Eckmann-Hilton Argument

The Eckmann-Hilton Argument states that for any two Binary Operations $\diamond,\star:A\times A\to A$ with units $1_{\diamond},1_{\star}:A$ such that $\star$ is a homomorphism with respect to $\diamond$ , then

$1_{\diamond}=1_{\star}$
$x\diamond y=x\star y$
$\star$ is Commutative
$\star$ is Associative

The proof is an exercise in some easy algebra:

First, note that $\star:A\times A\to A$ the statement that is homomorphism means that it must be a homomorphism from the Product Magma $A\times A$ to $A$ , and thus

$\diamond((w,x)\star(y,z))=\diamond(w,x)\star\diamond(y,z)$

If we unfold this, we get the following interchange law

$(w\diamond y)\star(x\diamond z)=(w\diamond x)\star(y\diamond z)$

Arumed with this observation, let us show that $1_{\diamond}=1_{\star}$

	$\displaystyle 1_{\diamond}$	$\displaystyle=1_{\diamond}\diamond 1_{\diamond}$
		$\displaystyle=(1_{\diamond}\star 1_{\star})\diamond(1_{\star}\star 1_{\diamond})$
		$\displaystyle=(1_{\diamond}\diamond 1_{\star})\star(1_{\star}\diamond 1_{% \diamond})$
		$\displaystyle=1_{\star}\star 1_{\star}$
		$\displaystyle=1_{\star}$

Next, $x\diamond y=x\star y$ .

	$\displaystyle x\diamond y$	$\displaystyle=(x\star 1_{\star})\diamond(1_{\star}\star y)$
		$\displaystyle=(x\diamond 1_{\star})\star(1_{\star}\diamond y)$
		$\displaystyle=(x\diamond 1_{\diamond})\star(1_{\diamond}\diamond y)$
		$\displaystyle=x\star y$

On to commutativity.

	$\displaystyle x\star y$	$\displaystyle=(1_{\diamond}\diamond x)\star(y\diamond 1_{\diamond})$
		$\displaystyle=(1_{\diamond}\star y)\diamond(x\star 1_{\diamond})$
		$\displaystyle=(1_{\star}\star y)\diamond(x\star 1_{\star})$
		$\displaystyle=y\diamond x$

Finally, associativity.

	$\displaystyle x\star(y\star z)$	$\displaystyle=(x\diamond 1_{\diamond})\star(y\star z)$
		$\displaystyle=(x\diamond 1_{\diamond})\star(y\diamond z)$
		$\displaystyle=(x\star y)\diamond(1_{\diamond}\star z)$
		$\displaystyle=(x\star y)\diamond(1_{\star}\star z)$
		$\displaystyle=(x\star y)\diamond z$
		$\displaystyle=(x\star y)\star z$

Corollaries

A Monoid Object in the Category of Monoids is a Commutative Monoid.