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[edit]

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

We can also directly relate fibrations and cofibrations: a fibration $p\colon E\to B$ is defined by having the homotopy lifting property, represented by the following diagram

and a cofibration $i\colon A\to X$ 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 $F\to E\to B$ we get the sequence

\cdots \to \Omega ^{2}B\to \Omega F\to \Omega E\to \Omega B\to F\to E\to B\,

and given a cofibration $A\to X\to X/A$ we get the sequence

A\to X\to X/A\to \Sigma A\to \Sigma X\to \Sigma \left(X/A\right)\to \Sigma ^{2}A\to \cdots .\,

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 $\pi _{n}(X,p)\cong \langle S^{n},X\rangle$ , 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 $K(G,n)$ and the relation