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The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model of an 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 E-spaces.

Specifically, I am interested in group-like E-spaces, which can be thought of as homotopy coherent commutative groups and are a model for connective spectra.

Another interesting case is E-ring spaces, which can be thought of as homotopy coherent commutative rings, and are a model for connective E-ring spectra.

As pointed out by Peter May in his answer, operads cannot model such structures because they do not allow for operations with multiple outputs, e.g., diagonal maps, so a part of the question is what type of structure one should use.

For example, simplicial algebraic theories (see http://ncatlab.org/nlab/show/(%E2%88%9E,1)-algebraic+theory) seem to be a viable option. In particular, the Barratt-Eccles construction admits a particularly elegant formulation in terms of a (2,1)-algebraic theory (i.e., a groupoid-valued algebraic theory), see http://ncatlab.org/nlab/show/(2,1)-algebraic+theory+of+E-infinity+algebras.

Is there an analog of the Barratt-Eccles construction for group-like E-spaces and E-ring spaces?

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1 Answer 1

There are serious problems making your ideas coherent here! The notion of operad was in large part intended to model kinds of algebras whose laws do not involve repeated variables, do not involve diagonal maps. Operads do not even model groups, in particular, and that was intended. Any $E_{\infty}$ operad models all spectra, not just connected ones, the essential point being that the zeroth space of the spectrum associated to an $E_{\infty}$-space $X$, no matter how constructed, must be a group completion of $X$. There is no "the" $E_{\infty}$ operad, rather there are many interesting ones. (There is an axiomatization of infinite loop space machines, due to Thomason and myself, that makes this precise.) If you want to model $E_{\infty}$ ring spectra using operads only, you can make a mistake by trying to use just one operad, as I did over 40 years ago, or you can do it right by using two interrelated operads, one for the addition and one for the multiplication, as I also did over 40 years ago. See http://www.math.uchicago.edu/~may/PAPERS/Final1.pdf for a modern recapitulation of that early theory.

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Indeed, it was stupid of me not to realize that the diagonal map cannot be expressed operadically! I edited the question to allow for generalizations of operads, e.g., properads, props, etc. –  Dmitri Pavlov Aug 31 at 10:45
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Properads and props still don't allow diagonal maps. You want a Lawvere theory. I don't know if anyone's thought about Lawvere theories valued in spaces. –  Qiaochu Yuan Sep 3 at 6:40
    
@QiaochuYuan: A lot of people thought about space-valued algebraic theories, see ncatlab.org/nlab/show(∞,1)-algebraic+theory. In particular, E_∞-spaces admit a particularly elegant description in terms of a (2,1)-algebraic theory, see ncatlab.org/nlab/show/…. In fact, this particular formulation of the Barratt-Eccles construction was one of the motivations for my question, so I will now add it to the main post. –  Dmitri Pavlov Sep 3 at 13:10
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@QiaochuYuan It seems to me that there is a lot of interest in enriched algebraic theories coming from the theoretical computer scientists. –  Zhen Lin Sep 3 at 15:32

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