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The condensed homotopy type of a scheme

Peter J. Haine, Tim Holzschuh, Marcin Lara, Catrin Mair, Louis Martini, Sebastian Wolf with an appendix by Bogdan Zavyalov

TL;DR

This work constructs and analyzes a condensed refinement of the étale homotopy type that unifies Friedlander–Artin–Mazur’s étale type with Bhatt–Scholze’s proétale fundamental group. It develops three equivalent formulations of the condensed homotopy type, proves descent and fiber-sequence properties, and computes explicit examples, notably rings of continuous functions. In the second part, it studies the condensed fundamental group, showing that π1cond(A¹ℂ) is nontrivial while its Noohi completion recovers the proétale π1, and that the quasiseparated quotient often behaves as a topological group with van Kampen and Künneth-type results. The results connect condensed exodromy to classical invariants and establish robust descent and base-change behavior, enabling practical computations and potential extensions to condensed coefficient rings.

Abstract

We study a condensed version of the étale homotopy type of a scheme, which refines both the usual étale homotopy type of Friedlander-Artin-Mazur and the proétale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a description of the condensed homotopy type using the Galois category of a scheme introduced by Barwick-Glasman-Haine. In the second part, we focus on the fundamental group of the condensed homotopy type in more detail. We show that, unexpectedly, the fundamental group of the condensed homotopy type of the affine line $\mathbf{A}^1_{\mathbf{C}}$ over the complex numbers is nontrivial. Nonetheless, its Noohi completion recovers the proétale fundamental group of Bhatt-Scholze. Moreover, we show that a mild correction, passing to the quasiseparated quotient, fixes most of this group's quirks. Surprisingly, this quotient is often a topological group.

The condensed homotopy type of a scheme

TL;DR

This work constructs and analyzes a condensed refinement of the étale homotopy type that unifies Friedlander–Artin–Mazur’s étale type with Bhatt–Scholze’s proétale fundamental group. It develops three equivalent formulations of the condensed homotopy type, proves descent and fiber-sequence properties, and computes explicit examples, notably rings of continuous functions. In the second part, it studies the condensed fundamental group, showing that π1cond(A¹ℂ) is nontrivial while its Noohi completion recovers the proétale π1, and that the quasiseparated quotient often behaves as a topological group with van Kampen and Künneth-type results. The results connect condensed exodromy to classical invariants and establish robust descent and base-change behavior, enabling practical computations and potential extensions to condensed coefficient rings.

Abstract

We study a condensed version of the étale homotopy type of a scheme, which refines both the usual étale homotopy type of Friedlander-Artin-Mazur and the proétale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a description of the condensed homotopy type using the Galois category of a scheme introduced by Barwick-Glasman-Haine. In the second part, we focus on the fundamental group of the condensed homotopy type in more detail. We show that, unexpectedly, the fundamental group of the condensed homotopy type of the affine line over the complex numbers is nontrivial. Nonetheless, its Noohi completion recovers the proétale fundamental group of Bhatt-Scholze. Moreover, we show that a mild correction, passing to the quasiseparated quotient, fixes most of this group's quirks. Surprisingly, this quotient is often a topological group.

Paper Structure

This paper contains 49 sections, 122 theorems, 329 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and overview
  3. Results about the condensed homotopy type
  4. Results about the condensed fundamental group
  5. Related work
  6. Linear overview
  7. Conventions
  8. Acknowledgments
  9. Preliminaries
  10. Recollection on condensed anima
  11. Pro-objects and completions
  12. Condensed ∞-categories
  13. Recollection on shape theory
  14. Recollection on proétale sheaves
  15. Basis of weakly contractible objects
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $f \colon \fromto{X}{S}$ be a morphism between qcqs schemes, and let $\fromto{\sbar}{S}$ be a geometric point of $S$. If $\dim(S) = 0$, then the naturally null sequence \begin{tikzcd}[sep=1.5em] \Picond(\Xsbar) \arrow[r] & \Picond(X) \arrow[r] & \Picond(S) \end{tikzcd}is a fiber sequence i

Figures (1)

  • Figure 1: Constructing the scheme $X^+$.

Theorems & Definitions (318)

  • Theorem 1.1: (fundamental fiber sequence, \ref{['cor:proetale_fundamental_fiber_sequence']})
  • Theorem 1.2: (\ref{['thm:smooth_fiber_sequ']})
  • Theorem 1.3: (integral hyperdescent, \ref{['cor:integral-hyperdescent-main-result']})
  • Theorem 1.4: (\ref{['cor:condensed_homotopy_type_of_rings_of_continuous_functions']})
  • Remark 1.5
  • Theorem 1.6: (\ref{['thm:description_of_pi_0', 'cor:pi0s_match_for_finitely_many_irr_comps']})
  • Remark 1.7: (see \ref{['ex:adic_disk']})
  • Theorem 1.8: (\ref{['cor:pionecond-of-A1-is-nontrivial']})
  • Theorem 1.9: (\ref{['thm:recovering_BS_fundamental_group']})
  • Theorem 1.10: (\ref{['thm:normal-implies-profinite']})
  • ...and 308 more