Very Schwartz coidempotents and continuous spectrum

We introduce the continuous version of the (unstable) smashing spectrum functor. In the stable case, it assigns to each dualizably symmetric monoidal stable presentable $\infty$-category a stably compact space whose open subsets correspond to very Schwartz idempotents -- a certain class of idempotents we define. As an application, we prove Tannaka duality for spectral sheaves on stably compact spaces, including the case of compact Hausdorff spaces.