ScalingStacks

Remark 3.2.3. Let $\mathcal{P}$ be a reduced $\infty$-operad. It is $d$-connected if and only if all the spaces $\mathcal{P}\left(n\right)$ in the underlying symmetric sequence of $\mathcal{P}$ are $d$-connected. If $\mathcal{P}$ is not equivalent to $\mathbb{E}_{0}$, then for some $n\geq 2$ we have $\mathcal{P}\left(n\right)\neq\varnothing$, and so there exists an $n$-ary operation $\mu\in\mathcal{P}\left(n\right)$ for $n\geq 2$. By composing $\mu$ with itself, we can obtain an operation in $\mathcal{P}$ of arbitrarily high arity and by composition with the unique nullary operation, we can obtain an operation of arbitrary arity. It follows that $\mathcal{P}\not\simeq\mathbb{E}_{0}$ if and only if $\mathcal{P}$ is $\left(-1\right)$-connected.