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**Homotopy theory and generalized duality for spectral sheaves.**
*(English)*
Zbl 0871.55006

We announce a Verdier-type duality theorem for sheaves of spectra on a topological space \(X\). Along the way we are led to develop the homotopy theory and stable homotopy theory of spectral sheaves. Such theories have been worked out in the past, most notably by K. Brown [Trans. Am. Math. Soc. 186(1973), 419-458 (1974; Zbl 0245.55007)] and S. M. Gersten [Lect. Notes Math. 341, 266-292 (1973; Zbl 0291.18017)], R. W. Thomason [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)], and J. F. Jardine [J. Pure Appl. Algebra 47, 35-87 (1987; Zbl 0624.18007)]. But for our purposes these theories are inappropriate. They also have not been developed as fully as they are here. The previous authors did not have at their disposal the especially good categories of spectra constructed in [A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May, Modern foundations of stable homotopy theory, in ‘Handbook of algebraic topology’, 213-253 (1995; Zbl 0865.55007)], which allow one to do all of homological algebra in the category of spectra. Because we want to work in one of these categories of spectra, we are led to consider sheaves of spaces (as opposed to simplicial sets), and this gives rise to some additional technical difficulties. As an application we compute an example from geometric topology of stratified spaces. In future work we intend to apply our theory to equivariant \(K\)-theory.