  This paper proves that the category of vacant n-tuple groupoids is equivalent
to the category of factorizations of groupoids by n subgroupoids. Moreover it
extends this equivalence to the category of maximally exclusive n-tuple
groupoids, that we define, and (n+1)-factorizations of groupoids with a normal
abelian subgroupoid. Finally it shows that in the smooth case, such a
factorization gives a presentation of the Poincar\'e group as a triple
groupoid.
