The following observation of Conduché is very useful when working with simplicial groupoids.
If is a simplicial group(oid), then decomposes as a multiple semidirect product:
The order of the terms corresponds to a lexicographic ordering of the indices ; 0; 1; 1,0; 2; 2,0; 2,1; 2,1,0; 3; 3,0; and so on, the term corresponding to being . 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.
If is a simplicial group(oid), then decomposes as a semidirect product:
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.
D. Conduché, Modules croisés généralisés de longueur 2 , J. Pure Appl. Alg., 34, (1984), 155–178.