Eckmann–Hilton argument

In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two monoid structures on a set where one is a homomorphism for the other. Given this, the structures can be shown to coincide, and the resulting monoid demonstrated to be commutative. This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who used it in a 1962 paper.

Let ${\displaystyle X}$ be a set equipped with two binary operations, which we will write ${\displaystyle \circ }$ and ${\displaystyle \otimes }$, and suppose:

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