Duality and Pro-Spectra

Christensen, J Daniel¹ ; Isaksen, Daniel C²

¹Dept of Math, University of Western Ontario, London, Ontario, Canada
²Department of Mathematics, Wayne State University, Detroit MI 48202, USA

Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 781-812

Résumé

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.

Détail

DOI : 10.2140/agt.2004.4.781

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

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