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Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead different expressions of the same concepts (similarly to the way in which different human languages can express the same thing). -wiki

Before I learned that category theory is said to be the unifying idea of mathematics which is based on noting that generally the act of studying a mathematical object is equivalent to studying it's relation with all other objects in the category of that mathematical object. So, what exactly is the idea in condensed mathematics which helps us unify mathematics beyond how we do it in Category theory?

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    $\begingroup$ Wow that is an atrocious Wikipedia page (and most math Wikipedia pages are pretty good). I hope someone knowledgeable improves it. $\endgroup$ Jun 28 at 2:17
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    $\begingroup$ The nlab page is a bit more concrete. That wikipedia page is awful. ncatlab.org/nlab/show/condensed+mathematics $\endgroup$ Jun 28 at 6:59
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    $\begingroup$ The most basic thing in condensed mathematics is a condensed set. nLab says, "A condensed set is a sheaf of sets on the pro-étale site of a point". My interpretation: this is not intended for beginners. $\endgroup$ Jun 28 at 14:24
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    $\begingroup$ @GeraldEdgar Pro-etale site of a point is simply the category of profinite sets with a pretty simple topology, so that alone is not a terribly complicated object. However, the general sentiment that it is not intended for beginners is definitely correct - the aim of the theory is development of homological algebra for topological objects, and in particular depends on a lot of classical homological algebra and, farther down the line, infinity categories. $\endgroup$
    – Wojowu
    Jun 28 at 14:30
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    $\begingroup$ I have rewritten most of the Wikipedia article linked. It is still not perfect but at least now it addresses the actual relevant points of the theory. I have edited the question to include the link to an older version of the article which the OP has cited. $\endgroup$
    – Wojowu
    Jun 28 at 15:31

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I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written that you can get some sense of the main idea just from studying the first few pages.

The starting point is the observation that the traditional way of endowing something with both a topology and an algebraic structure has some shortcomings. The simplest example is that topological abelian groups do not form an abelian category. We know from long experience with category theory that an excellent indication of whether you have a "good" definition of an object is that the category of all your objects (and the maps between them) has nice properties, and topological abelian groups fail to have some of those nice properties. In condensed mathematics, the category of topological abelian groups is replaced by the category of condensed abelian groups, which is an abelian category.

Although the goal is easy enough to state and motivate, the method of achieving it was initially not obvious even to Scholze, one of the architects (along with Clausen) of condensed mathematics. A key role is played by a category that might seem unpromising at first glance: the category $\mathcal{S}$ of profinite sets, a.k.a. totally disconnected compact Hausdorff spaces (or Stone spaces), with finite jointly surjective families of maps as covers. Most people's first impression of totally disconnected spaces is that they're weird, and they have some trouble even thinking of examples other than the discrete topology. However, categorically speaking, $\mathcal{S}$ has some nice properties. A condensed abelian group, roughly speaking, a special kind of (contravariant) functor from $\mathcal{S}$ to the category of abelian groups (somewhat more precisely, it is a sheaf of abelian groups on $\mathcal{S}$). That is, the way a topological structure is imposed on abelian groups is not in the classical way (i.e., by taking a set and defining a group operation and a topology on it in isolation), but by taking certain functors from this funny-looking topological category $\mathcal{S}$ to your algebraic category.

There's nothing about this story that is peculiar to abelian groups; by replacing "abelian group" with "set" or "ring" you get condensed sets and condensed rings and so forth.

The benefit of this shift from classical structures to condensed structures is not just aesthetic. One of the nicest applications is that it leads to new proofs of certain classical results in algebraic geometry. A longstanding puzzle (if you want to call it that) is that certain theorems about complex varieties that "feel algebraic" seem to be provable only via "transcendental methods"; i.e., by invoking analysis in a seemingly essential way. Condensed mathematics provides new proofs of some of these classical theorems that are more algebraic. See Condensed Mathematics and Complex Geometry by Clausen and Scholze for more details.

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    $\begingroup$ While you have linked to the two sets of notes by Scholze, let me also suggest you include a link to his and Clausen's notes on the just-finished course on complex geometry, which I believe your final paragraph is about. $\endgroup$
    – Wojowu
    Jun 28 at 14:33
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    $\begingroup$ @Wojowu Excellent suggestion. I have added a link. $\endgroup$ Jun 28 at 15:54
  • $\begingroup$ How does this relate to the idea of unifying math? $\endgroup$ Jun 30 at 0:40
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    $\begingroup$ @EthakkaappamwithChai "Unifying math" sounds like journalistic hyperbole to me, the kind of clickbait that makes money but that infuriates mathematicians. Having said that, I think that my final paragraph may be the closest thing to an answer. Instead of having to give separate algebraic and analytic arguments for different but closely analogous theorems, one can kill two birds with one condensed stone. $\endgroup$ Jun 30 at 3:39

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