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  "body": "\n## Summary\n\nAn $E_\\infty$ operad is a topological operad whose spaces of $n$-ary operations are contractible and carry a free action of the [[symmetric group|0515A6D2Y1]] $\\Sigma_n$.\nAn algebra over it is a space or spectrum with a multiplication that is commutative and associative up to a coherent system of homotopies.\nSuch algebras model [[infinite loop spaces|A7VB2NSGJ2]] and $E_\\infty$ ring spectra, the homotopy-theoretic version of commutative rings.\n\n## Explanations\n\n### L1 human\n\nStrictly commutative multiplications are too rigid in homotopy theory: many natural products are only commutative up to homotopy.\nAn $E_\\infty$ operad packages all ways of multiplying $n$ elements into a space, and asks that this space be contractible.\nThen any two ways are homotopic, any two homotopies are homotopic, and so on.\nThe free action of $\\Sigma_n$ is what keeps the structure homotopically correct instead of strictly commutative.\nA standard example has as $n$-ary operations the linear isometric embeddings of $(\\R^\\infty)^n$ into $\\R^\\infty$.\n",
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