Rigidification of cubical quasicategories (Q6614590)

The paper constructs a cubical analogue of the rigidification functor from quasicategories to simplicial categories, originally developed by Joyal and Lurie. The authors define a functor from the category of cubical sets (as introduced by Doherty, Kapulkin, Lindsey, and Sattler) to the category of small simplicial categories. This functor establishes a Quillen equivalence between the Joyal model structure on cubical sets and Bergner's model structure on simplicial categories. The paper adapts the framework of necklaces, initially developed by Dugger and Spivak, to the cubical setting. This adaptation allows the authors to extend the rigidification process to cubical quasicategories, providing a method for converting between these two models of higher categories. The authors demonstrate that this rigidification functor preserves the essential homotopical properties, ensuring that the resulting simplicial categories retain the same homotopy type as the original cubical quasicategories.