Moduli stack of stable curves from a stratified homotopy viewpoint

In 1984, Charney and Lee defined a category of stable curves and exhibited a rational homology equivalence from its geometric realisation to (the analytification of) the moduli stack of stable curves, also known as the Deligne-Mumford-Knudsen compactification. We strengthen this result by showing that, in fact, this category captures the stratified homotopy type of the moduli stack. In particular, it classifies constructible sheaves via an exodromy equivalence.