[Paper Review] Cubical models of $(\infty, 1)$-categories

Brandon Doherty, Chris Kapulkin+2 more|arXiv (Cornell University)|May 11, 2020

Homotopy and Cohomology in Algebraic Topology11 references4 citations

TL;DR

This paper constructs a model structure on cubical sets with connections, where cofibrations are monomorphisms and fibrant objects are defined by filling inner open boxes—cubical analogues of inner horns. It establishes a Quillen equivalence between this cubical model structure and the Joyal model structure on simplicial sets via the triangulation functor, and shows that weak equivalences between fibrant cubical objects are precisely Dwyer-Kan equivalences, enabling a well-behaved notion of mapping space.

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.

Motivation & Objective

• To develop a cubical model structure for (∞,1)-categories analogous to the Joyal model structure on simplicial sets.
• To define fibrant objects in cubical sets via the right lifting property with respect to inner open boxes, the cubical analogue of inner horns.
• To establish a Quillen equivalence between the new cubical model structure and the Joyal model structure on simplicial sets.
• To provide a natural notion of mapping space in cubical quasicategories, enabling characterization of weak equivalences as DK-equivalences.
• To demonstrate that the triangulation functor induces a Quillen equivalence using the straightening-over-the-point functor as a key technical tool.

Proposed method

• Construct a model structure on marked cubical sets, where edges are marked to represent equivalences, using Jeff Smith’s theorem.
• Use the minimal marking functor to left-induce a model structure on cubical sets from the marked model structure.
• Define inner open boxes as the cubical analogues of inner horns, capturing the correct filling conditions for fibrant objects.
• Introduce a cubical theory of cones to relate simplicial and cubical shapes, generalizing the straightening-over-the-point construction.
• Establish Quillen equivalence via the straightening functor Q: sSet → cSet, showing it is a Quillen equivalence and that its derived functors are inverses of those of the triangulation functor T.
• Use the presence of connections in the box category to ensure compatibility between cubical and simplicial mapping spaces.

Experimental results

Research questions

• RQ1 Can a model structure on cubical sets be constructed such that fibrant objects are characterized by filling inner open boxes, analogous to inner horns in simplicial sets?
• RQ2 Is this cubical model structure Quillen equivalent to the Joyal model structure on simplicial sets?
• RQ3 Does the cubical model admit a well-defined notion of a mapping space that characterizes weak equivalences between fibrant objects as DK-equivalences?
• RQ4 Can the triangulation functor serve as a Quillen equivalence between the cubical and simplicial models of (∞,1)-categories?
• RQ5 What role do connections in the box category play in ensuring compatibility between cubical and simplicial homotopy theories?

Key findings

• A model structure on cubical sets with connections is constructed where cofibrations are monomorphisms and fibrant objects are defined by the right lifting property with respect to inner open boxes.
• The model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor T: cSet → sSet.
• Weak equivalences between fibrant cubical objects are precisely DK-equivalences, confirming the correctness of the homotopy theory.
• The triangulation functor T is a Quillen equivalence, established indirectly by proving that the straightening-over-the-point functor Q: sSet → cSet is a Quillen equivalence with inverse derived functors.
• The cubical theory of cones provides a general framework for relating simplicial and cubical shapes, with potential applications beyond this work.
• The construction works for all four variants of the box category (with no connections, one connection, or both), though the full comparison between cubical and simplicial mapping spaces requires both connections.