  We define compactifications of vector spaces which are functorial with
respect to certain linear maps. These "many-body" compactifications are
manifolds with corners, and the linear maps lift to b-maps in the sense of
Melrose. We derive a simple criterion under which the lifted maps are in fact
b-fibrations, and identify how these restrict to boundary hypersurfaces. This
theory is an application of a general result on the iterated blow-up of cleanly
intersecting submanifolds which extends related results in the literature.
