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
An example is given by currying, which tells us that for any object
We can also directly relate fibrations and cofibrations: a fibration
and a cofibration
The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration
and given a cofibration
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
A formalization of the above informal relationships is given by Fuks duality. Though the link to Fuks duality seems to lead nowhere, the link in the References that follow to Eckmann-Hilton duality leads to an explanation of the phrase.