---
tag: T78YG90VVT
kind: concept
name: E-infinity operad
msc: [55P48, 18M75]
sources:
  - url: https://en.wikipedia.org/wiki/E∞-operad
same_as:
  wikidata: Q5423545
  wikipedia: E∞-operad
---

## Summary

An $E_\infty$ operad is a topological operad whose spaces of $n$-ary operations are contractible and carry a free action of the [[symmetric group|0515A6D2Y1]] $\Sigma_n$.
An algebra over it is a space or spectrum with a multiplication that is commutative and associative up to a coherent system of homotopies.
Such algebras model [[infinite loop spaces|A7VB2NSGJ2]] and $E_\infty$ ring spectra, the homotopy-theoretic version of commutative rings.

## Explanations

### L1 human

Strictly commutative multiplications are too rigid in homotopy theory: many natural products are only commutative up to homotopy.
An $E_\infty$ operad packages all ways of multiplying $n$ elements into a space, and asks that this space be contractible.
Then any two ways are homotopic, any two homotopies are homotopic, and so on.
The free action of $\Sigma_n$ is what keeps the structure homotopically correct instead of strictly commutative.
A standard example has as $n$-ary operations the linear isometric embeddings of $(\R^\infty)^n$ into $\R^\infty$.