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.

This is a shorter version. We relegated the treatment of homotopy d-categories and d-operads to a separate note