Eckmann–Hilton duality

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In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category. A significantly deeper form argues that the dual notion of a limit is a colimit allows us to change the Eilenberg–Steenrod axioms for homology to give axioms for cohomology. It is named after Beno Eckmann and Peter Hilton.

Discussion[edit]

An example is given by currying, which tells us that for any object , a map is the same as a map , where is the exponential object, given by all maps from to . In the case of topological spaces, if we take to be the unit interval, this leads to a duality between and which then gives a duality between the reduced suspension which is a quotient of and the loop space which is a subspace of . This then leads to the adjoint relation which allows the study of spectra which give rise to cohomology theories.

We can also directly relate fibrations and cofibrations: a fibration is defined by having the homotopy lifting property, represented by the following diagram

Homotopy lifting property.svg

and a cofibration is defined by having the dual homotopy extension property, represented by dualising the previous diagram:

Homotopy extension property.svg

The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration we get the sequence

and given a cofibration we get the sequence

and more generally, the duality between the exact and coexact Puppe sequences.

This also allows us to relate homotopy and cohomology: we know that homotopy groups are homotopy classes of maps from the n-sphere to our space, written , 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–MacLane spaces and the relation .

A formalization of the above informal relationships is given by Fuks duality[1].

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