- Eckmann–Hilton argument
In
mathematics , the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is anargument about twomonoid structures on a set where one is ahomomorphism for the other. Given this, the structures can be shown to coincide, and the resultingmonoid demonstrated to becommutative . This can then be used to prove the commutativity of the higherhomotopy group s.The Eckmann–Hilton result
Let X be a set equipped with two binary operations, which we will write . and *, and suppose:
1. * and . are both unital, with the same unit 1, say, and
2. forall a,b,c,d in X, (a*b).(c*d) = (a.c)*(b.d).Then * and . are the same and in fact commutative.
Remarks
The operations * and . are often referred to as multiplications, but this might imply they are associative, a property which is not required for the proof. In fact associativity follows; moreover, condition 1 above can be weakened to the assertion that both operations are unital, since it can be proved from condition 2 that the units must then coincide. If the operations are associative, each one defines the structure of a monoid on X, and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism with respect to . (or vice versa). An even more abstract way of stating the theorem is: If X is a monoid object in the monoidal category of monoids, then X is in fact a commutative monoid.
Proof
The proof is not hard, although it is much more conceptually clear if geometric diagrams are used. In ordinary algebra notation, the proof is as follows:
Let a,b in X. Then a.b = (1*a).(b*1) = (1.b)*(a.1) = b*a = (b.1)*(1.a) = (b*1).(1*a) = b.a ,
References
* [http://math.ucr.edu/home/baez/week89.html John Baez: Eckmann–Hilton principle (week 89)]
* [http://math.ucr.edu/home/baez/week100.html John Baez: Eckmann–Hilton principle (week 100)]External links
* [http://www.youtube.com/watch?v=Rjdo-RWQVIY]
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