#### Volume 19, issue 6 (2019)

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The $\infty$–categorical Eckmann–Hilton argument

### Tomer M Schlank and Lior Yanovski

Algebraic & Geometric Topology 19 (2019) 3119–3170
##### Abstract

We define a reduced $\infty$–operad $\mathsc{P}$ to be $d$–connected if the spaces $\mathsc{P}\left(n\right)$ of $n$–ary operations are $d$–connected for all $n\ge 0$. Let $\mathsc{P}$ and $\mathsc{Q}$ be two reduced $\infty$–operads. We prove that if $\mathsc{P}$ is ${d}_{1}$–connected and $\mathsc{Q}$ is ${d}_{2}$–connected, then their Boardman–Vogt tensor product $\mathsc{P}\otimes \mathsc{Q}$ 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.

##### Mathematical Subject Classification 2010
Primary: 18D05, 18D50, 55P48
##### Publication
Revised: 16 February 2019
Accepted: 26 February 2019
Published: 20 October 2019
##### Authors
 Tomer M Schlank Einstein Institute of Mathematics Hebrew University of Jerusalem Jerusalem Israel Lior Yanovski Einstein Institute of Mathematics Hebrew University of Jerusalem Jerusalem Israel