nLab
Barratt-Eccles operad

Contents

Idea

The Barratt-Eccles operad \mathcal{E} is a specific realization of an E-infinity operad, a cofibrant resolution of the commutative operad (in the model structure on operads).

As a topological operad it is given by n:=EΣ n\mathcal{E}_n := E \Sigma_n, the universal principal bundle for the symmetric group Σ n\Sigma_n. As an sSet-operad it has n=N(Σ n//Σ n)\mathcal{E}_n = N(\Sigma_n // \Sigma_n), the nerve of the action groupoid of Σ n\Sigma_n acting on itself.

Definition

As a simplicial operad

We give the definition of the Barratt-Eccles operad as an object of the category of multi-colored symmetric simplicial operads (sSet-enriched symmetric multicategories). (See Berger-Fress, section 1.1.5.)

The Barratt-Eccles operad \mathcal{E} is the operad defined as follows.

It has a single color.

For nn \in \mathbb{N}, its simplicial set (n)\mathcal{E}(n) of nn-ary operations is the nerve N(Σ n//Σ n)N(\Sigma_n//\Sigma_n) of the action groupoid Σ n//Σ n\Sigma_n//\Sigma_n of the symmetric group Σ n\Sigma_n permuting nn elements acting by right multiplication on itself.

(n):=N(Σ n//Σ n). \mathcal{E}(n) := N(\Sigma_n // \Sigma_n) \,.

Explicitly, this is the simplicial set whose kk-cells are (k+1)(k+1)-tuples of group elements

(n) k= iso(Σ n) ×k+1. \mathcal{E}(n)_k =_{iso} (\Sigma_n)^{\times k+1} \,.

Regarded as the nerve of the action groupoid, the face maps on (n)\mathcal{E}(n) are given by multiplication in Σ n\Sigma_n

d i(g 0,g 1,,g k)=g 0,,g i1,g ig i+1,g i+2,,g k0ik. d_i (g_0, g_1, \cdots, g_k) = g_0, \cdots, g_{i-1}, g_i \cdot g_{i+1}, g_{i+2}, \cdots, g_k \;\;\;\;\; 0 \leq i \leq k \,.

But, alternatively, we can parameterize (n) k\mathcal{E}(n)_k by the tuples

(w 0,w 1,,w k):=(g 0,g 0g 1,g 0g 1g 2,, i=0 kg i). (w_0, w_1, \cdots, w_k) := (g_0, g_0 \cdot g_1, g_0 \cdot g_1 \cdot g_2, \cdots, \prod_{i=0}^{k} g_i) \,.

In terms of this the iith face map is given simply by omitting the iith entry

d i(w 0,,w k)=(w 0,,w^ i,,w k). d_i(w_0, \cdots, w_k) = (w_0, \cdots, \hat w_i, \cdots, w_k) \,.

The iith degeneracy map is given by repeating the iith entry

s i(w 0,,w n):=(w 0,,w i1,w i,w i,w i+1,,w n)0in. s_i (w_0, \cdots, w_n) := (w_0, \cdots, w_{i-1}, w_i, w_i, w_{i+1}, \cdots, w_n) \;\;\; 0 \leq i \leq n \,.

In terms of this, the Σ n\Sigma_n-action on (n)\mathcal{E}(n) (giving the structure of a symmetric operad) is then the diagonal action

σ(w 0,w 1,,w k):=(σw 0,σw 1,,σw k). \sigma \cdot (w_0, w_1, \cdots, w_k) := (\sigma \cdot w_0, \sigma \cdot w_1, \cdots, \sigma \cdot w_k) \,.

The composition operations in the operad

(r)×((n 1)××(n r))(n 1++n r) \mathcal{E}(r) \times (\mathcal{E}(n_1) \times \cdots \times \mathcal{E}(n_r)) \to \mathcal{E}(n_1 + \cdots + n_r)

are the morphisms of simplicial sets which in degree kk are maps on tuples, which in each degree ii are given by the natural function

Σ r×(Σ n 1××Σ n r)Σ n 1++n r \Sigma_r \times (\Sigma_{n_1} \times \cdots \times \Sigma_{n_r}) \to \Sigma_{n_1 + \cdots + n_r}

that composes rr permutations with a permutation of rr elements to a permutation of i=0 rn r\sum_{i = 0}^r n_r elements.

(This function is in fact that which gives the composition in Assoc when regarded as a symmetric operad, Assoc:=Symm(*)Assoc := Symm(*).)

As a dg-operad

(…)

See (Berger-Fresse).

Properties

As a simplicial operad

Each of the simplicial sets (n)\mathcal{E}(n) for nn \in \mathbb{N} is contractible. One way to see this is to observe that Σ n//Σ n\Sigma_n // \Sigma_n is (the nerve of) the pullback

Σ n//Σ n * (BΣ n) I BΣ n, \array{ \Sigma_n // \Sigma_n &\to& * \\ \downarrow && \downarrow \\ (\mathbf{B} \Sigma_n)^I &\to& \mathbf{B}\Sigma_n } \,,

where BΣ n\mathbf{B}\Sigma_n is one-object groupoid with Σ n\Sigma_n as its morphisms, (BΣ n) I(\mathbf{B}\Sigma_n)^I is its arrow category and the bottom vertical map is evaluation at the source. Since this is an acyclic fibration, so is the top vertical morphism.

It follows that the canonical morphism of simplicial operads

Comm \mathcal{E} \to Comm

to the commutative operad (which has Comm(n)=*Comm(n) = * for all nn \in \mathbb{N}) is a weak equivalence (in the model structure on operads). In fact, it is a cofibrant resolution.

As a dg-operad

(…)

References

As a simplicial operad, the Barratt-Eccles operad was introduced in

  • M. Barratt, P. Eccles, On Γ +\Gamma^+-structures I. A free group functor for stable homotopy theory, Topology 13 (1974), 25-45.

Its realization as a dg-operad is discussed in detail in

Elmendorf and Mandell show that the Barratt-Eccles operad EΣ *E\Sigma_* is obtained by applying functor EE to certain operad Σ *\Sigma_* in

  • A.D. Elmendorf, M.A. Mandell, Rings, modules, and algebras in infinite loop space theory, Advances in Mathematics 205, n.1, 2006, pp 163–228, arxiv, doi

Revised on October 15, 2013 19:37:05 by Urs Schreiber (80.237.234.132)