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  "body": "\n## Definition\n\nAn *operad* $\\mathcal{O}$ in [[topological spaces|0CNQXVBHCX]] consists of spaces $\\mathcal{O}(n)$ for $n \\ge 0$ and the following data.\nThere is a right [[action|0YSE802YCQ]] of the [[symmetric group|0RJMNEN3BJ]] $\\Sigma_n$ on each $\\mathcal{O}(n)$ and an identity element $1 \\in \\mathcal{O}(1)$.\nThere are continuous composition maps\n$$\\mathcal{O}(k) \\times \\mathcal{O}(j_1) \\times \\cdots \\times \\mathcal{O}(j_k) \\to \\mathcal{O}(j_1 + \\cdots + j_k),$$\nwhich are associative, unital and equivariant in the standard sense.\nAn operad $\\mathcal{O}$ is an *$E_\\infty$ operad* if, for every $n \\ge 0$:\n\n1. the space $\\mathcal{O}(n)$ is [[contractible|R5WP8Z69Q8]], and\n2. the action of $\\Sigma_n$ on $\\mathcal{O}(n)$ is free.\n\n## Remarks\n\nOften one also asks the action to be nice enough that $\\mathcal{O}(n) \\to \\mathcal{O}(n)/\\Sigma_n$ is a principal $\\Sigma_n$-bundle, so that $\\mathcal{O}(n)/\\Sigma_n$ is a model of $B\\Sigma_n$.\n\nExample: the linear isometries operad, with $\\mathcal{L}(n)$ the space of linear isometric embeddings $(\\R^\\infty)^n \\to \\R^\\infty$ and composition by composing maps.\nEach $\\mathcal{L}(n)$ is contractible, and $\\Sigma_n$ acts by permuting the $n$ copies of $\\R^\\infty$.\nThe action is free: an isometric embedding sends the $n$ summands to mutually orthogonal subspaces, so it is not invariant under a nontrivial permutation.\n\nExample: the Barratt–Eccles operad has $\\mathcal{O}(n) = E\\Sigma_n$, which is contractible with a free $\\Sigma_n$-action.\n\nNon-example: the commutative operad, with $\\mathcal{O}(n)$ a single point for every $n$, satisfies clause 1 but not clause 2 when $n \\ge 2$, since $\\Sigma_n$ fixes the point.\nIts algebras are strictly commutative monoids, which are too restrictive in homotopy theory.\n",
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