I am trying to follow the beautiful notes by Peter Scholze on condensed mathematics (https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)

I am noting that the hard time that I am getting is a lack of basis in Grothendieck topologies, sites, and so on, things that are not from the theory of condensed math itself but prerequisites.

So I would like to know if is there a standard book, or nice notes used for the introduction on such matters, with more concrete examples of the objects defined.

Thank you very much for any suggestions.


3 Answers 3


Dagur Ásgeirsson has written a text to fill this gap:

We discuss in some detail the prerequisites for each of the first four chapters of Scholze's "Lectures on Condensed Mathematics". Some proofs are given in more detail or slightly altered, but all the main ideas are the same. In particular, I claim no originality of the results. It is my hope, however, that I have succeded in creating a more accessible presentation of the foundations of condensed mathematics, along with the prerequisites, than has previously been available.

  • 18
    $\begingroup$ Should call the notes "Condensed Math: Expanded" $\endgroup$ Apr 12 at 12:58
  • 2
    $\begingroup$ Nah, "Condensed Math: Decondensed" @SamHopkins $\endgroup$ Apr 13 at 11:49
  • 1
    $\begingroup$ I suggest "Condensed Math: Evaporated" @TrystwithFreedom. $\endgroup$
    – Danu
    Apr 14 at 9:02

Blowing my own trumpet a bit: http://math.commelin.net/files/liquid_notes.pdf

These are notes for a talk that I gave this week for an audience of complex geometers.

I take a somewhat unorthodox approach, which hopefully complements other accounts a bit. Two disclaimers:

  • These notes do not mention profinite sets, Grothendieck topologies, sites, topoi, etc... In fact, they don't contain a definition of condensed sets. Instead I describe a large subcategory in very down-to-earth terms.
  • If you really want to understand what is going on, there is no realistic way around the categorical machinery. Indeed, towards the end of the notes this already leaks into my presentation.
  • Everything in the notes is already contained in some form or other in the lecture notes by Clausen and Scholze. I do not claim any contributions. I've tried to give some pointers into the lecture notes.

I hope that these notes can be some kind of appetizer.

  • $\begingroup$ Does your description of condensed sets in terms of compactological spaces (with formal quotients) correspond to some kind of general category-theoretic completion? In other words, is the category of condensed sets equivalent to, say, the ex/reg completion of the category of compactological spaces? $\endgroup$ Sep 23 at 18:14

There are some useful references:

  1. Dagur Ásgeirsson's Master thesis, cited in other responses (https://dagur.sites.ku.dk/files/2022/01/condensed-foundations.pdf): Beautifully written, I think you just need to know the basics of category theory and topology. (For the latter sections, some experience with homological algebra is welcome).

  2. The first 3 chapters of Catrin Mair's Master thesis (https://arxiv.org/pdf/2105.07888.pdf): A very nice text that contains information about basic topos theory.

  3. If you know Portuguese, I have written my undergraduate thesis to fill this gap too: (https://repositorio.ufsc.br/bitstream/handle/123456789/245189/TCC%20Luiz%20Felipe%20Garcia.pdf?sequence=1)

Some seminar notes and the Condensed Masterclass are also very useful!

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    $\begingroup$ Thank you, I can read in Portuguese, and is awesome that you made such work at the undergrad level. $\endgroup$
    – User43029
    Apr 17 at 17:48

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