Homotopy Theory and Generalized Duality for Spectral Sheaves

In this paper 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 [Br], [BG], [T], and [J]. 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 [EKM], 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.