Eckmann-Hilton argument

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In mathematics, the Eckmann-Hilton argument (or Eckmann-Hilton principle or Eckmann-Hilton theorem) is an argument about 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.

[edit] Presentation

As will be evident later, it is very inconvenient to postulate existence of identities in the basic treatment of the argument. So we begin with magmas, with the goal of looking toward commutative monoidal structures.

[edit] Non-monoidal case

Let Mag denote the magma category, i.e. the category whose objects are magmas and whose morphisms are magma homomorphisms (defined similar to group homomorphisms). We consider the conditions implied for the sole existence of magma objects, which gives place to Med, the medial category.

(expressions such as "abelian", "centered", "affine", "medial", "dichotomic" or "preconvex" are magma object language generalizations but we drop quotation marks)

A basic example of pure mediality or abelianity (old) (i.e. an auto magma object with a binary operation T which fulfills (x T y) T (u T z) = (x T u) T (y T z)) is

x T y = a(x) + b(y) + t in a commutative semigroup (not necessarily with identity) with a and b endomorphisms which commute and t a fixed element in the semigroup. This generalizes to commutative semigroups the notion of linear and affine combination.

We say that an medial operation is centered if admits some (two-sided) cancellative idempotent (a center).

Now, if we have a centered medial operation (let c be a center), define a(x) = x T c and b(y) = c T y. As cancellative entails, we have retracts d and e such that d(a(x)) = x and e(b(y)) = y, if d and e are bijective maps, we can define x + y = d(x) T e(y) this is medial too, c is its identity and reconstruct x T y = a(x) + b(y), so a case of the basic example.

But in Mag we can extend an injective endomorphism, so the extension of b o a = a o b gives an extension to a basic example.

Conversely, assume that the basic example is over a commutative monoid with x T y = a(x) + b(y), as a(0) = 0 = b(0) then 0 T 0 = 0 i.e. idempotent and x T 0 = a(x), 0 T y = b(y).

Definitions:

  • A linear combination in real numbers a.x + b.y is called affine if and only if a + b = 1, but that, of course, means a.x + b.x = x for all x.
  • We say that an affine operation is central if all elements are (two-sided) cancellative.
  • We say that an affine operation is dichotomic if it is commutative.
  • The only affine combination in real numbers which is commutative is x T y = ½x + ½y.
  • We say that a central dichotomic operation is preconvex. General Convex Reconstruction In J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp

This ideas can be used to begin characterization of real numbers. see Escardó, Simpson on ½x + ½y

And, which is more important, resolves the metric-category problem: metric morphisms are short maps (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 linear short maps, only Banach unit balls of linear short maps! But unit balls are not additive closed, just ½x + ½y (convex) closed.

But we are showing that closure need not be monoidal, just medial in the auto magma object sense, which is the true Eckmann-Hilton sense. (0 is not an identity for ½x + ½y, just a (kind of) center). But the reconstructive extension just presented is, exactly, the Banach space with its monoidal structure.

See also:

[edit] External links

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