The smashing spectrum of sheaves

TL;DR

This paper classifies smashing localizations in the $mbda$-category of sheaves valued in derived vector spaces for any $mbda$-topos, revealing new structural insights and a Tannaka-type reconstruction.

Contribution

It provides the first example of a nonzero presentably symmetric monoidal stable $mbda$-category with a smashing spectrum that has no points, using Boolean covers.

Findings

Classifies all smashing localizations as restrictions to closed subtoposes.

Establishes the existence of a Boolean cover for the proof.

Derives a Tannaka-type categorical reconstruction for locales.

Abstract

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.