Weakly Globular n-Fold Categories as a Model of Weak n-Categories

Abstract

In this chapter we prove the main results of this work: the equivalence after localization of the categories $\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ and T a ⁿ and the proof of the homotopy hypothesis for $\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$. This exhibits weakly globular n-fold categories as a model of weak n-categories. The two comparison functors are the rigidification functor $Q_n:\mbox{{$\mathsf {Ta}^{\mathsf {n}}$}} \rightarrow \mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ built in Chap. 10 and the discretization functor $Disc_{n}:\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}\rightarrow \mbox{{$\mathsf {Ta}^{\mathsf {n}}$}}$ of this chapter. The construction of the latter uses the category $\mbox{{$\mathsf {FCat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ of Chap. 11. The proof of the homotopy hypothesis requires the introduction of a groupoidal version of the three Segal-type models.