On bifibrations of model categories

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.