The $\infty$–categorical Eckmann–Hilton argument

Abstract

We define a reduced $\infty$–operad $\mathcal{P}$ to be $d$–connected if the spaces $\mathcal{P}\left(n\right)$ of $n$–ary operations are $d$–connected for all $n\ge 0$. Let $\mathcal{P}$ and $\mathcal{Q}$ be two reduced $\infty$–operads. We prove that if $\mathcal{P}$ is $d_1$–connected and $\mathcal{Q}$ is $d_2$–connected, then their Boardman–Vogt tensor product $\mathcal{P}\otimes \mathcal{Q}$ is $\left(d_1+d_2+2\right)$–connected. We consider this to be a natural $\infty$–categorical generalization of the classical Eckmann–Hilton argument.