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Derived Stone Embedding

Amos Kaminski

TL;DR

The paper extends the classical Stone embedding to a derived, higher-categorical setting by embedding the pro-category of $ au$-finite spaces into the category of pyknotic spaces via $ extbf{Pyk}( extbf{S})$. Utilizing $ extinfty$-topos machinery, it develops a Postnikov-like framework in which homotopy data are organized as sheaves and Eilenberg–MacLane layers, and analyzes finiteness via coherent and localic topoi, solidification, and group actions. The main contribution is a partial, precise characterization of the essential image of the derived Stone embedding: a connected, truncated pyknotic space lies in the image if and only if its pyknotic homotopy groups are profinite, with key lemmas ensuring the gluing of profinite Eilenberg–MacLane pieces. This advances profinite homotopy theory in the derived setting, offering a robust topos-theoretic framework and a concrete criterion for recognizing objects arising from pro-$ au$-finite spaces within $ extbf{Pyk}( extbf{S})$.

Abstract

A classical result, the Stone embedding, characterizes profinite sets as totally disconnected, compact Hausdorff spaces. Building on "Pyknotic objects, I. Basic notions", which introduced a derived Stone embedding of the pro-category of $π$-finite spaces into pyknotic spaces, this paper uses the $\infty$-topoi machinery to partially characterize the essential image of this embedding, extending the classical characterization to the derived setting.

Derived Stone Embedding

TL;DR

The paper extends the classical Stone embedding to a derived, higher-categorical setting by embedding the pro-category of -finite spaces into the category of pyknotic spaces via . Utilizing -topos machinery, it develops a Postnikov-like framework in which homotopy data are organized as sheaves and Eilenberg–MacLane layers, and analyzes finiteness via coherent and localic topoi, solidification, and group actions. The main contribution is a partial, precise characterization of the essential image of the derived Stone embedding: a connected, truncated pyknotic space lies in the image if and only if its pyknotic homotopy groups are profinite, with key lemmas ensuring the gluing of profinite Eilenberg–MacLane pieces. This advances profinite homotopy theory in the derived setting, offering a robust topos-theoretic framework and a concrete criterion for recognizing objects arising from pro--finite spaces within .

Abstract

A classical result, the Stone embedding, characterizes profinite sets as totally disconnected, compact Hausdorff spaces. Building on "Pyknotic objects, I. Basic notions", which introduced a derived Stone embedding of the pro-category of -finite spaces into pyknotic spaces, this paper uses the -topoi machinery to partially characterize the essential image of this embedding, extending the classical characterization to the derived setting.

Paper Structure

This paper contains 17 sections, 39 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. $\infty$-topoi
  3. Homotopy Groups
  4. Powering over $\infty$-groupoids
  5. Homotopy Groups
  6. Finiteness Properties
  7. Coherent Topoi
  8. Bounded And Localic Topoi
  9. Group Actions
  10. Pyknotic Spaces
  11. Introduction
  12. Basic Concepts
  13. Solidification and the Derived Stone Embedding
  14. Coherent Objects in $\textbf{Pyk}(\mathcal{S})$
  15. Characterization Of The Essential Image
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X$ be a simply connected space with finitely generated homotopy groups. Then $X \simeq X_{\mathbb{Q}} \times_{\left(\prod_p \widehat{X}_p\right)_{\mathbb{Q}}} \prod_p \widehat{X}_p.$

Theorems & Definitions (67)

  • Theorem 1.1: lurie_mit
  • Theorem 1.2: stonem
  • Theorem 1.3: pyknotic
  • Theorem 1.4: Partial Characterization of the Essential Image -- Theorem 4.6
  • Corollary 1.5: Corollary 3.12
  • Lemma 1.6: Lemma 4.9
  • Lemma 1.7: Lemma 2.15
  • Definition 2.1: Powering over $\infty$-groupoids
  • Definition 2.2: Final Geometric Morphism
  • Definition 2.3: Locally Constant Stacks
  • ...and 57 more