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arxiv: 2605.03658 · v1 · submitted 2026-05-05 · 🧮 math.NT · math.AG· math.CT· math.FA· math.GN

Lectures on Condensed Mathematics

Peter Scholze This is my paper

Pith reviewed 2026-05-07 13:37 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.CTmath.FAmath.GN
keywords condensed mathematicscondensed setslecturesnumber theoryalgebraic geometrycategory theoryprofinite sets
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The pith

Condensed mathematics supplies a new categorical framework for treating topological spaces in algebraic geometry and number theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper delivers lecture notes from a 2019 course that lay out condensed mathematics, the theory developed jointly with Dustin Clausen. It seeks to replace ordinary topological spaces with objects that behave more like algebraic ones, so that continuity conditions align naturally with the operations common in number theory and geometry. A reader with the right background would see this as a way to make previously awkward limits and completions into straightforward categorical constructions. The notes serve as a stable reference for anyone wanting to apply the framework in arithmetic settings.

Core claim

Condensed mathematics, as presented here, defines condensed sets as sheaves on the site of profinite sets satisfying a gluing condition for surjective maps, thereby providing a replacement for topological spaces that integrates cleanly with the algebraic structures appearing in number theory and algebraic geometry.

What carries the argument

Condensed sets, which act as the basic objects that encode topology while remaining amenable to algebraic operations and limits.

If this is right

  • Continuous maps between spaces become morphisms that respect the algebraic operations without extra continuity checks.
  • Limits and colimits of spaces can be formed inside the category of condensed sets and still carry the expected topology.
  • Sheaf cohomology and other homological constructions on spaces become computable using the same tools as in algebraic geometry.
  • Profinite completions and other arithmetic completions acquire natural interpretations as condensed objects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same replacement of spaces by condensed sets could streamline arguments that currently mix topological and algebraic language in rigid geometry.
  • Extending the framework to include additional sites beyond profinite sets might yield versions adapted to other geometric contexts.
  • If the theory scales to higher categories, it could supply a uniform language for both topological and derived algebraic geometry.

Load-bearing premise

The audience already commands the category theory, algebraic geometry, and number theory needed to follow the definitions and proofs.

What would settle it

A concrete topological phenomenon arising in number theory or p-adic geometry that cannot be captured by any condensed set would show the framework misses essential cases.

read the original abstract

This is an updated version of the lectures notes for a course on condensed mathematics taught in the summer term 2019 at the University of Bonn. The material presented is joint work with Dustin Clausen. This is intended as a stable citable version of the original lectures, with mostly cosmetic changes to the original document, together with some small corrections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. This manuscript is an updated version of lecture notes from a 2019 course on condensed mathematics at the University of Bonn. The material is joint work with Dustin Clausen; the text is presented as a stable, citable version of the original lectures, incorporating mostly cosmetic changes and small corrections.

Significance. If the exposition accurately reflects the joint development of condensed mathematics, the notes provide a structured introduction to a framework that reinterprets topological and algebraic structures via condensed sets and related categories. This could aid researchers in algebraic geometry and number theory by offering direct access to foundational ideas from one of the primary developers, complementing the original research papers.

minor comments (1)
  1. [Abstract] The abstract mentions 'small corrections' but does not list them; a brief changelog or footnote indicating the nature of the corrections would improve traceability for readers comparing versions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report correctly identifies the work as an updated version of the 2019 Bonn lectures on condensed mathematics, prepared as a stable citable reference in collaboration with Dustin Clausen.

Circularity Check

0 steps flagged

No significant circularity: expository lecture notes with no derivation chain

full rationale

The document consists of lecture notes presenting foundational material on condensed mathematics as joint work with Dustin Clausen. No novel theorems, quantitative predictions, or load-bearing derivations are advanced. The text is purely expository and does not contain any self-definitional steps, fitted inputs renamed as predictions, or self-citation chains that reduce claims to their own inputs. All content relies on external category theory, algebraic geometry, and number theory background, with no internal reductions by construction. This matches the default expectation for non-circular expository work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; it is an overview of course material.

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