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](https://mathoverflow.net/a/404304) 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](https://mathoverflow.net/questions/280980), 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](https://www.jstor.org/stable/1993859?seq=1#metadata_info_tab_contents); see also [MO 230790](https://mathoverflow.net/questions/230790).)




---


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



Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$
There are no left-unital multiplications on E. If there were, then for any element $x$ in $\pi\_n(E)$, we would have $x+x = 1 \cdot x + 1 \cdot x = (1+1) \cdot x = 0$ because all elements in $ \pi\_0 E$ are 2-torsion. This is not satisfied by the homotopy groups in your table.


