Cubical models of $(\infty, 1)$categories
Abstract
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 a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DKequivalences.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2005.04853
 arXiv:
 arXiv:2005.04853
 Bibcode:
 2020arXiv200504853D
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 Primary: 18N60;
 18N40;
 Secondary: 55U35;
 55U40
 EPrint:
 109 pages