n-category = (n,n)-category
n-groupoid = (n,0)-category
The Moore complex of a simplicial group (or, more generally, any simplicially enriched groupoid), carries a lot more structure that just a chain complex of group(oid)s. This structure is its structure as a hypercrossed complex. The notion was introduced by Pilar Carrasco in her thesis.
Pilar Carrasco, 1987, Complejos Hipercruzados, Cohomologia y Extensiones, Ph.D. thesis, Universidad de Granada.