The Eckmann-Hilton Argument is a classical result in algebraic topology saying that given a set \(X\) with two unital binary operations satisfying a certain "interchange" law, the two operations must be the same and moreover, this one operation endows \(X\) with a commutative (and associative) monoid structure. This result, though remarkably easy to prove, has many surprisingly interesting applications (the most famous one is showing that higher homotopy groups are always abelian).
The Eckmann-Hilton Argument can be reformulated as a statement about operads. These are categorical gadgets which allow us to systematically study various algebraic structures and the way they interact (much like the notion of a group allows us to systematically study symmetries). For example, there is an operad \(U\) classifying the structure of a unital binary operation, such that endowing a set \(X\) with a unital binary operation can be encoded as giving \(X\) the structure of an algebra over \(U\). Moreover, there is a notion of "tensor product" on operads. And the structure of two unital binary operations satisfying the interchange law can be encoded as an algebra structure over the tensor product of \(U\) with \(U\). The statement of the Eckmann-Hilton argument then becomes that this tensor product is the operad Com classifying the structure of a commutative monoid.
Operads are particularly useful when trying to mix algebra and homotopy in order to study homotopy coherent algebraic structures. This is most elegantly done using infinity-operads. In a joint work with Tomer Schlank we formulated and proved an infinity-categorical generalization of the Eckmann-Hilton argument in terms of infinity-operads. This result assumes the form of a connectivity bound on the spaces of operations of the tensor product of two infinity-operads.
In this talk I will start by recalling the classical Eckmann-Hilton argument, explain briefly what operads are and the way we can use them to reformulate the classical result. Then I will sketch the infinity-categorical generalization of the setup and the statement. No prior knowledge of operads or infinity-categories is required.