  The finite stable homotopy category S_0 has been suggested as a candidate for
a category of perfect complexes over the monoid scheme Spec F_1. We apply a
reconstruction theorem from algebraic geometry to S_0, and show that one
recovers the one point topological space. We also classify filtering subsets of
the set of principal thick subcategories of S_0, and of its p-local versions.
This is motivated by a result saying that the analogous classification for the
category of perfect complexes over an affine scheme provides topological
information.
