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Stone duality between condensed mathematics and algebraic geometry

Rok Gregoric

Abstract

We extend Stone duality to a fully faithful embedding of condensed sets into fpqc sheaves over an arbitrary field, which preserves colimits and finite limits. We study how familiar notions from condensed mathematics/topology and algebraic geometry correspond to each other under this form of Stone duality.

Stone duality between condensed mathematics and algebraic geometry

Abstract

We extend Stone duality to a fully faithful embedding of condensed sets into fpqc sheaves over an arbitrary field, which preserves colimits and finite limits. We study how familiar notions from condensed mathematics/topology and algebraic geometry correspond to each other under this form of Stone duality.
Paper Structure (24 sections, 56 theorems, 139 equations)

This paper contains 24 sections, 56 theorems, 139 equations.

Table of Contents

  1. Preliminaries on Stone algebras
  2. Stone duality
  3. The pearl of an algebra
  4. The pearl and connected components
  5. Stone algebras and the fpqc topology
  6. Stone algebras over fields
  7. Comparison between condensed sets and fpqc sheaves
  8. The setup of condensed sets and fpqc sheaves
  9. The main theorem
  10. The pétra space, enhanced k-points, and Stone core
  11. The enhanced $k$-points of an fpqc sheaf
  12. The pétra space of a condensed set
  13. The Stone core of an fpqc sheaf
  14. Stone duality: condensed sets to algebraic geometry
  15. Global functions and affinization
  16. ...and 9 more sections

Key Result

Theorem A

The equivalence of categories Classic Stone induces an adjunction between condensed sets and fpqc sheaves over $\mathbf F_2$ whose left adjoint constituent preserves all finite limits and is fully faithful.

Theorems & Definitions (164)

  • Theorem A: Stone duality for condensed sets; $\mathbf F_2$-version
  • Theorem B: Theorems \ref{['condensed is geometric']}, \ref{['Shv = QCoh']}, Propositions \ref{['non-algebraic']}, \ref{['Global functions']}, \ref{['Closed prop']}, \ref{['Open prop']}, Corollary \ref{['points are points']}
  • Theorem C: Stone duality for condensed sets; $k$-version -- Theorem \ref{['FF on topoi']}
  • Definition 1.1
  • Theorem 1.2: Stone duality
  • proof
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • ...and 154 more