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The sheaves--spectrum adjunction

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arxiv 2302.04069 v2 pith:U4V4FHRZ submitted 2023-02-08 math.CT math.AGmath.AT

Ko Aoki This is my paper · ORCID
classification math.CTmath.AGmath.AT
keywords smashingspectruminftyadjunctioncategoryfunctorlocalizationsresult
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Smashing, Balmer, Zariski spectra: an ideal approach

    math.AT 2026-07 accept novelty 7.0 of 10

    A single frame-theoretic construction, the Zariski frame of radical categorical ideals, unifies Zariski spectra, Balmer spectra, and smashing frames in higher algebra.

  2. Homology manifolds via six functor formalisms

    math.AT 2026-06 unverdicted novelty 7.0 of 10

    Using six functor formalisms, the authors prove that hypercomplete locally compact ANR homology manifolds are cohomologically smooth, that compact ANR homology manifolds are Poincaré duality complexes with Spivak tang...

  3. A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve

    math.AG 2024-12 accept novelty 7.0 of 10

    A 6-functor formalism for Z_p-linear solid quasi-coherent sheaves on small v-stacks is constructed, yielding Poincare duality for pro-etale Q_p-cohomology.

  4. On the Bauer--Furuta construction

    math.AT 2024-12 conditional novelty 7.0 of 10

    Using six-functor sheaf theory, the Bauer-Furuta invariant is defined as the proper pushforward f_* f^!(1), with f^!(1) computed as the Thom spectrum of the family index.

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