The sheaves--spectrum adjunction

This paper demystifies the notion of the smashing spectrum of a stable presentably symmetric monoidal $\infty$-category, defined as a locale whose opens correspond to smashing localizations. Previously, this concept was studied in tensor-triangular geometry in the compactly generated rigid setting. Our main result identifies the smashing spectrum functor as the right adjoint to the spectral sheaves functor, providing in particular an external characterization that avoids explicit reference to objects, ideals, or localizations. The sheaves--spectrum adjunction formalizes the intuition that the smashing spectrum constitutes the best approximation of a given $\infty$-category by $\infty$-categories of sheaves. We establish an unstable generalization of this result by identifying the correct unstable analog of the smashing spectrum, which parametrizes smashing colocalizations instead. As an application, we give a categorical presentation of Clausen--Scholze's categorified locales.