E-infinity operad

Remarks

Often 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 $B\Sigma_n$.

Example: 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. Each $\mathcal{L}(n)$ is contractible, and $\Sigma_n$ acts by permuting the $n$ copies of $\R^\infty$. The action is free: an isometric embedding sends the $n$ summands to mutually orthogonal subspaces, so it is not invariant under a nontrivial permutation.

Example: the Barratt–Eccles operad has $\mathcal{O}(n) = E\Sigma_n$, which is contractible with a free $\Sigma_n$-action.

Non-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. Its algebras are strictly commutative monoids, which are too restrictive in homotopy theory.