Condensed mathematics is a theory developed by Dustin Clausen and Peter Scholze which replaces a topological space by a certain sheaf of sets, in order to solve some technical problems of doing homological algebra on topological groups.
[citation needed] The fundamental idea in the development of the theory is given by replacing topological spaces by condensed sets, defined below.
In 2013, Bhargav Bhatt and Peter Scholze introduced a general notion of pro-étale site associated to an arbitrary scheme.
In 2018, Dustin Clausen and Scholze arrived at the conclusion that the pro-étale site of a single point, which is isomorphic to the site of profinite sets introduced above, already has rich enough structure to realize large classes of topological spaces as sheaves on it.
[4] In 2020 Scholze completed a proof of their results which would enable the incorporation of functional analysis as well as complex geometry into the condensed mathematics framework, using the notion of liquid vector spaces.
The argument has turned out to be quite subtle, and to get rid of any doubts about the validity of the result, he asked other mathematicians to provide a formalized and verified proof.
[8] Coincidentally, in 2019 Barwick and Haine introduced a similar theory of pyknotic objects.
This theory is very closely related to that of condensed sets, with the main differences being set-theoretic in nature: pyknotic theory depends on a choice of Grothendieck universes, whereas condensed mathematics can be developed strictly within ZFC.