Homotopy limits of model categories, revisited

TL;DR

This paper revisits the concept of homotopy limits in model categories, clarifying their definition, applications, and the limitations of using right Quillen functors exclusively.

Contribution

It provides a comprehensive review of homotopy limits of model categories and discusses the theoretical constraints of using right Quillen functors.

Findings

Clarifies the definition of homotopy limits in model categories

Summarizes key applications of homotopy limits

Explains why right Quillen functors are preferable in certain contexts

Abstract

The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this paper we review its definition and summarize some of these applications. We conclude with a discussion of why we could work with right Quillen functors instead, but cannot work with a combination of the two.