Duality and Pro-spectra
J. Daniel Christensen and Daniel C. Isaksen
jdc@uwo.ca isaksen@math.wayne.edu
Keywords: Spectrum, pro-spectrum, Spanier-Whitehead duality,
closed model category, colocalization
Arxiv: math.AT/0403451
MSC-class: 55P42 (Primary); 55P25, 18G55, 55U35, 55Q55 (Secondary)
Abstract:
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.