Eckmann–Hilton duality

Discussion

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 ${\displaystyle \mathrm{\Sigma }X}$, which is a quotient of ${\displaystyle X\times I}$, and the loop space ${\displaystyle \mathrm{\Omega }Y}$, which is a subspace of ${\displaystyle Y^I}$. This then leads to the adjoint relation ${\displaystyle ⟨\mathrm{\Sigma }X,Y⟩=⟨X,\mathrm{\Omega }Y⟩}$, which allows the study of spectra, which give rise to cohomology theories.

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

and a cofibration ${\displaystyle i:A\to X}$ 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 ${\displaystyle F\to E\to B}$ we get the sequence

${\displaystyle \cdots \to {\mathrm{\Omega }}^{2}B\to \mathrm{\Omega }F\to \mathrm{\Omega }E\to \mathrm{\Omega }B\to F\to E\to B}$

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

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

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 ${\displaystyle {\pi }_n(X,p)\cong ⟨S^n,X⟩}$, 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 ${\displaystyle K(G,n)}$ and the relation

${\displaystyle H^n(X;G)\cong ⟨X,K(G,n)⟩.}$

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