Nerves and cones of free loop-free {\omega}-categories

TL;DR

This paper demonstrates that the complicial nerve construction preserves homotopy equivalences with cone constructions in free, loop-free -categories, linking simplices to orientals.

Contribution

It establishes homotopical compatibility of complicial nerves with cone constructions in free, loop-free -categories, generalizing known equivalences for simplices and orientals.

Findings

Complicial nerve is homotopically compatible with cone constructions.

Standard m-simplex is equivalent to the complicial nerve of the m-oriental.

Results apply to free, loop-free -categories.

Abstract

We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $\omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that the standard $m$-simplex is equivalent to the complicial nerve of the $m$-oriental.