A model-categorical foundation for filtered and stratified spaces and simplicial sets is constructed, with filtered homotopy groups and a filtered Whitehead theorem, while the full filtered Kan-Quillen equivalence remains open.
On bifibrations of model categories
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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.
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2019 1verdicts
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Stratified Homotopy Theory
A model-categorical foundation for filtered and stratified spaces and simplicial sets is constructed, with filtered homotopy groups and a filtered Whitehead theorem, while the full filtered Kan-Quillen equivalence remains open.