisfy coherence conditions, guaranteeing $\mathrm{Hom}_{\mathcal{C}}(A^{\otimes n},A)$ is just a point.
• An "involutory object" $A$ of $\mathcal{C}$ should similarly include an isomorphism $A\otimes A\cong\mathbf{1}_{\mathcal{C}}$ as part of the data, and again there are coherence issues: both $\mathrm{Hom}_{\mathcal{C}}(A^{\otimes2n},\mathbf{1}_{\mathcal{C}})$ and $\mathrm{Hom}_{\mathcal{C}}(A^{\otimes2n+1},A)$ should be punctual sets.

For this pair of definitions, I believe that:

• The "universal monoidal category with an idempotent object" is $\mathbb{B}_{\mathsf{disc}}$, characterised by an isomorphism of categories $$\mathsf{Fun}^{\otimes}(\mathbb{B}_{\mathsf{disc}},\mathcal{C})\cong\!\!\!\!\!/\color{red}{\approx}\mathrm{Idem}(\mathcal{C});$$ Edit: There's a small problem here: this is true only if instead we consider strictly unitary strong monoidal functors $F\colon\mathbb{B}_{\mathsf{disc}}\to\mathcal{C}$, as then there's no extra room left for the lax monoidal unity constraint $\mathbf{1}_{\mathcal{C}}\overset{\sim}{\dashrightarrow}F(A)$ to be a non-identity isomorphism.
• Similarly the "universal monoidal category with an involutory object" is $(\mathbb{Z}/2)_{\mathsf{disc}}$, again characterised by an isomorphism of categories $$\mathsf{Fun}^{\otimes}((\mathbb{Z}/2)_{\mathsf{disc}},\mathcal{C})\cong\mathrm{Inv}(\mathcal{C});$$
• However, the "universal braided/symmetric monoidal category with an idempotent/involutory object" have to be different than $\mathbb{B}_{\mathsf{disc}}$ and $(\mathbb{Z}/2)_{\mathsf{disc}}$, so we should again get new, more complicated monoidal categories.

Returning to the $\infty$-setting, the answer for the $\mathbb{E}_1$-monoidal case is clear:

• For idempotent objects, we have Lurie's $\infty$-category $\mathsf{Idem}$ (HTT 4.4.5.2), which is the same as the classifying space $\mathbf{B}\mathbb{B}$ of $\mathbb{B}$. The universal monoidal $\infty$-category with an idempotent object is then $(\mathbf{B}\mathbb{B})_{\mathsf{disc}}$.
• For involutory objects, we have $(\mathbf{B}\mathbb{Z}/2)_{\mathsf{disc}}$.

Similarly to the $1$-categorical case, the answers will be different for $\mathbb{E}_{k}$-monoidal categories ($k\geq2$), so we might expect to possibly obtain objects (provided they indeed exist) that are to $\mathbb{B}$ and $\mathbb{Z}/2$ as the sphere spectrum is to $\mathbb{Z}$. (Possibly, these might also carry semi/ring structures, so we can wonder if we even get something like "the Boolean semiring space" and "the ring space $\mathbb{S}/2$")

Finally, here's a slightly more precise restatement of the question: Are there

1. $\mathbb{E}_k$-monoidal $\infty$-categories "$\mathbb{B}_k$" and "$(\Omega^k S^k)/2$";
2. Symmetric monoidal $\infty$-categories "$\mathbb{B}_\infty$" and "$\mathbb{S}/2$"

such that, similarly to the correspondence

$$\left\{\begin{gathered}\text{strong $\mathbb{E}_{k}$-monoidal}\\\text{functors $\Omega^kS^k\to\mathcal{C}$}\end{gathered}\right\}\cong\left\{\text{invertible objects of $\mathcal{C}$}\right\}\quad\text{(for $\mathcal{C}$ $\mathbb{E}_{k}$-monoidal)},$$ we have bijections $$\left\{\begin{gathered}\text{strong $\mathbb{E}_k$-monoidal}\\\text{functors $\mathbb{B}_k\to\mathcal{C}$}\end{gathered}\right\}\cong\left\{\text{idempotent objects of $\mathcal{C}$}\right\}\quad\text{(for $\mathcal{C}$ $\mathbb{E}_{k}$-monoidal)},$$ and similarly for "$\mathbb{B}_\infty$", "$(\Omega^k S^k)/2$", and "$\mathbb{S}/2$"?

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 $a^2=1_A$). Moreover, this refines to a corepresentation of the functor $$\mathrm{Inv} \colon \mathsf{CMon} \to \mathsf{Ab},$$ making $\mathbb{Z}/2$ into an abelian group.

A similar story is true for $\mathbb{Z}$ and invertible elements, but let's 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 $\mathbb{E}_{\infty}$, then we get $\Omega^kS^k$ instead of $\mathbb{S}$.

Finally, 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 $$\mathsf{Inv}\colon\mathsf{Mon}_{\mathbb{E}_{k}}(\mathsf{Cats}_\infty)\to\mathsf{Grp}_{\mathbb{E}_{k}}(\mathcal{S})$$ corepresentable by an $\mathbb{E}_{k}$-space for $2\leq k\leq\infty$?

Inverses in "categorical level" 0.

Inverses in "categorical level" 1.

Inverses in "categorical level" $\infty$.

Inverses in monoids.

Inverses in monoidal categories.

Inverses in monoidal $\infty$-categories.