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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an
argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
about two unital magma structures on a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
where one is a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
for the other. Given this, the structures are the same, and the resulting
magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...
is a
commutative monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
. This can then be used to prove the commutativity of the higher
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s. The principle is named after
Beno Eckmann Beno Eckmann (31 March 1917 – 25 November 2008) was a Switzerland, Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry. Life Born to a Jewish family in Bern, Eckmann r ...
and
Peter Hilton Peter John Hilton (7 April 1923Peter Hilton, "On all Sorts of Automorphisms", ''The American Mathematical Monthly'', 92(9), November 1985, p. 6506 November 2010) was a British mathematician, noted for his contributions to homotopy theory and f ...
, who used it in a 1962 paper.


The Eckmann–Hilton result

Let X be a set equipped with two
binary operations In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation o ...
, which we will write \circ and \otimes, and suppose: # \circ and \otimes are both unital, meaning that there are identity elements 1_\circ and 1_\otimes of X such that 1_\circ \circ a= a =a \circ 1_\circ and 1_\otimes \otimes a= a =a \otimes 1_\otimes, for all a\in X.
# (a \otimes b) \circ (c \otimes d) = (a \circ c) \otimes (b \circ d) for all a,b,c,d \in X . Then \circ and \otimes are the same and in fact commutative and associative.


Remarks

The operations \otimes and \circ are often referred to as
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
structures or multiplications, but this suggests they are assumed to be associative, a property that is not required for the proof. In fact, associativity follows. Likewise, we do not have to require that the two operations have the same neutral element; this is a consequence.


Proof

First, observe that the units of the two operations coincide: 1_\circ = 1_\circ \circ 1_\circ = (1_\otimes \otimes 1_\circ) \circ (1_\circ \otimes 1_\otimes) = (1_\otimes \circ 1_\circ) \otimes (1_\circ \circ 1_\otimes) = 1_\otimes \otimes 1_\otimes = 1_\otimes. Now, let a,b \in X. Then a \circ b = (1 \otimes a) \circ (b \otimes 1) = (1 \circ b) \otimes (a \circ 1) = b \otimes a = (b \circ 1) \otimes (1 \circ a) = (b \otimes 1) \circ (1 \otimes a) = b \circ a. This establishes that the two operations coincide and are commutative. For associativity, (a \otimes b) \otimes c = (a \otimes b) \otimes (1 \otimes c) = (a \otimes 1) \otimes (b \otimes c) = a \otimes (b \otimes c).


Two-dimensional proof

The above proof also has a "two-dimensional" presentation that better illustrates the application to higher
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s. For this version of the proof, we write the two operations as vertical and horizontal juxtaposition, i.e., \begina\\ 60pt\end and a \ b. The interchange property can then be expressed as follows: For all a,b,c,d\in X, \begin(a \ b)\\ 60ptc \ d)\end = \begina\\ 60pt\end \beginb\\ 60pt\end, so we can write \begin a \ b \\ 60ptc \ d \end without ambiguity. Let \bullet and \circ be the units for vertical and horizontal composition respectively. Then \bullet = \begin\bullet \\ 60pt\bullet \end = \begin \bullet \ \ \circ \\ 60pt\circ \ \ \bullet \end = \circ \ \circ = \circ , so both units are equal. Now, for all a,b\in X, a \ b = \begin a \ \ \bullet\\ 60pt\bullet \ \ b \end = \begina \\ 60ptb\end = \begin \bullet \ \ a \\ 60ptb \ \ \bullet \end = b \ a = \begin b \ \ \bullet \\ 60pt\bullet \ \ a \end = \beginb \\ 60pta\end , so horizontal composition is the same as vertical composition and both operations are commutative. Finally, for all a,b,c\in X, a \ (b \ c) = a\ \beginb\\ 60pt\end = \begin a \ \ b\\ 60pt\bullet \ \ c \end = \begin (a \ b) \\ 60ptc \end = (a \ b) \ c , so composition is associative.


Remarks

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 \otimes is a monoid homomorphism (X,\circ)\times(X,\circ)\to(X,\circ) (or vice versa). An even more abstract way of stating the theorem is: If X is a
monoid object In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''η ...
in the
category of monoids In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''� ...
, then X is in fact a commutative monoid. It is important that a similar argument does ''not'' give such a trivial result in the case of monoid objects in the categories of small categories or of groupoids. Instead the notion of group object in the category of
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...
s turns out to be equivalent to the notion of
crossed module In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, (g,h) \mapsto g \cdot h , and a homomorphism of groups : d\colon H \longrighta ...
. This leads to the idea of using multiple groupoid objects in homotopy theory. More generally, the Eckmann–Hilton argument is a special case of the use of the interchange law in the theory of (strict) double and multiple categories. A (strict) double category is a set, or class, equipped with two category structures, each of which is a morphism for the other structure. If the compositions in the two category structures are written \circ, \otimes then the interchange law reads : (a \circ b) \otimes (c \circ d) = (a \otimes c) \circ (b \otimes d) whenever both sides are defined. For an example of its use, and some discussion, see the paper of Higgins referenced below. The interchange law implies that a double category contains a family of abelian monoids. The history in relation to
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s is interesting. The workers in topology of the early 20th century were aware that the nonabelian
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
was of use in geometry and analysis; that abelian
homology groups In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian group ...
could be defined in all dimensions; and that for a connected space, the first homology group was the fundamental group made abelian. So there was a desire to generalise the nonabelian fundamental group to all dimensions. In 1932,
Eduard Čech Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...
submitted a paper on higher
homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about Loop (topology), loops in a Mathematic ...
to the International Congress of Mathematics at Zürich. However, Pavel Alexandroff and
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...
quickly proved these groups were abelian for n > 1, and on these grounds persuaded Čech to withdraw his paper, so that only a small paragraph appeared in the ''Proceedings''. It is said that
Witold Hurewicz Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician who worked in topology. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on th ...
attended this conference, and his first work on higher homotopy groups appeared in 1935. Thus the dreams of the early topologists have long been regarded as a mirage. Cubical higher homotopy groupoids are constructed for filtered spaces in the book
Nonabelian algebraic topology
' cited below, which develops basic algebraic topology, including higher analogues to the Seifert–Van Kampen theorem, without using
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension n, the n-dimensional ...
or simplicial approximation.


References


John Baez: Eckmann–Hilton principle (week 89)
*. * . *. *. * * Murray Bremner and Sara Madariaga. (2014
Permutation of elements in double semigroups


External links


Eugenia Cheng of 'the Catsters' video team explains the Eckmann–Hilton argument.


{{DEFAULTSORT:Eckmann-Hilton argument Category theory Theorems in abstract algebra