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.

Notes on file formats

J. Daniel Christensen, Daniel C. Isaksen

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


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