The Definition of the Three Segal-Type Models

Abstract

In this chapter we introduce the three Segal-type models: the category $\mbox{{$\mathsf {Ta}_{\mathsf {wg}}^{\mathsf {n}}$}}$ of weakly globular Tamsamani n-categories and its subcategories T a ⁿ (Tamsamani n-categories) and $\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ (weakly globular n-fold categories). We also define the notion of n-equivalence in these models, as part of their inductive definition. We discuss some elementary properties of the three models, and we illustrate some examples in low dimensions.