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 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.

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

...
Wikipedia