Corollary 5.3.6 (05ZT) · ScalingStacks

[05ZT]

Corollary 5.3.6. Let $\left(\mathcal{P}_{i}\right)_{i=1}^{\infty}$ be a sequence of reduced $\infty$ -operads not equivalent to $\mathbb{E}_{0}$ . There is an equivalence of $\infty$ -operads $\bigotimes\limits_{i=1}^{\infty}\mathcal{P}_{i}\simeq\mathbb{E}_{\infty}$ .

[05ZU]

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