Brandon Doherty: Constructing connections in fibrant cubical sets

Abstract

Via the cubical Joyal model structure, \((\infty,1)\)-categories may be modeled as cubical sets with connections having fillers for specified open boxes. Likewise, marked cubical sets with connections model \((\infty,n)\)-categories for arbitrary n, via the comical model structures. In fact, these model structures may be defined in the presence of only faces and degeneracies, but previous proofs of their Quillen equivalence with established simplicial models made use of connections in an essential way. In this talk I will discuss a method by which model structures on cubical sets lacking connections may be shown to be Quillen equivalent to those with connections, using combinatorial techniques which allow the construction of connections in their fibrant objects via open-box filling.