The $\infty$-Categorical Eckmann-Hilton Argument

TL;DR

This paper generalizes the classical Eckmann-Hilton argument to the setting of $$-categories, showing that the tensor product of two reduced $$-operads increases their connectivity by a specific amount.

Contribution

It introduces a notion of $d$-connectivity for reduced $$-operads and proves a formula for the connectivity of their tensor product, extending classical results to higher categorical contexts.

Findings

The tensor product of two $d$-connected $$-operads is $(d_1+d_2+2)$-connected.

This generalizes the classical Eckmann-Hilton argument to $$-categories.

Provides a new tool for understanding the structure of $$-operads in higher category theory.

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 $n\ge0$. 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.