Segal-type algebraic models of $n$–types

Abstract

For each $n\ge 1$, we introduce two new Segal-type models of $n$–types of topological spaces: weakly globular $n$–fold groupoids and their lax version. 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 $\left(k-1\right)$–connected $n$–types.