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**The \(\infty\)-categorical Eckmann-Hilton argument.**
*(English)*
Zbl 1444.18026

The Eckmann-Hilton argument [B. Eckmann and 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 [A. Joyal and R. Street, Adv. Math. 102, No. 1, 20–78 (1993; Zbl 0817.18007)]. For a vast generalisation, see [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.

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.

Reviewer: Ross H. Street (Sydney)

### MSC:

18N70 | \(\infty\)-operads and higher algebra |

18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |

55P48 | Loop space machines and operads in algebraic topology |

18D40 | Internal categories and groupoids |

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\textit{T. Schlank} and \textit{L. Yanovski}, Algebr. Geom. Topol. 19, No. 6, 3119--3170 (2019; Zbl 1444.18026)

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