  The assignment of classifying spectra to saturated fusion systems was
suggested by Linckelmann and Webb and has been carried out by Broto, Levi and
Oliver. A more rigid (but equivalent) construction of the classifying spectra
is given in this paper. It is shown that the assignment is functorial for
fusion-preserving homomorphisms in a way which extends the assignment of stable
p-completed classifying spaces to finite groups, and admits a transfer theory
analogous to that for finite groups. Furthermore the group of homotopy classes
of maps between classifying spectra is described, and in particular it is shown
that a fusion system can be reconstructed from its classifying spectrum
regarded as an object under the stable classifying space of the underlying
p-group.
