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 ${\displaystyle X}$, 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.

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

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

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

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

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

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

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

A formalization of the above informal relationships is given by .^[1]

See also[]

• Model category

References[]

• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.
• "Eckmann-Hilton duality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]