Conditions de Kan sur les nerfs des $\omega$-cat\'egories

TL;DR

This paper characterizes when the Street nerve of a strict ω-category forms a Kan complex or a quasi-category based on the invertibility of its cells, and introduces a saturated complicial set structure related to orientals.

Contribution

It provides a precise criterion linking weak invertibility of cells in ω-categories to Kan and quasi-category conditions of their nerves, and constructs a saturated complicial set structure.

Findings

Street nerve of strict ω-category is a Kan complex iff cells are weakly invertible.

Street nerve is a quasi-category iff cells of dimension > 1 are weakly invertible.

The nerve can be structured as a saturated complicial set with morphisms from orientals.

Abstract

We show that the Street nerve of a strict $\omega$-category $C$ is a Kan complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for $n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip $\mathcal{N}(C)$ with a structure of saturated complicial set where the $n$-simplices correspond to morphisms from the $n^{th}$ oriental to $C$ sending the unique non-trivial $n$-cell of the domain to a weakly invertible cell of $C$.