## 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

### Citations:

Zbl 0099.02101; Zbl 0817.18007; Zbl 1138.18003
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### References:

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