In this paper, we prove that the category of vacant $n$-tuple groupoids is equivalent to the category of factorizations of groupoids by $n$ factors that satisfy some Yang-Baxter type equation. Moreover we extend this equivalence to the category of maximally exclusive $n$-tuple groupoids, which we define, by dropping one assumption. The paper concludes by a note on how these results could tell us more about some Lie groups of interest.
Keywords: factorization, groupoid, cubical categories
2010 MSC: 18D05,18A32,18B40
Theory and Applications of Categories, Vol. 28, 2013, No. 12, pp 304-331.