Eckmann-Hilton argument

Eckmann-Hilton argument

In abstract algebra, the Eckmann-Hilton argument 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.

Table of contents

1 Presentation
2 Non monoidal case
3 External links

Presentation

Let C the category of binary operations, it has direct products, so the concept of an internal operation, say

T': (X,T)x(X,T) -> (X,T),

as T' is a morphism it must give

(x T' y) T (u T' z) = (x T u) T' (y T z)

so if we wont take the original operation, this will be allowed only if

(x T y) T (u T z) = (x T u) T (y T z)

this operation can have two-sided identity only if it is a monoidal operation! (iff, of course).

Non monoidal case

we must stop now

Eckmann-Hilton principle

Eckmann-Hilton theorem

External links

M.A. Batanin The Eckmann-Hilton argument, higher operads and E_n-spaces.

John Baez: Eckmann-Hilton principle (week 89)

John Baez: Eckmann-Hilton principle (week 100)