On bifibrations of model categories

2017/09/29 by Cagne, Pierre, Melliès, Paul-André
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO)

paper · doi:10.48550/arxiv.1709.10484

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→\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.

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