Title: 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.