# Eckmann–Hilton duality

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 fact 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

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

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

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].

## See also

## References

- Hatcher, Allen (2002),
*Algebraic Topology*, Cambridge: Cambridge University Press, ISBN 0-521-79540-0. - Hazewinkel, Michiel, ed. (2001) [1994], "Eckmann-Hilton duality",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4