Stellar stratifications on classifying spaces

Abstract

We extend Björner’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 Nanda et al (Discrete Morse theory and classifying spaces, arXiv:1612.08429 [15]). As another application, we show that the exit-path category Exit(X), in the sense of Lurie (Higher algebra, http://www.math.harvard.edu/~lurie/papers/HA.pdf [11]), of a finite cylindrically normal CW stellar complex X is a quasi-category.