ScalingStacks

Proof. By 3.2.3, all $\mathcal{P}_{i}$-s are $\left(-1\right)$-connected. By induction on $k$ and 5.3.3, the $\infty$-operad $\mathcal{P}_{1}\otimes\cdots\otimes\mathcal{P}_{k}$ is $\left(k-2\right)$-connected. For every $n\in\mathbb{N}$ we get

$$\left(\bigotimes\limits_{i=1}^{\infty}\mathcal{P}_{i}\right)\left(n\right)\simeq\operatorname*{colim}\limits_{k}\left(\mathcal{P}_{1}\otimes\cdots\otimes\mathcal{P}_{k}\right)\left(n\right)\simeq\text{pt}$$

and therefore $\bigotimes\limits_{i=1}^{\infty}\mathcal{P}_{i}\simeq\mathbb{E}_{\infty}$. ∎