The \(\infty\)-categorical Eckmann-Hilton argument (Q2279080)

The Eckmann-Hilton argument [\textit{B. Eckmann} and \textit{P. J. Hilton}, Math. Ann. 145, 227--255 (1962; Zbl 0099.02101)] is an abstraction of the proof that the homotopy groups $\pi_n(X;x_0)$ of a pointed space $(X;x_0)$ are commutative for $n > 1$. The fact is that $\pi_n(X;x_0)$ has two unital multiplications on it arising in different ways, yet one a morphism for the other; the argument yields that the two operations agree, are associative, and are commutative. The term is sometimes applied to arguments which identify models of one structure in a (perhaps higher) category of models of another (possibly the same) structure as an extra property or equipment of the first structure. For example, provision of a monoidal structure on an object in the 2-category of monoidal categories and strong monoidal functors amounts to equipping the object with a braiding; see [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. For a vast generalisation, see [\textit{M. A. Batanin}, Adv. Math. 217, No. 1, 334--385 (2008; Zbl 1138.18003)]. In other words, the argument is about tensor products of theories. In particular, as in the present paper, the theories might be operads. The main theorem states that, for reduced $\infty$-operads $\mathcal{P}$ and $\mathcal{Q}$, 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 $d_1+d_2+2$-connected. The terms ``reduced'' and ``$d$-connected'' are defined in the paper.