Model category structures \`a la Thomason on 2-Cat

TL;DR

This paper establishes a connection between basic localizers of 2-Cat and Thomason-style model category structures, showing they model localized homotopy theories of simplicial sets.

Contribution

It proves that for each basic localizer of 2-Cat, there exists a corresponding Thomason-style model structure, linking 2-categorical homotopy theories to simplicial set localizations.

Findings

Existence of Thomason-like model structures for all basic localizers of 2-Cat

These structures model combinatorial left Bousfield localizations

Bridges the gap between 2-categorical and simplicial homotopy theories

Abstract

In his paper "Th\'eories homotopiques des 2-cat\'egories", Jonathan Chiche studies homotopy theories on 2-Cat, the category of small strict 2-categories, given by classes of weak equivalences which he calls basic localizers of 2-Cat. These basic localizers of 2-Cat are a 2-categorical generalization of the notion of a basic localizer introduced by Grothendieck in "Pursuing stacks". In this paper, we deduce from the results of Jonathan Chiche and results we have obtained with Georges Maltsiniotis that for essentially every basic localizer W of 2-Cat, there exists a model category structure \`a la Thomason on 2-Cat whose weak equivalences are given by W. We show that these model category structures model exactly combinatorial left Bousfield localization of the classical homotopy theory of simplicial sets.