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Macquarie University  Department of Mathematics

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Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

July 18-21 2005, Australian National University, Canberra

Mon, 18 July:  11:00 - 11:40

Localizations in stable homotopy theory
Rosický, Jiri (Masaryk University)

\thanks{Joint work with C. Casacuberta and J. J. Gutiérrez.}
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Localizing subcategories are very important in stable homotopy theory and the authors of \bibref{2} mention that they do not know any example of a localizing subcategory $\mathcal L$ without a localization functor, which is the same thing as $\mathcal L$ not being coreflective. We will justify their suspicion that the answer may depend on set theory by showing that, assuming Vop\v enka's principle, every localizing subcategory $\mathcal L$ of the homotopy category $\mathcal S$ of spectra is coreflective. Moreover, $\mathcal L$ is generated by a single object and, dually, every colocalizing subcategory $\mathcal C$ of $\mathcal S$ is reflective and generated by a single object. The consequence is that every localizing subcategory of $\mathcal S$ is a cohomological Bousfield class.

Our results are true for monogenic stable homotopy categories $\mathrm{Ho}(\mathcal K)$ (in the sense of \bibref{2}) more general than $\mathcal S$ --- we have to assume at least that $\mathcal K$ is a combinatorial model category. This means that $\mathcal K$ is locally presentable and the model-category structure is cofibrantly generated. Our proofs are based on techniques developed in \bibref{1} and transfered to homotopy theory in \bibref{3}.

\begin{references}
\bibitem J.\ Adámek and J.\ Rosický, {\em Locally Presentable and Accessible Categories}, Cambridge University Press 1994.
\bibitem M.\ Hovey, J.\ H.\ Palmieri and N.\ P.\ Strickland, {\em Axiomatic Stable Homotopy Theory}, Mem.\ Amer.\ Math.\ Soc.\ 610 (1997).
\bibitem C.\ Casacuberta, D.\ Scevenels and J.\ H.\ Smith, {\em Implications of large-cardinal principles in homotopical localizations}, to appear in Adv.\ Math.
\end{references}

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