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.
Keywords
spectrum, pro-spectrum, Spanier-Whitehead duality, closed
model category, colocalization
