ScalingStacks

Proof. By A.5.2.6.10 and A.5.2.6.12, we have a commutative diagram of $\infty$-categories

in which $U$ is the forgetful functor. Since the $k$-fold loop space functor restricts to a functor $\mathcal{C}_{*}^{\left[k,2k+d\right]}\to\tau_{\leq k+d}\mathcal{C}_{*}$, we can restrict the above diagram to

The $\infty$-category $\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}^{\operatorname{\scriptsize{grp}}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right)$ is a full subcategory of $\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right)$, which is itself equivalent to $\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}\left(\tau_{\leq k+d}\mathcal{C}\right)$. The $\infty$-category $\tau_{\leq k+d}\mathcal{C}$ is a $\left(k+d+1\right)$-topos (with the Cartesian symmetric monoidal structure) and $\mathbb{E}_{k}$ is $\left(k-2\right)$-connected. Thus, 5.1.4 implies that for every $X$ in $\underline{\operatorname{Alg}}_{\mathbb{E}_{k}}^{\operatorname{\scriptsize{grp}}}\left(\tau_{\leq k+d}\mathcal{C}_{*}\right)$, the reduced endomorphism operad of $X$ is an essentially $\left(d+1\right)$-operad.

Let $d=-1$. We recall from 3.2.3 that if $\mathcal{P}\not\simeq\mathbb{E}_{0}$, then it is $\left(-1\right)$-connected. Therefore, if $\mathcal{P}$ is an essentially $0$-operad, then $\mathcal{P}\simeq\mathbb{E}_{\infty}$. Hence, $\mathcal{P}$ is either $\mathbb{E}_{0}$ or $\mathbb{E}_{\infty}$.

Let $d=-2$. We get that $\mathcal{P}$ is an essentially $\left(-1\right)$-operad and hence equivalent to $\mathbb{E}_{\infty}$. ∎