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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.

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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