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2004 Duality and Pro-Spectra
J Daniel Christensen, Daniel C Isaksen
Algebr. Geom. Topol. 4(2): 781-812 (2004). DOI: 10.2140/agt.2004.4.781

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.

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J Daniel Christensen. Daniel C Isaksen. "Duality and Pro-Spectra." Algebr. Geom. Topol. 4 (2) 781 - 812, 2004. https://doi.org/10.2140/agt.2004.4.781

Information

Received: 7 August 2004; Accepted: 31 August 2004; Published: 2004
First available in Project Euclid: 21 December 2017

zbMATH: 1054.55010
MathSciNet: MR2100680
Digital Object Identifier: 10.2140/agt.2004.4.781

Subjects:
Primary: 55P42
Secondary: 18G55, 55P25, 55Q55, 55U35

Rights: Copyright © 2004 Mathematical Sciences Publishers

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