  This paper attempts to demystify the notion of the smashing spectrum of a
presentably symmetric monoidal stable $\infty$-category, which is a locale
whose opens correspond to smashing localizations. It has been studied in
tensor-triangular geometry in the compactly generated rigid case. Our main
result states that the smashing spectrum functor is right adjoint to the
spectral sheaves functor; it in particular gives an external definition using
neither objects, ideals, nor localizations. This sheaves-spectrum adjunction
informally means that the smashing spectrum gives the best approximation of a
given $\infty$-category by $\infty$-categories of sheaves. We in fact prove the
unstable version of this result by finding a correct unstable extension of
smashing spectrum, which instead parameterizes smashing colocalizations.
  As an application, we give a categorical presentation of Clausen--Scholze's
categorified locales.
