Stellar Stratifications on Classifying Spaces

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 $\mathrm{E}\mathrm{x}\mathrm{i}\mathrm{t}(X)$ of a finite cylindrically normal CW stellar complex $X$ is a quasicategory.