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Classifying anima of condensed $\infty$-categories of points

Peter J. Haine

Abstract

We compare the classifying anima of two natural condensed $\infty$-categories associated to a coherent $\infty$-topos. One from our work with Barwick and Glasman on exit-path categories in algebraic geometry, and the other from Lurie's work on ultracategories. The key consequence of our comparison is a connection between algebraic geometry and model theory: up to a mild completion, the proétale fundamental group of a scheme and the Lascar group of a complete first-order theory are both special cases of the same construction.

Classifying anima of condensed $\infty$-categories of points

Abstract

We compare the classifying anima of two natural condensed -categories associated to a coherent -topos. One from our work with Barwick and Glasman on exit-path categories in algebraic geometry, and the other from Lurie's work on ultracategories. The key consequence of our comparison is a connection between algebraic geometry and model theory: up to a mild completion, the proétale fundamental group of a scheme and the Lascar group of a complete first-order theory are both special cases of the same construction.
Paper Structure (15 sections, 18 theorems, 72 equations)

This paper contains 15 sections, 18 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Linear overview
  3. Notational conventions
  4. Acknowledgments
  5. Around the Łoś ultraproduct theorem
  6. Ultraproducts via sheaf theory
  7. The Łoś ultraproduct theorem & consequences
  8. Classifying anima of condensed ∞-categories of points
  9. Condensed ∞-categories of points
  10. Classifying anima of free colimit completions
  11. The case of spectral ∞-topoi
  12. Complements on ∞-topoi
  13. Bounded and Postnikov complete ∞-topoi
  14. Coherent ∞-topoi and ∞-pretopoi
  15. Spectral ∞-topoi

Key Result

Theorem 3

Let $\Xcal$ be a spectral . Then for each extremally disconnected profinite set $K$, the inclusion admits a left adjoint. As a consequence, the inclusion of condensed $\mathbf{Pt}^{\mathrm{coh}}(\Xcal) \inclusion \mathbf{Pt}(\Xcal)$ induces an equivalence on condensed classifying anima.

Theorems & Definitions (56)

  • Example 1
  • Example 2
  • Theorem 3: (\ref{['thm:extra_adjoint_for_spectral_topoi']})
  • Proposition 1.6
  • proof
  • Proposition 1.7: (fundamental Łoś ultraproduct theorem, SAG)
  • Corollary 1.8: (categorical logic formulation of the Łoś ultraproduct theorem)
  • Remark 1.9: (the usual formulation of the Łoś ultraproduct theorem)
  • Corollary 1.10: (sheaf-theoretic formulation of the Łoś ultraproduct theorem)
  • proof
  • ...and 46 more