We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} (-)^\times &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the $\mathbb{E}_\infty$-groups $QS^0$ and $\Omega Q\mathbb{RP}^\infty$. Passing to spectra via the equivalence between $\mathbb{E}_\infty$-groups and connective spectra, we obtain the sphere spectrum $\mathbb{S}$ corresponding to $QS^0$ and a spectrum $E$ corresponding to $\Omega Q\mathbb{RP}^\infty$.

(One possible name for $E$ might be "$\mathbb{S}/2$" since it satisfies an analogous universal property to that of $\mathbb{Z}/2$, corepresenting "involutory objects". However, that notation already usually denotes the mod 2 Moore spectrum, so let's write $E$ for it instead.)

For comparison, their first $8$ homotopy groups are as follows: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(E) &\cong \mathbb{Z}/2,\\ \pi_1(E) &\cong \mathbb{Z}/2,\\ \pi_2(E) &\cong \mathbb{Z}/8,\\ \pi_3(E) &\cong \mathbb{Z}/2,\\ \pi_4(E) &\cong 0,\\ \pi_5(E) &\cong \mathbb{Z}/2,\\ \pi_6(E) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(E) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $E$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $E$?

We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} \mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $(A,\cdot_A,1)\mapsto A$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the $\mathbb{E}_\infty$-groups $QS^0$ and $\Omega Q\mathbb{RP}^\infty$. Passing to spectra via the equivalence between $\mathbb{E}_\infty$-groups and connective spectra, we obtain the sphere spectrum $\mathbb{S}$ corresponding to $QS^0$ and a spectrum $E$ corresponding to $\Omega Q\mathbb{RP}^\infty$.

(One possible name for $E$ might be "$\mathbb{S}/2$" since it satisfies an analogous universal property to that of $\mathbb{Z}/2$, corepresenting "involutory objects". However, that notation already usually denotes the mod 2 Moore spectrum, so let's write $E$ for it instead.)

For comparison, their first $8$ homotopy groups are as follows: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(E) &\cong \mathbb{Z}/2,\\ \pi_1(E) &\cong \mathbb{Z}/2,\\ \pi_2(E) &\cong \mathbb{Z}/8,\\ \pi_3(E) &\cong \mathbb{Z}/2,\\ \pi_4(E) &\cong 0,\\ \pi_5(E) &\cong \mathbb{Z}/2,\\ \pi_6(E) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(E) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $E$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $E$?

We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} (-)^\times &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the $\mathbb{E}_\infty$-groups $QS^0$ and $\Omega Q\mathbb{RP}^\infty$. Passing to spectra via the equivalence between $\mathbb{E}_\infty$-groups and connective spectra, we obtain the sphere spectrum $\mathbb{S}$ and a spectrum $E$ corresponding to $\Omega Q\mathbb{RP}^\infty$  (one possible name for $E$ would be "$\mathbb{S}/2$" since it has an analogous universal property to $\mathbb{Z}/2$, but that notation already usually denotes the mod 2 Moore spectrum). For comparison, their first $8$ homotopy groups are as follows: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(E) &\cong \mathbb{Z}/2,\\ \pi_1(E) &\cong \mathbb{Z}/2,\\ \pi_2(E) &\cong \mathbb{Z}/8,\\ \pi_3(E) &\cong \mathbb{Z}/2,\\ \pi_4(E) &\cong 0,\\ \pi_5(E) &\cong \mathbb{Z}/2,\\ \pi_6(E) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(E) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $E$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $E$?

We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} (-)^\times &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the $\mathbb{E}_\infty$-groups $QS^0$ and $\Omega Q\mathbb{RP}^\infty$. Passing to spectra via the equivalence between $\mathbb{E}_\infty$-groups and connective spectra, we obtain the sphere spectrum $\mathbb{S}$ corresponding to $QS^0$ and a spectrum $E$ corresponding to $\Omega Q\mathbb{RP}^\infty$.

(One possible name for $E$ might be "$\mathbb{S}/2$" since it satisfies an analogous universal property to that of $\mathbb{Z}/2$, corepresenting "involutory objects". However, that notation already usually denotes the mod 2 Moore spectrum, so let's write $E$ for it instead.)

For comparison, their first $8$ homotopy groups are as follows: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(E) &\cong \mathbb{Z}/2,\\ \pi_1(E) &\cong \mathbb{Z}/2,\\ \pi_2(E) &\cong \mathbb{Z}/8,\\ \pi_3(E) &\cong \mathbb{Z}/2,\\ \pi_4(E) &\cong 0,\\ \pi_5(E) &\cong \mathbb{Z}/2,\\ \pi_6(E) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(E) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $E$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $E$?

We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} (-)^\times &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the sphere spectrum $\mathbb{S}$ and $``\mathbb{S}/2\text{''}\overset{\mathrm{def}}{=}\Omega Q\mathbb{RP}^\infty$. Here are the first $8$ homotopy groups of $\mathbb{S}$ and $\mathbb{S}/2$, for comparison: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_1(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}/2) &\cong \mathbb{Z}/8,\\ \pi_3(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_4(\mathbb{S}/2) &\cong 0,\\ \pi_5(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_6(\mathbb{S}/2) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(\mathbb{S}/2) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $\mathbb{S}/2$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $\mathbb{S}/2$?

We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} (-)^\times &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the $\mathbb{E}_\infty$-groups $QS^0$ and $\Omega Q\mathbb{RP}^\infty$. Passing to spectra via the equivalence between $\mathbb{E}_\infty$-groups and connective spectra, we obtain the sphere spectrum $\mathbb{S}$ and a spectrum $E$ corresponding to $\Omega Q\mathbb{RP}^\infty$ (one possible name for $E$ would be "$\mathbb{S}/2$" since it has an analogous universal property to $\mathbb{Z}/2$, but that notation already usually denotes the mod 2 Moore spectrum). For comparison, their first $8$ homotopy groups are as follows: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(E) &\cong \mathbb{Z}/2,\\ \pi_1(E) &\cong \mathbb{Z}/2,\\ \pi_2(E) &\cong \mathbb{Z}/8,\\ \pi_3(E) &\cong \mathbb{Z}/2,\\ \pi_4(E) &\cong 0,\\ \pi_5(E) &\cong \mathbb{Z}/2,\\ \pi_6(E) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(E) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $E$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $E$?