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.

中文翻译：

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关于模型类别的双纤维

在本文中，我们开发了 Quillen 双纤化的概念，它结合了 Grothendieck 双纤化和 Quillen 模型结构的两个概念。特别地，给定双纤维 $p:\mathcal E\to\mathcal B$，我们描述了纤维 $\mathcal E_A$ 上的模型结构族和基于类别 $\mathcal B$ 组合成模型结构的时间在总范畴 $\mathcal E$ 上，使得函子 $p$ 保留了共纤化、纤维化和弱等价。使用这种 Grothendieck 构造模型结构，我们重新审视 Reedy 模型结构的传统定义和可能的概括，并展示它们的双纤维性质。