The smashing spectrum of sheaves

For an arbitrary $\infty$-topos, we classify the smashing localizations in the $\infty$-category of sheaves valued in derived vector spaces: Any of them is the restriction functor to a (unique) closed subtopos. Our proof is based on the existence of a Boolean cover. This result in particular gives us the first example of a nonzero presentably symmetric monoidal stable $\infty$-category whose smashing spectrum has no points. Combining this with the sheaves-spectrum adjunction, we obtain a Tannaka-type categorical reconstruction result for locales.