  A number of compactifications familiar in complex-analytic geometry, in
particular, the Baily-Borel compactification and its toroidal variants, as well
as the Deligne-Mumford compactifications, can be covered by open subsets whose
nonempty intersections are Eilenberg-MacLane spaces. We exploit this fact to
describe the (rational) homotopy type of these spaces and the natural maps
between them in terms of the simplicial sets attached to certain categories. We
thus generalize an old result of Charney-Lee on the Baily-Borel
compactification of A_g and recover (and rephrase) a more recent one of
Ebert-Giansiracusa on the Deligne-Mumford compactifications. We also describe
an extension of the period map for Riemann surfaces (going from the
Deligne-Mumford compactification to the Baily-Borel compactification of the
moduli space of principally polarized varieties) in these terms.
