Homotopy limits of model categories and more general homotopy theories

TL;DR

This paper generalizes the concept of homotopy fiber products to define homotopy limits of diagrams of model categories, establishing their equivalence to the usual homotopy limits within a broader homotopy theory framework.

Contribution

It introduces a new definition of homotopy limits for diagrams of left Quillen functors, extending previous work on homotopy fiber products to a more general setting.

Findings

Homotopy limits correspond to the usual homotopy limits in a generalized model.

The paper extends the concept of homotopy fiber products to broader homotopy limits.

Provides a unified framework for understanding homotopy limits in model categories.

Abstract

Generalizing a definition of homotopy fiber products of model categories, we give a definition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homotopy fiber products, we prove that such a homotopy limit does in fact correspond to the usual homotopy limit, when we work in a more general model for homotopy theories in which they can be regarded as objects of a model category.