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

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.