Functoriality of Homotopically Discrete Objects

Abstract

In this chapter we introduce the new category $\mbox{{$\mathsf {FCat}_{\mathsf {wg}}^{\mathsf {n}}$}}$. This is a refinement of the category $\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ with better behaved homotopically discrete substructures, admitting functorial sections. The main result of this chapter is that there is a functor G _n from $\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ to $\mbox{{$\mathsf {FCat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ which approximates up to n-equivalence any object of $\mbox{{$\mathsf {Cat}_{\mathsf {wg}}^{\mathsf {n}}$}}$ with one of $\mbox{{$\mathsf {FCat}_{\mathsf {wg}}^{\mathsf {n}}$}}$. This result is used crucially in Chap. 12 to build the discretization functor from weakly globular n-fold categories to Tamsamani-n categories.