The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an operad morphism from the normalized chains of $E$ onto the Surjection operad, another very canonical $E_{\infty}$-operad.
I am looking to understand this morphism better. The authors simple give an algorithm but do not hint at the origin of or motivation for their description. Any clues? Thanks!
As written in the paper that you cite, this construction originates from two papers of McClure–Smith, and a geometric interpretation of the table reduction morphism is given in:
Clemens Berger and Benoit Fresse. "Une décomposition prismatique de l'opérade de Barratt-Eccles." (French) C. R. Math. Acad. Sci. Paris 335 (2002), no. 4, pp. 365–370.
In particular, I think Lemme 3.3 is enlightening. Let me very quickly summarize it.
Recall that $\mathcal{E}(n) = \Bbbk[E\Sigma_n]$ is the Barratt–Eccles operad, where $\mathcal{E}(n)_d = \Bbbk[\Sigma_n^{d+1}]$. It is the linearization of a simplicial operad $\mathcal{W}$. On the other hand, $\mathcal{X}$ is the surjection operad, where $\mathcal{X}(n)_d$ has for basis the set of surjections $u : \{1,\dots,n+d\} \to \{1,\dots,n\}$ such that $u(i) \neq u(i+1)$ for all $i$.
For a surjection $u \in \mathcal{X}(n)_d$, let $d_k = \# u^{-1}(k)$. There is an associated prism $\tau_u : \Delta^{d_1-1} \times \dots \times \Delta^{d_n-1} \to \mathcal{W}(n)$ defined combinatorially. Among all these prisms, there is a maximal one, called the fundamental simplex. Then:
Lemma 3.3. The map $TR : \mathcal{E} \to \mathcal{X}$ is characterized by the following properties:
- it is a chain map;
- if $\sigma$ is the maximal simplex of a prism $\tau_u$, then: $$TR(\sigma) = \begin{cases} u & \text{if } \sigma \text{ is the fundamental simplex of } \tau_u; \\ 0 & \text{otherwise.} \end{cases}$$
