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.
Stellar Stratifications on Classifying Spaces
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abstract
We extend Bj\"orner's characterization of the face poset of finite CW complexes to a certain class of stratified spaces, called cylindrically normal stellar complexes. As a direct consequence, we obtain a discrete analogue of cell decompositions in smooth Morse theory, by using the classifying space model introduced in arXiv:1612.08429. As another application, we show that the exit-path simplicial set $\mathrm{Exit}(X)$ of a finite cylindrically normal CW stellar complex $X$ is a quasicategory.
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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.