Homotopy $n$-types of cubical sets and graphs

TL;DR

This paper develops a new model structure for homotopy n-types on cubical sets and graphs, extending existing simplicial set models to cubical contexts and providing a fibration category framework for graphs.

Contribution

It introduces a novel construction of the homotopy n-type model structure on cubical sets and applies it to graphs, establishing a new categorical framework.

Findings

Model structure on cubical sets for homotopy n-types

Fibration category structure on graphs with discrete n-equivalences

Extension of simplicial set models to cubical sets and graphs

Abstract

We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an analogous model structure can be constructed on the category of cubical sets, and use it to equip the category of (simple) graphs with a fibration category structure whose weak equivalences are discrete $n$-equivalences.