Volume 4, issue 2 (2004)

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Duality and Pro-Spectra

J Daniel Christensen and Daniel C Isaksen

Algebraic & Geometric Topology 4 (2004) 781–812

arXiv: math.AT/0403451


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.

spectrum, pro-spectrum, Spanier-Whitehead duality, closed model category, colocalization
Mathematical Subject Classification 2000
Primary: 55P42
Secondary: 55P25, 18G55, 55U35, 55Q55
Forward citations
Received: 7 August 2004
Accepted: 31 August 2004
Published: 23 September 2004
J Daniel Christensen
Dept of Math
University of Western Ontario
Daniel C Isaksen
Department of Mathematics
Wayne State University
Detroit MI 48202