Cubical Models of Higher Categories
Abstract
This thesis concerns model structures on presheaf categories, modeling the theory of infinity-categories. We introduce the categories of simplicial and cubical sets, and review established examples of model structures on these categories for infinity-groupoids and (infinity, 1)-categories, including the Quillen and Joyal model structures on simplicial sets, and the Grothendieck model structure on cubical sets. We also review the complicial model structure on marked simplicial sets, which presents the theory of (infinity, n)-categories. We then construct a model structure on the category of cubical sets whose cofibrations are the monomor- phisms 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. To do this, we develop a theory of cones in cubical sets. 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. We also develop model structures for (infinity,n)-categories on marked cubical sets, and show that these are equivalent to the complicial model structures on marked simplicial sets.