Cubical models of higher categories without connections

TL;DR

This paper establishes equivalences between different models of higher categories using cubical and simplicial sets, demonstrating that models without connections are fundamentally comparable to those with connections.

Contribution

It proves Quillen equivalences between cubical models without connections and both with connections and simplicial models, unifying various higher category frameworks.

Findings

Cubical models without connections are Quillen equivalent to those with connections.

The cubical Joyal model structure is equivalent to the simplicial Joyal model.

Any connection-free comical set can be equipped with connections compatibly.

Abstract

We prove that each of the model structures for ($n$-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure for ($n$-trivial, saturated) complicial sets on the category of marked simplicial sets, as well as to the corresponding comical model structures on cubical sets with connections. As a consequence, we show that the cubical Joyal model structure on cubical sets without connections is equivalent to its analogues on cubical sets with connections and to the Joyal model structure on simplicial sets. We also show that any comical set without connections may be equipped with connections via lifting, and that this can be done compatibly on the domain and codomain of any fibration or cofibration of comical sets.