We show that cellular approximations of nilpotent Postnikov stages are always
nilpotent Postnikov stages, in particular classifying spaces of nilpotent
groups are turned into classifying spaces of nilpotent groups. We use a
modified Bousfield-Kan homology completion tower z_k X whose terms we prove are
all X-cellular for any X. As straightforward consequences, we show that if X is
K-acyclic and nilpotent for a given homology theory K, then so are all its
Postnikov sections, and that any nilpotent space for which the space of pointed
self-maps map_*(X,X) is "canonically" discrete must be aspherical.Comment: 19 page
