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Fractured Structures in Condensed Mathematics

Nima Rasekh, Qi Zhu

Abstract

We construct a fractured structure, in the sense of Lurie, on the $\infty$-topos of condensed anima. This fractured structure allows us to better comprehend various properties of condensed anima - we use it to exhibit an explicit collection of jointly conservative points for condensed anima. To rule out further candidates for fractured structures, we analyze limits in the category of extremally disconnected spaces. In particular, we show that it does not admit all fibers, answering a question from Clausen.

Fractured Structures in Condensed Mathematics

Abstract

We construct a fractured structure, in the sense of Lurie, on the -topos of condensed anima. This fractured structure allows us to better comprehend various properties of condensed anima - we use it to exhibit an explicit collection of jointly conservative points for condensed anima. To rule out further candidates for fractured structures, we analyze limits in the category of extremally disconnected spaces. In particular, we show that it does not admit all fibers, answering a question from Clausen.
Paper Structure (14 sections, 37 theorems, 34 equations, 1 figure)

This paper contains 14 sections, 37 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. From petit vs. gros to fractured topoi
  3. Condensed anima as a gros topos
  4. Fractured structure on condensed anima
  5. The magical topology of extremally disconnected spaces
  6. Fractured topoi and condensed anima
  7. Fractured topoi
  8. Condensed mathematics
  9. A fractured structure on condensed anima
  10. Constructing the fractured structure
  11. From condensed anima to sheaves of spaces
  12. Further candidates for fractured structures
  13. Compact Hausdorff spaces, profinite sets and (open) embeddings
  14. Extremally disconnected spaces and embeddings

Key Result

Theorem A

The left Kan extension of the restriction functor induces a fractured $\infty$-topos structure.

Figures (1)

  • Figure 1: The Fractured Mind by Rob Crane from https://newirishart.com/artworks/rob-crane-the-fractured-mind-rc0109.htm.

Theorems & Definitions (84)

  • Theorem A: \ref{['thm:main']}
  • Theorem B: \ref{['prop:condopen over yoS']}
  • Theorem C: \ref{['thm:formula enough points']}
  • Theorem D: \ref{['thm:no fibers']}
  • Definition 2.1: lurie2018sag
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 74 more