On bifibrations of model categories

TL;DR

This paper introduces Quillen bifibrations, combining Grothendieck bifibrations with model structures, and explores their applications to Reedy model structures and their generalizations.

Contribution

It develops a framework for Quillen bifibrations that unify bifibration concepts with model category theory, enabling new insights into Reedy structures.

Findings

Characterization of when a family of model structures forms a model structure on the total category

Revisiting Reedy model structures through the lens of bifibrations

Potential generalizations of Reedy model structures

Abstract

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family of model structures on the fibers $\mathcal E_A$ and on the basis category $\mathcal B$ combines into a model structure on the total category $\mathcal E$, such that the functor $p$ preserves cofibrations, fibrations and weak equivalences. Using this Grothendieck construction for model structures, we revisit the traditional definition of Reedy model structures, and possible generalizations, and exhibit their bifibrational nature.