Cubical models of $(\infty, 1)$-categories

Brandon Doherty, Chris Kapulkin, Zachery Lindsey, Christian Sattler

Published 2020-05-11Version 1

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit an elegant and canonical notion of a mapping space between two objects.