N-tuple groups and matched n-tuples of groups

TL;DR

This paper establishes a categorical equivalence between vacant n-tuple groupoids and factorizations of groupoids by n subgroupoids, extending to maximally exclusive cases and relating to the Poincaré group in the smooth setting.

Contribution

It introduces the concept of maximally exclusive n-tuple groupoids and links them to (n+1)-factorizations of groupoids with a normal abelian subgroupoid.

Findings

Vacant n-tuple groupoids are equivalent to factorizations of groupoids by n subgroupoids.

Maximally exclusive n-tuple groupoids correspond to (n+1)-factorizations with a normal abelian subgroupoid.

In the smooth case, such factorizations present the Poincaré group as a triple groupoid.

Abstract

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.