# The table reduction morphism of operads from Barratt-Eccles to Surjection

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!

• Have you tried contacting the authors? I'm sure they'll be pleased to answer your questions. – Fernando Muro Mar 21 '19 at 21:13

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:

1. it is a chain map;
2. if $$\sigma$$ is the maximal simplex of a prism $$\tau_u$$, then: $$TR(\sigma) = \begin{cases} u & \text{if } \sigma \text{ is the fundamental simplex of } \tau_u; \\ 0 & \text{otherwise.} \end{cases}$$