n-Fold groupoids, n-types and n-track categories

For each n ≥ 1, we introduce two new Segal-type models of ntypes of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit algebraic fundamental n-fold groupoid functor. We compare these models to Tamsamani’s weak n-groupoids, and extract from them a model for (k − 1)-connected n-types.