Eckmann-Hilton argument
Encyclopedia
In mathematics
, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument
about two monoid
structure
s 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 group
s. The principle is named after Beno Eckmann
and Peter Hilton
, who used it in a 1962 paper.
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.
.
Then * and . are the same and in fact commutative.
, and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism 
(or vice versa). An even more abstract way of stating the theorem is: If 
is a monoid object in the monoidal category of monoids, then 
is in fact a commutative monoid.
Let
. Then 
It is sometimes represented as a 'clock'. In this image, "0" is the unit for p⊕q and "1" is the unit for ⊗. Starting from any position on the clock we can move to the next by some use of the unital character of "0" and "1" or the distributive rule:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
about two monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
structure
Structure (category theory)
In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications...
s on a set where one is a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
for the other. Given this, the structures can be shown to coincide, and the resulting monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
demonstrated to be commutative. This can then be used to prove the commutativity of the higher homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s. The principle is named after Beno Eckmann
Beno Eckmann
Beno Eckmann was a Swiss mathematician who was a student of Heinz Hopf.Born in Bern, Eckmann received his master's degree from Eidgenössische Technische Hochschule Zürich in 1931. Later he studied there under Heinz Hopf, obtaining his Ph.D. in 1941...
and Peter Hilton
Peter Hilton
Peter John Hilton was a British mathematician, noted for his contributions to homotopy theory and for code-breaking during the Second World War.-Life:Hilton was born in London, and educated at St Paul's School...
, who used it in a 1962 paper.
The Eckmann–Hilton result
Let 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.
.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, and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism (or vice versa). An even more abstract way of stating the theorem is: If is a monoid object in the monoidal category of monoids, then 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
. Then It is sometimes represented as a 'clock'. In this image, "0" is the unit for p⊕q and "1" is the unit for ⊗. Starting from any position on the clock we can move to the next by some use of the unital character of "0" and "1" or the distributive rule: