Algebraic and Geometric Topology 4 (2004), paper no. 34, pages 781-812.

Duality and Pro-Spectra

J. Daniel Christensen, Daniel C. Isaksen


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.

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

AMS subject classification. Primary: 55P42. Secondary: 55P25, 18G55, 55U35, 55Q55.

DOI: 10.2140/agt.2004.4.781

E-print: arXiv:math.AT/0403451

Submitted: 7 August 2004. Accepted: 31 August 2004. Published: 23 September 2004.

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J. Daniel Christensen, Daniel C. Isaksen

Department of Mathematics, University of Western Ontario, London, Ontario, Canada
and
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Email: jdc@uwo.ca, isaksen@math.wayne.edu

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