#### 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

and

Department of Mathematics, Wayne State
University, Detroit, MI 48202, USA

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

AGT home page

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