Cubical models of $(\infty, 1)$-categories
Algebraic Topology
2022-02-08 v3 Category Theory
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 DK-equivalences.
Cite
@article{arxiv.2005.04853,
title = {Cubical models of $(\infty, 1)$-categories},
author = {Brandon Doherty and Chris Kapulkin and Zachery Lindsey and Christian Sattler},
journal= {arXiv preprint arXiv:2005.04853},
year = {2022}
}
Comments
109 pages; to appear in Mem. Amer. Math. Soc