Groupoidal 2-quasi-categories and homotopy 2-types

TL;DR

This paper introduces groupoidal 2-quasi-categories, establishes their model structure, and demonstrates their equivalence to simplicial sets, providing a new framework for modeling homotopy 2-types.

Contribution

It defines groupoidal 2-quasi-categories, constructs their model structure, and proves their Quillen equivalence to simplicial sets, linking higher category theory with homotopy theory.

Findings

Groupoidal 2-quasi-categories form the fibrant objects in a new model structure.

The model category of these quasi-categories is Quillen equivalent to simplicial sets.

2-truncated groupoidal 2-quasi-categories model homotopy 2-types.

Abstract

We define a notion of groupoidal 2-quasi-categories and show that they are the fibrant objects of a model structure on the category of $\Theta_2$-sets. We show that this model category is Quillen equivalent to the Kan-Quillen model category of simplicial sets and that 2-truncated groupoidal 2-quasi-categories are models for homotopy 2-types.