The Eckmann-Hilton Argument

Feb 13, 2026

The Eckmann-Hilton argument causes many seemingly more complex structures to collapse into simpler ones. For instance, the fundamental group of any topological space is always Abelian.

Theorem

Let be a set which is a magma under two unital binary operations, and . Suppose one is a homomorphism for the other. Then and moreover is a commutative monoid under these operations.

We usually speak of homomorphisms as unary functions compatible with a structure, e.g. with . Binary homomorphisms look like this:

It seems strange that the and “swap” positions, but this is the binary analog to the notion of the homomorphism being a compatible operation, such that times plus times is the same as plus first times plus .

By the way

A set with a unital binary operation is called a unital magma. Unital means that the operation has a unique left and right identity, and a simple argument shows that these identities in fact coincide.

Proof. The key step is to show that the identities of both operations coincide. Let and be the identities of and respectively, then

The rest of the proof should follow easily. The setup

gives that and similar arguments give associativity and commutativity, which show that under the operations is indeed a commutative monoid and concludes the proof. ⁠

Another consequence of Eckmann-Hilton is that a monoid object in the category of monoids Mon is a commutative monoid, in fact this can be taken as a category theoretic formulation of the argument itself.