Pseudo-Functors Modelling Higher Structures

Abstract

In this chapter we introduce another main structure of this work: the category ${\mathsf {SegPs}}{[\varDelta ^{{n-1}^{op}},{\mathsf {Cat}\,}]}$ of Segalic pseudo-functors. This category is a full subcategory of the category ${\mathsf {Ps}}[\varDelta ^{{n-1}^{op}},{\mathsf {Cat}\,}]$ of pseudo-functors. The main result of this chapter is that the classical strictification functor form pseudo-functors ${\mathsf {Ps}}{[\varDelta ^{{n-1}^{op}},{\mathsf {Cat}\,}]}$ to strict functors ${[\varDelta ^{{n-1}^{op}},{{\mathsf {Cat}\,}}]}$ restrict to a functor from Segalic pseudo-functors to weakly globular n-fold categories. This result is crucially used in Chap. 10 to construct the rigidification functor Q _n from weakly globular Tamsamani n-categories to weakly globular n-fold categories.