Eckmann–Hilton argument

In mathematics, the argument (or principle or theorem) of Eckmann-Hilton is an argument about of pairs of monoid structures on a set where one is a homomorphism to the other. Given this, the structures can be shown to coincide, and the resulting monoid is demonstrably commutative. This can be used to prove the commutativity of higher homotopy groups.

Presentation

As will become apparent later, it is highly inconvenient to postulate the existence of identities in the basic treatment of the argument. Therefore we start with magmas, with the aim of aiming for commutative monoidal structures.

Non-Monoidal Case

Let Mag be the category of magmas (i..e. binary operations), we consider the conditions implied by the mere existence of magma objects that give rise to Med the media category.

(expressions such as "abelian", "centered", "affine", "medial", "dichotomous" or "preconvex" are generalizations for magma objects but we remove the quotes) Basic example of pure mediality: x T y = a(x) + b(y) + t in a commutative semigroup (not necessarily with identity element) with a and b endomorphisms commuting with each other and t a fixed element of the semigroup. In this example 0 T 0 = t if 0 is neutral of the semigroup.

Basic example of mediality or abelianity (old) (i.e. a self-magma object with a binary operation T satisfying (x T y) T (u T z) = (x T u) T (y T z)) x T y = a(x) + b(y) + t pure in a commutative semigroup (not necessarily with identity) with a and b commuting endomorphisms and t a fixed element in the semigroup. This generalizes to commutative semigroups the notion of linear and affine combination.

We say that a medial operation is centered if it admits some bilateral canceling idempotent (a center).

Now, if we have a centered medial operation (let c be a center), let's define a(x) = x T c and b(y) = c T y, as cancelativity requires, we have contractions d and e such that d(a(x)) = x y e(b(y)) = y, if d and e are bijectives, we can define x + y = d(x) T e(y), this is medial as well, c is its identity and reconstructs x T y = a(x) + b(y), thus a basic example case. But in Mag we can extend an injective endomorphism, so the extension of b or a = a or b gives an extension to a basic example. Conversely, assume the basic example is about a commutative monoid with x T y = a(x) + b(y), since a(0) = 0 = b(0) then 0 T 0 = 0 i.e. idempotent y x T 0 = a(x), 0 T y = b(y).

Definitions:

• An operation is afin if it's medial and idempotent.
• A linear combination of real numbers a.x + b.y is called aphin if and only if a + b = 1, but this, of course, means a.x + b.x = x for all x.
• We say an affin operation is central if all the elements are cancelative bilateral.
• We say an affin operation is dichotomous if it's commutative.
• The only affin combination of real numbers that is switching is x T and = 1⁄2x + 1⁄2y. We say a central dichotomous operation is preconvex.

These ideas can be used to begin the characterization of real numbers. (see Escardó, Simpson on ½x + ½y), most importantly, it solves the metric category problem: metric morphisms are short functions (or weak contractions or 1-Lipschitz); So far, so good. But for Banach spaces this gives a contradictio in adjecto: there are no Banach spaces of continuous short linear functions, only Banach unit balls of short linear functions! But the unit balls are not additively closed, just ½x + ½y closed (convex). But we have shown that the closure need not be monoidal, just medial in the magma self-object sense, which is the true Eckmann-Hilton sense. (0 is not an identity for ½x + ½y, just a (sort of) "center"). But the reconstructive extension just presented is exactly the Banach space with its monoidal structure.