Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories

TL;DR

This paper establishes a Quillen equivalence between cubical and simplicial models of $( , n)$-categories, demonstrating their fundamental similarity through advanced model category theory.

Contribution

It proves the equivalence of cubical and simplicial approaches to $( , n)$-categories using the theory of cones and model structures.

Findings

Marked triangulation functor is a Quillen equivalence.

Cubical and simplicial models are fundamentally equivalent.

Uses cone theory to establish the equivalence.

Abstract

We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for ($n$-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for ($n$-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler.