---
tag: 4RPERPNBK1
kind: definition
name: Eckmann–Hilton duality
msc: [55P30]
defines: 1EK3355WMC
uses: [T9DG8DDE75, Q7RFNJH76Q, 2JFF5F3EGF, W5WS5EY8ZM, 5JSWM1HBX2]
sources:
  - url: https://en.wikipedia.org/wiki/Eckmann–Hilton_duality
---

## Definition

*Eckmann–Hilton duality* is an informal correspondence between notions of homotopy theory of [[pointed spaces|T9DG8DDE75]].
It reverses the direction of all maps and exchanges the members of each of the following pairs:

1. maps out of the cylinder $X \times [0, 1]$, and maps into the path space $Y^{[0, 1]}$, as two descriptions of a homotopy;
2. the reduced [[suspension|Q7RFNJH76Q]] $\Sigma X$, and the [[loop space|2JFF5F3EGF]] $\Omega Y$;
3. [[cofibrations|W5WS5EY8ZM]], defined by extending homotopies, and fibrations, defined by lifting homotopies;
4. the [[wedge sum|5JSWM1HBX2]] $X \vee Y$, and the product $X \times Y$;
5. mapping cones and cofibre sequences, and homotopy fibres and fibre sequences;
6. the homotopy groups $\pi_n(Y) = [S^n, Y]_*$, and the cohomology groups $H^n(X; G) \cong [X, K(G, n)]$ of a CW complex $X$, with $K(G, n)$ an Eilenberg–MacLane space.

## Remarks

The pair in clause 2 is linked by the natural bijection $[\Sigma X, Y]_* \cong [X, \Omega Y]_*$.
Example: the homotopy extension property of a cofibration corresponds under clause 3 to the homotopy lifting property of a fibration.
Likewise the cofibre sequence of a map corresponds to its fibre sequence.
The correspondence is not a theorem: the dual of a true statement need not be true, and each dual statement needs its own proof.
The duality was named after Beno Eckmann and Peter Hilton.