Since the sphere spectrum is connective, we may view it as an $\mathbb{E}_{\infty}$-group in spaces, namely $QS^0$. This space, as any simplicial set, has a homotopy category $\mathsf{Ho}(QS^0)$, and since $\mathsf{Ho}$ is symmetric monoidal, $\mathsf{Ho}(QS^0)$ acquires the structure of an $\mathbb{E}_{\infty}$-group in categories, making it into a "grouplike" symmetric monoidal category (more commonly called a symmetric groupoidal category or an abelian $2$-group).

Is there a known explicit description of the symmetric groupoidal category $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)$?

Is there a known explicit description of the abelian $2$-group $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?

Since the sphere spectrum is connective, we may view it as an $\mathbb{E}_{\infty}$-group in spaces, namely $QS^0$. This space, as any simplicial set, has a homotopy category $\mathsf{Ho}(QS^0)$, and since $\mathsf{Ho}$ is symmetric monoidal, $\mathsf{Ho}(QS^0)$ acquires the structure of an $\mathbb{E}_{\infty}$-group in categories, making it into a "grouplike" symmetric monoidal category (more commonly called a symmetric groupoidal category or an abelian $2$-group).

Question. Is there a known explicit description of the symmetric groupoidal category $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)$?

Since the sphere spectrum is connective, we may view it as an $\mathbb{E}_{\infty}$-group in spaces, namely $QS^0$. This space, as any simplicial set, has a homotopy category $\mathsf{Ho}(QS^0)$, and since $\mathsf{Ho}$ is symmetric monoidal, $\mathsf{Ho}(QS^0)$ acquires the structure of an $\mathbb{E}_{\infty}$-group in categories, making it into a "grouplike" symmetric monoidal category (more commonly called a symmetric groupoidal category or an abelian $2$-group).

Is there a known explicit description of the symmetric groupoidal category $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)$?