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Very Schwartz coidempotents and continuous spectrum

Ko Aoki

TL;DR

The paper develops a continuous analogue of the smashing spectrum, connecting it to Tannaka duality for spectral sheaves on stably compact spaces and recovering classical results in the compact Hausdorff case. It introduces a robust framework of idealoids and restricted Ind-objects to control Ind-constructions, and defines Schwartz and very Schwartz objects to study continuity and nuclearity in presentable ∞-categories. Central constructions include the continuous spectrum $\operatorname{Sm}^{\mathrm{con}}$ as the right adjoint to $\operatorname{Shv}(-;\mathsf{Sp})$ and the classification of open subsets by very Schwartz coidempotents, yielding a Tannaka-type fully faithful correspondence for shuttling between spaces and their categories of sheaves. The paper also develops the rigid spectrum $\operatorname{Sm}^{\mathrm{rig}}$, analyzes trace-class morphisms and very nuclear idempotents, and proves concrete computations in qcqs schemes and for spectral sheaves on stably compact spaces. Overall, it provides new dualities and computations that connect categorical, topological, and sheaf-theoretic perspectives, with notable implications for non-noetherian and noncompact settings.

Abstract

We introduce the continuous version of the (unstable) smashing spectrum functor. In the stable case, it assigns to each dualizably symmetric monoidal stable presentable $\infty$-category a stably compact space whose open subsets correspond to very Schwartz idempotents -- a certain class of idempotents we define. As an application, we prove Tannaka duality for spectral sheaves on stably compact spaces, including the case of compact Hausdorff spaces.

Very Schwartz coidempotents and continuous spectrum

TL;DR

The paper develops a continuous analogue of the smashing spectrum, connecting it to Tannaka duality for spectral sheaves on stably compact spaces and recovering classical results in the compact Hausdorff case. It introduces a robust framework of idealoids and restricted Ind-objects to control Ind-constructions, and defines Schwartz and very Schwartz objects to study continuity and nuclearity in presentable ∞-categories. Central constructions include the continuous spectrum as the right adjoint to and the classification of open subsets by very Schwartz coidempotents, yielding a Tannaka-type fully faithful correspondence for shuttling between spaces and their categories of sheaves. The paper also develops the rigid spectrum , analyzes trace-class morphisms and very nuclear idempotents, and proves concrete computations in qcqs schemes and for spectral sheaves on stably compact spaces. Overall, it provides new dualities and computations that connect categorical, topological, and sheaf-theoretic perspectives, with notable implications for non-noetherian and noncompact settings.

Abstract

We introduce the continuous version of the (unstable) smashing spectrum functor. In the stable case, it assigns to each dualizably symmetric monoidal stable presentable -category a stably compact space whose open subsets correspond to very Schwartz idempotents -- a certain class of idempotents we define. As an application, we prove Tannaka duality for spectral sheaves on stably compact spaces, including the case of compact Hausdorff spaces.
Paper Structure (30 sections, 74 theorems, 44 equations, 1 figure, 2 tables)

This paper contains 30 sections, 74 theorems, 44 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Tannaka duality for compact Hausdorff spaces
  3. Stably compact spaces and continuous spectrum
  4. Idealoids and restricted Ind-objects
  5. Idealoids
  6. Restricted Ind-objects
  7. Compact morphisms and continuous categories
  8. Compact morphisms
  9. (Very) Schwartz Ind-objects
  10. Continuous categories
  11. Tensor products of continuous categories
  12. Stable continuous categories
  13. Stably compact spaces and sheaves thereon
  14. Stably compact spaces
  15. Operations on stably compact spaces
  16. ...and 15 more sections

Key Result

Theorem A

We have an adjunction \begin{tikzcd}[column sep=huge] \Cat{CH}^{\op}\ar[r,shift left,"\Shv(\X;\Cat{Sp})"]& \CAlg(\Cat{Pr}_{\st})_{\rig} \subset \CAlg(\Cat{Pr}_{\st}) \rlap,\ar[l,shift left,"\Sm^{\rig}"] \end{tikzcd}where the right adjoint $\operatorname{Sm}^{\textno

Figures (1)

  • Figure 1: The class of stably compact spaces contains both the classes of spectral spaces and compact Hausdorff spaces. The intersection of these two is the class of Stone spaces.

Theorems & Definitions (251)

  • Theorem A
  • Theorem B
  • Remark 1.3
  • Theorem C
  • Remark 1.4
  • Theorem D
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • ...and 241 more