Orientals as free weak $\omega$-categories

TL;DR

This paper demonstrates that orientals, originally defined as free strict $$-categories on simplices, also serve as free weak $$-categories, linking strict and weak $$-category theories via fibrant replacements.

Contribution

It establishes that orientals are the free weak $$-categories on the same data, extending their role from strict to weak $$-categories using model structures.

Findings

Orientals are fibrant replacements of simplices in Verity's model structure.

They serve as free weak $$-categories on the same generating data.

The paper connects strict and weak $$-category theories through model structures.

Abstract

The orientals are the free strict $\omega$-categories on the simplices introduced by Street. The aim of this paper is to show that they are also the free weak $\omega$-categories on the same generating data. More precisely, we exhibit the Street nerves of the orientals as fibrant replacements of the simplices in Verity's model structure for complicial sets.