Homotopy fiber products of homotopy theories

TL;DR

This paper investigates the properties of homotopy fiber products of homotopy theories, demonstrating their well-behaved nature when translated into the setting of complete Segal spaces, thus ensuring their correctness.

Contribution

It establishes that homotopy fiber products of homotopy theories are well-behaved in the context of complete Segal spaces, confirming their correctness beyond model categories.

Findings

Homotopy pullbacks are well-behaved in complete Segal spaces.

Homotopy fiber products are consistent across different homotopy theory frameworks.

The notion of homotopy fiber product is validated in a broader setting.

Abstract

Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks.