A model structure on prederivators for $(\infty,1)$-categories

TL;DR

This paper characterizes which prederivators correspond to $( abla,1)$-categories and establishes a model structure on prederivators that is Quillen equivalent to the Joyal model for quasicategories.

Contribution

It provides a characterization of prederivators arising from quasicategories and constructs a model structure on prederivators that aligns with the Joyal model.

Findings

Prederivators associated with quasicategories are characterized.

A model structure on prederivators is established.

A Quillen equivalence with the Joyal model structure is proven.

Abstract

By theorems of Carlson and Renaudin, the theory of $(\infty,1)$-categories embeds in that of prederivators. The purpose of this paper is to give a two-fold answer to the inverse problem: understanding which prederivators model $(\infty,1)$-categories, either strictly or in a homotopical sense. First, we characterize which prederivators arise on the nose as prederivators associated to quasicategories. Next, we put a model structure on the category of prederivators and strict natural transformations, and prove a Quillen equivalence with the Joyal model structure for quasicategories.