Tannaka duality for spectral sheaves on compact Hausdorff spaces

Tannaka duality asks to what extent a geometric object can be recovered from a category naturally associated with it. In this talk, the geometric objects are topological spaces, and the associated categories are their categories of sheaves of spectra.
I showed that, for compact Hausdorff spaces X and Y, every colimit-preserving symmetric monoidal functor from Shv(X; Sp) to Shv(Y; Sp) arises uniquely from a continuous map from Y to X, where Shv(X; Sp) denotes the (∞, 1)-category of sheaves of spectra on X. The main ingredient is an adjunction between the construction of sheaves and a new variant of the smashing spectrum. I will also discuss variants of this machinery to broader classes of spaces, including ongoing work on toposes.