  Cofiltered diagrams of spectra, also called pro-spectra, have arisen in
diverse areas, and to date have been treated in an ad hoc manner. The purpose
of this paper is to systematically develop a homotopy theory of pro-spectra and
to study its relation to the usual homotopy theory of spectra, as a foundation
for future applications. The surprising result we find is that our homotopy
theory of pro-spectra is Quillen equivalent to the opposite of the homotopy
theory of spectra. This provides a convenient duality theory for all spectra,
extending the classical notion of Spanier-Whitehead duality which works well
only for finite spectra. Roughly speaking, the new duality functor takes a
spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite
subcomplexes. In the other direction, the duality functor takes a cofiltered
diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of
the spectra in the diagram. We prove the equivalence of homotopy theories by
showing that both are equivalent to the category of ind-spectra (filtered
diagrams of spectra).
  To construct our new homotopy theories, we prove a general existence theorem
for colocalization model structures generalizing known results for cofibrantly
generated model categories.
