nLab Barratt-Eccles operad

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 $\mathcal{E}_n := E \Sigma_n$, the universal principal bundle for the symmetric group $\Sigma_n$. As an sSet-operad it has $\mathcal{E}_n = N(\Sigma_n // \Sigma_n)$, the nerve of the action groupoid of $\Sigma_n$ 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 $n \in \mathbb{N}$, its simplicial set $\mathcal{E}(n)$ of $n$-ary operations is the nerve $N(\Sigma_n//\Sigma_n)$ of the action groupoid $\Sigma_n//\Sigma_n$ of the symmetric group $\Sigma_n$ permuting $n$ elements acting by right multiplication on itself.

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

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

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

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

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

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

$$(w_0, w_1, \cdots, w_k) := (g_0, g_0 \cdot g_1, g_0 \cdot g_1 \cdot g_2, \cdots, \prod_{i=0}^{k} g_i) \,.$$

In terms of this the $i$th face map is given simply by omitting the $i$th entry

$$d_i(w_0, \cdots, w_k) = (w_0, \cdots, \hat w_i, \cdots, w_k) \,.$$

The $i$th degeneracy map is given by repeating the $i$th entry

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

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

$$\sigma \cdot (w_0, w_1, \cdots, w_k) := (\sigma \cdot w_0, \sigma \cdot w_1, \cdots, \sigma \cdot w_k) \,.$$

The composition operations in the operad

$$\mathcal{E}(r) \times (\mathcal{E}(n_1) \times \cdots \times \mathcal{E}(n_r)) \to \mathcal{E}(n_1 + \cdots + n_r)$$

are the morphisms of simplicial sets which in degree $k$ are maps on tuples, which in each degree $i$ are given by the natural function

$$\Sigma_r \times (\Sigma_{n_1} \times \cdots \times \Sigma_{n_r}) \to \Sigma_{n_1 + \cdots + n_r}$$

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

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

As a dg-operad

(…)

See (Berger-Fresse).

Properties

As a simplicial operad

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

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

where $\mathbf{B}\Sigma_n$ is one-object groupoid with $\Sigma_n$ as its morphisms, $(\mathbf{B}\Sigma_n)^I$ 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

$$\mathcal{E} \to Comm$$

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

As a dg-operad

(…)

Related concepts

• E-infinity algebra

• (2,1)-algebraic theory of E-infinity algebras

References

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

• M. Barratt, P. Eccles, On $\Gamma^+$-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

• C. Berger, Combinatorial models for real configuration spaces and $E_n$-operads, pdf
• Clemens Berger, Benoit Fresse Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174, arXiv:math.AT/0109158, doi

• Clemens Berger, Benoit Fresse, A prismatic decomposition of the Barratt-Eccles operad, math.AT/0204326

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