The sheaves--spectrum adjunction

TL;DR

This paper clarifies the concept of the smashing spectrum in stable monoidal $$-categories, showing it as a right adjoint to a sheaves functor and extending the idea to unstable contexts with applications to categorified locales.

Contribution

It formally establishes the sheaves--spectrum adjunction in the $$-category setting and introduces an unstable analogue, broadening the theoretical framework.

Findings

Identifies the smashing spectrum as a right adjoint to the sheaves functor.

Provides an external characterization avoiding explicit objects or ideals.

Offers a categorical presentation of Clausen--Scholze's categorified locales.

Abstract

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.