A Quillen model structure for Gray-categories

TL;DR

This paper establishes a Quillen model structure on Gray-categories where weak equivalences are triequivalences, and demonstrates Gray-groupoids model homotopy 3-types, linking higher category theory with homotopy theory.

Contribution

It introduces a new model structure on Gray-categories and Gray-groupoids, providing a functorial proof that Gray-groupoids model homotopy 3-types, and conjectures an equivalence with tricategories.

Findings

Gray-groupoids model homotopy 3-types

The model structure restricts to Gray-groupoids

Conjecture of Quillen equivalence with tricategories

Abstract

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.