If two unital operations on one set share their identity and satisfy , then the operations coincide and are commutative.
For , pointwise multiplication of based cube maps into a topological group induces the ordinary addition in its homotopy group . Concatenation and pointwise multiplication share an identity and satisfy the interchange rule, so the Eckmann-Hilton argument identifies the two operations. In particular, the -th power map induces multiplication by on every homotopy group, for all integers .
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The Eckmann–Hilton argument is a concept in category theory and homotopy theory that plays a role in the context of algebraic structures such as monoids and operads. It particularly addresses the interactions between two operations defined on a space or an algebraic structure when these operations are defined in a certain way, especially in relation to commutativity and associativity.