  An old theorem of Charney and Lee says that the classifying space of the
category of stable nodal topological surfaces and isotopy classes of
degenerations has the same rational homology as the Deligne-Mumford
compactification. We give an integral refinement: the classifying space of the
Charney-Lee category actually has the same homotopy type as the moduli stack of
stable curves, and the etale homotopy type of the moduli stack is equivalent to
the profinite completion of the classifying space of the Charney-Lee category.
