The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model for the E_∞-operad; algebras in spaces over the Barratt-Eccles operad model E_∞-spaces, i.e., homotopy coherent commutative monoids in spaces. It can be described concretely by applying the nerve functor componentwise to an operad in groupoids, which itself is obtained by applying the codiscrete groupoid functor componentwise to an operad Σ in sets such that Σ(n) is the symmetric group of order n and the operadic composition Σ(n)×(Σ(a₁)×⋯×Σ(aₙ))→Σ(a₁+⋯+aₙ) is given by stacking the permutations in Σ(aᵢ) together and composing them with the block permutation in Σ(a₁+⋯+aₙ) induced by the permutation in Σ(n). Here the codiscrete groupoid functor sends a set X to the groupoid with X as the set of objects and exactly one morphism between any pair of objects; it is the right adjoint to the forgetful functor from groupoids to sets that sends a groupoid to its underlying set of objects.

I am interested in similarly spirited constructions for various cousins of the E_∞-operad. Specifically, I am interested in the E_∞-group operad, which models homotopy coherent commutative groups, for example, algebras in spaces over this operad are a model for connective spectra; it is obtained from the E_∞-operad by adding homotopy coherent inverses. Another interesting case is the E_∞-ring operad, which models homotopy coherent commutative rings, for example, algebras in spaces over this operad are a model for connective E_∞-ring spectra; it is obtained from the E_∞-group operad by adding a homotopy coherent commutative multiplication operation with the appropriately coherent distributive property.

Is there an analog of the Barratt-Eccles construction for the E_∞-group and E_∞-ring operads?