---
tag: 1EK3355WMC
kind: concept
name: Eckmann–Hilton duality
msc: [55P30]
sources:
  - url: https://en.wikipedia.org/wiki/Eckmann–Hilton_duality
same_as:
  wikidata: Q5332931
  wikipedia: Eckmann–Hilton duality
---

## Summary

Eckmann–Hilton duality is a correspondence in homotopy theory between pairs of dual notions, obtained by reversing the direction of maps.
Under it, the [[suspension|Q7RFNJH76Q]] corresponds to the [[loop space|2JFF5F3EGF]], [[cofibrations|W5WS5EY8ZM]] to fibrations, and the [[wedge sum|5JSWM1HBX2]] to the product.
It is a guiding principle, not a theorem: it suggests dual statements and dual proofs, but each dual statement must be checked on its own.

## Explanations

### L1 human

Many constructions in homotopy theory come in pairs that mirror each other.
One member of a pair is about maps going out of a space, and the other is about maps coming into a space.
For example, a homotopy can be seen as a map out of a cylinder $X \times [0, 1]$ or as a map into a space of paths.
Eckmann–Hilton duality is the habit of turning each result about one side into a guess about the other side.