  The notion of cellular stratified spaces was introduced in a joint work of
the author with Basabe, Gonz{\'a}lez, and Rudyak with the aim of constructing a
cellular model of the configuration space of a sphere. Although the original
aim was not achieved in the project, the notion of cellular stratified spaces
turns out to be useful, at least, in the study of configuration spaces of
graphs. In particular, the notion of totally normal cellular stratified spaces
was used successfully in a joint work with the former students of the author
arXiv:1312.7368 to study the homotopy type of configuration spaces of graphs
with a small number of vertices. Roughly speaking, totally normal cellular
stratified spaces correspond to acyclic categories in the same way regular cell
complexes correspond to posets. In this paper, we extend this correspondence by
replacing cells by stellar cells and acyclic categories by topological acyclic
categories.
