Rigidification of cubical quasi-categories

TL;DR

This paper introduces a cubical analogue of the rigidification functor, establishing a Quillen equivalence between cubical sets and simplicial categories, extending the work of Joyal and Lurie to a cubical context.

Contribution

It constructs a new rigidification functor for cubical sets and proves it induces a Quillen equivalence with simplicial categories, adapting existing frameworks to cubical structures.

Findings

Established a Quillen equivalence between cubical sets and simplicial categories.

Constructed a cubical rigidification functor analogous to the simplicial case.

Extended the framework of necklaces to the cubical setting.

Abstract

We construct a cubical analogue of the rigidification functor from quasi-categories to simplicial categories present in the work of Joyal and Lurie. We define a functor from the category of cubical sets of Doherty-Kapulkin-Lindsey-Sattler to the category of (small) simplicial categories. We show that this rigidification functor establishes a Quillen equivalence between the Joyal model structure on cubical sets (as it is called by the four authors) and Bergner's model structure on simplicial categories. We follow the approach to rigidification of Dugger and Spivak, adapting their framework of necklaces to the cubical setting.