nLab decomposition theorem for simplicial groups

Discussion

The following observation of Conduché is very useful when working with simplicial groupoids.

Proposition

If $G$ is a simplicial group(oid), then $G_n$ decomposes as a multiple semidirect product:

$$G_n \cong NG_n \rtimes s_0NG_{n-1}\rtimes s_1NG_{n-1}\rtimes s_1s_0NG_{n-2} \rtimes s_2NG_{n-1}\rtimes \ldots s_{n-1}s_{n-2}\ldots s_0NG_0$$

The order of the terms corresponds to a lexicographic ordering of the indices $\emptyset$; 0; 1; 1,0; 2; 2,0; 2,1; 2,1,0; 3; 3,0; $\ldots$ and so on, the term corresponding to $i_1 \gt \ldots \gt i_p$ being $s_{i_1}\ldots s_{i_p}NG_{n-p}$. The actions involved are clear once the following lemma is examined.

The proof of the result is an induction based on a simple lemma, which is easy to prove.

Lemma

If $G$ is a simplicial group(oid), then $G_n$ decomposes as a semidirect product:

$$G_n \cong Ker d^n_n \rtimes s^{n-1}_{n-1}(G_{n-1}).$$

This decomposition generalises the one used in the classical Dold-Kan correspondence. It is extremely useful when analysing the Moore complex of a simplicial group and the relationship between that complex and the original simplicial group. It plays a crucial role in the theory of hypercrossed complexes.

Reference

D. Conduché, Modules croisés généralisés de longueur 2 , J. Pure Appl. Alg., 34, (1984), 155–178.