We define a Levi category to be a weakly orthogonal category equipped with a suitable length functor and prove two main theorems about them. First, skeletal cancellative Levi categories are precisely the categorical versions of graphs of groups with a given orientation. Second, the universal groupoid of a skeletal cancellative Levi category is the fundamental groupoid of the corresponding graph of groups. These two results can be viewed as a co-ordinate-free refinement of a classical theorem of Philip Higgins.
Keywords: Graphs of groups, self-similar groupoid actions, cancellative categories
2010 MSC: 18B40, 20E06
Theory and Applications of Categories, Vol. 32, 2017, No. 23, pp 780-802.