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The sheaves--spectrum adjunction

Ko Aoki

TL;DR

The paper establishes a canonical, adjoint relationship between spectral sheaves and the unstable smashing spectrum, providing a conceptual origin for the smashing spectrum in terms of idempotent coalgebras (smashing colocalizations) and extending the construction to an unstable setting. It shows that the functor Shv: Loc^op → CAlg(Pr) has a right adjoint Sm, with Sm(C) described pointwise by the locale of idempotent coalgebras, and it develops a lattice-theoretic framework that mirrors frame theory. It also develops a Tannaka-style viewpoint for categorified locales and proves a Künneth-type formula for Shv, along with an application to categorical presentations of Clausen–Scholze's categorified locales. The results unify and generalize the notion of smashing spectra across stable and unstable contexts, and provide a robust categorical toolkit for understanding locales, sheaves of spectra, and their interplay in condensed mathematics. Practically, this yields structural insights and reconstruction principles for categorified locales and clarifies when and how smashing localizations control the ambient monoidal/categorical landscape.

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.

The sheaves--spectrum adjunction

TL;DR

The paper establishes a canonical, adjoint relationship between spectral sheaves and the unstable smashing spectrum, providing a conceptual origin for the smashing spectrum in terms of idempotent coalgebras (smashing colocalizations) and extending the construction to an unstable setting. It shows that the functor Shv: Loc^op → CAlg(Pr) has a right adjoint Sm, with Sm(C) described pointwise by the locale of idempotent coalgebras, and it develops a lattice-theoretic framework that mirrors frame theory. It also develops a Tannaka-style viewpoint for categorified locales and proves a Künneth-type formula for Shv, along with an application to categorical presentations of Clausen–Scholze's categorified locales. The results unify and generalize the notion of smashing spectra across stable and unstable contexts, and provide a robust categorical toolkit for understanding locales, sheaves of spectra, and their interplay in condensed mathematics. Practically, this yields structural insights and reconstruction principles for categorified locales and clarifies when and how smashing localizations control the ambient monoidal/categorical landscape.

Abstract

This paper demystifies the notion of the smashing spectrum of a stable presentably symmetric monoidal -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 -category by -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.
Paper Structure (15 sections, 28 theorems, 25 equations)

This paper contains 15 sections, 28 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Smashing spectrum and categorified locales
  3. Results
  4. Organization of this paper
  5. (Co)idempotent algebras and smashing (co)localizations
  6. Idempotent objects
  7. Coidempotent objects
  8. The stable case
  9. The unstable smashing spectrum
  10. Lattice-theoretic properties of cIdem
  11. Smallness
  12. Sheaves on a locale
  13. The sheaves--spectrum adjunction
  14. Categorified locales
  15. A categorical presentation

Key Result

Theorem A

The spectral sheaves functor from the opposite of the category of locales to the $\infty$-category of stable presentably symmetric monoidal $\infty$-categories has a right adjoint, which is given by restricting the smashing spectrum functor e:sm.

Theorems & Definitions (82)

  • Remark 1.2
  • Theorem A
  • Remark 1.5
  • Theorem B
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Corollary 1.10
  • Remark 1.11
  • Theorem C
  • ...and 72 more