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Notes on condensed mathematics

Abstract

These are notes from a graduate topics course

¹

kskedlaya.org/math205-fall24/

given at UC San Diego during the fall 2024 quarter; recordings of the lectures can be found at the same web site. The notes were typeset using PreTeXt

²

pretextbook.org

so as to produce matching HTML

³

kskedlaya.org/condensed/

and PDF

⁴

kskedlaya.org/papers/condensed.pdf

versions; the source code is available from this GitHub repository

⁵

github.com/kedlaya/condensed

. (Beware that the recordings do not correspond precisely to the notes as you see them due to numerous corrections and remarks having been incorporated into the latter.)

As of this writing, condensed mathematics is a moving target: the first version of the theory appears in [6], but this theory is set up in a way that exposes it to a sensitive dependence on large cardinal axioms. We will instead follow the approach sketched in [8], which limits this exposure without too much of a cost when it comes to practical applications; unfortunately, there is not yet an entirely satisfactory print reference (these notes included).

Our goal here is not to give a sweeping overview; for this we defer to the other references. Rather, we try to err on the side of filling in some details in the basic development, and explaining some concepts that are not specific to condensed mathematics but are crucial for understanding what is going on.

These notes have benefited from feedback from numerous readers, including: Hargun Bhatia, Alina Bucur, Justin Carel, Yusuf Cherni, Rex Cheung, Dimitri Dine, JJ Garzella, Christian Klevdal, Emmanuel Lepage, Shubhankar Sahai, Ehsan Shahoseini, Fukuhiro Ueda, Lucas Miguel Valle Thiele, Nathan Wenger, Chris Xu, David Zhang.