Algebraic & Geometric Topology

Duality and Pro-Spectra

J Daniel Christensen and Daniel C Isaksen

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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.

Article information

Algebr. Geom. Topol., Volume 4, Number 2 (2004), 781-812.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra
Secondary: 55P25: Spanier-Whitehead duality 18G55: Homotopical algebra 55U35: Abstract and axiomatic homotopy theory 55Q55: Cohomotopy groups

spectrum pro-spectrum Spanier-Whitehead duality closed model category colocalization


Christensen, J Daniel; Isaksen, Daniel C. Duality and Pro-Spectra. Algebr. Geom. Topol. 4 (2004), no. 2, 781--812. doi:10.2140/agt.2004.4.781.

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