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{{For|the argument about certain monoids having to be commutative|Eckmann–Hilton argument}}
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In the mathematical disciplines of [[algebraic topology]] and [[homotopy theory]], '''Eckmann&ndash;Hilton duality''' in its most basic form, consists of taking a given [[diagram (category theory)|diagram]] for a particular concept and reversing the direction of all arrows, much as in [[category theory]] with the idea of the [[opposite category]].
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It is named after [[Beno Eckmann]] and [[Peter Hilton]].
 
For example, the fact that the dual notion of a [[limit (category theory)|limit]] is a [[colimit]] allows us to change the [[Eilenberg&ndash;Steenrod axioms]] for [[homology (mathematics)|homology]] to give axioms for [[cohomology]]. <!-- explain more-->
 
Another example is given by [[currying]], which tells us that for any object <math>X</math>, a map <math> X \times I \to Y</math> is the same as a map <math> X \to Y^I</math>, where <math> Y^I</math> is the [[exponential object]], given by all maps from <math> I </math> to <math> Y </math>. In the case of [[topological spaces]], if we take <math> I </math> to be the unit interval, this leads to a duality between <math> X \times I</math> and <math> Y^I</math> which then gives a duality between
the [[suspension (topology)|reduced suspension]] <math> \Sigma X</math> which is a quotient of <math>X \times I</math> and
the [[loop space]] <math> \Omega Y</math> which is a subspace of <math> Y ^ I</math>.
This then leads to the [[adjoint relation]] <math> \langle \Sigma X, Y \rangle = \langle X, \Omega Y \rangle</math> which allows the study of [[spectrum (homotopy theory)|spectra]] which give rise to [[cohomology|cohomology theories]].
 
We can also directly relate [[fibration]]s and [[cofibration]]s: a fibration <math> p \colon E \to B </math> is defined by having the [[homotopy lifting property]], represented by the following diagram
 
[[Image:Homotopy lifting property.svg|175px|center]]
 
and a cofibration <math> i \colon A \to X</math> is defined by having the dual [[homotopy extension property]], represented by dualising the previous diagram:
 
[[Image:Homotopy extension property.svg|175px|center]]
 
The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration <math> F \to E \to B</math> we get the sequence
 
: <math>\cdots \to \Omega^2 B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B \, </math>
 
and given a cofibration <math> A \to X \to X/A</math> we get the sequence
 
: <math> A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma \left (X/A \right ) \to \Sigma^2 A \to \cdots. \, </math>
 
This also allows us to relate [[homotopy]] and cohomology: we know that [[homotopy group]]s are [[homotopy class]]es of maps from the [[n-sphere|''n''-sphere]] to our space, written <math> \pi_n(X,p) \cong \langle S^n,X \rangle</math>, and we know that the sphere has a single nonzero (reduced) [[cohomology group]]. On the other hand, cohomology groups are homotopy classes of maps to spaces with a single nonzero homotopy group. This is given by the [[Eilenberg&ndash;MacLane space]]s <math> K(G,n)</math> and the relation <math> H^n(X;G) \cong \langle X,K(G,n) \rangle </math>.
 
== References ==
*{{Hatcher AT}}
 
{{DEFAULTSORT:Eckmann-Hilton Duality}}
[[Category:Duality theories]]

Latest revision as of 14:18, 5 May 2014

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