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There is apparently cutting-edge research by Dustin Clausen & Peter Scholze (and probably others) under the name Condensed Mathematics, which is meant to show that the notion of Topological Space is not so well-chosen, and that Condensed Sets lead to better behaved structures.

What is a simple low-tech example to see the difference?

I am looking for some explicit construction with quite simple topological spaces where some bad behaviour occur, and how their condensed analog fix that.

I am aware of the nlab entry and of an introductory text by F. Deglise on this page but it goes quite far too quickly and I am missing knowledge to grasp it.

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    $\begingroup$ Condensed Sets lead to better behaved structures --- I think "better behaved" is rather field specific, as I don't see how this notion adds much insight to most anyone who works in general topology rather than in category-theoretic fields (of which I realize there is some overlap with general topology, but in my opinion even this seems mostly of category-theoretic interest). Nonetheless (+1), I think it would be of interest for someone to give an "elementary level" example of what you're asking. $\endgroup$ Commented Mar 1, 2021 at 18:17
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    $\begingroup$ On the first page of Scholze's notes is a list of three problems which the theory is designed to address. I suggest you start here. $\endgroup$
    – Tyrone
    Commented Mar 1, 2021 at 18:36

2 Answers 2

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Here's a one-paragraph answer: Topological spaces formalize the idea of spaces with a notion of "nearness" of points. However, they fail to handle the idea of "points that are infinitely near, but distinct" in a useful way. Condensed sets handle this idea in a useful way.

Let me give a few examples. Say you have a nice space like the line $X=\mathbb R$, and acting on it you have a group $G$. If $G$ acts nicely, for example $G=\mathbb Z$ through translations, then the quotient space $X/G$ is well-behaved as a topological space, giving the circle $S^1$ in this case. However, if $G$ has dense orbits, for example $G=\mathbb Q$, then $X/G$ is not well-behaved as a topological space. In fact, in this example $\mathbb R/\mathbb Q$ has many distinct points, but they are all infinitely near: any neighborhood of one point $x$ will also contain any other point $y$. Thus, as a topological space $\mathbb R/\mathbb Q$ is indiscrete; in other words, it is not remembering any nontrivial topological structure.

One could also consider nonabelian examples, like the quotient $\mathrm{GL}_2(\mathbb R)/\mathrm{GL}_2(\mathbb Z[\tfrac 12])$.

Similar things also happen in functional analysis. For example, one can embed summable sequences

$$\ell^1(\mathbb N)=\{(x_n)_n\mid x_n\in \mathbb R, \sum_n |x_n|<\infty\}$$

into square-summable sequences

$$\ell^2(\mathbb N)=\{(x_n)_n\mid x_n\in \mathbb R, \sum_n |x_n|^2<\infty\}$$

and consider the quotient $\ell^2(\mathbb N)/\ell^1(\mathbb N)$. As a topological vector space, this is indiscrete, and so has no further structure than the abstract $\mathbb R$-vector space. For this reason, quotients of this type are usually avoided, although they may come up naturally!

Without repeating the definition of condensed sets here, let me just advertise that they can formalize the idea of "points that are infinitely close, but distinct", and all the above quotients can be taken in condensed sets and are reasonable. As we have seen, this means that one must abandon "neighborhoods" as the primitive concept, as in these examples all neighborhoods of one point already contain all other points. Roughly, what is formalized instead is the notion of convergence, possibly allowing that one sequence converges in several different ways.

So in the condensed world, it becomes possible to consider quotients like $\ell^2(\mathbb N)/\ell^1(\mathbb N)$, and inside all condensed $\mathbb R$-vector spaces they are about as strange as torsion abelian groups like $\mathbb Z/2\mathbb Z$ are inside all abelian groups. However, there are some surprising new phenomena: For example, there is a nonzero map of condensed $\mathbb R$-vector spaces

$$\ell^1(\mathbb N)\to \ell^2(\mathbb N)/\ell^1(\mathbb N)$$

induced by the map

$$(x_n)_n\mapsto (x_n \log |x_n|)_n,$$

in other words $x\mapsto x\log |x|$ pretends to be a linear map! (Let me leave this as a (fun) exercise.)

(These are liquid $\mathbb R$-vector spaces, and the presence of such strange maps makes it hard to set up the basic theory of liquid vector spaces; but arguably makes it more interesting!)

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    $\begingroup$ Dear Professor Scholze, thank you very much for this extremely enlightening answer, that's exactly the informal intuition I was looking for! (I accepted an answer already so cannot accept yours but perhaps you will win a gold 'populist' badge this way). $\endgroup$
    – Archie
    Commented Jul 15, 2021 at 17:22
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    $\begingroup$ @Archie Actually you can unaccept my answer and accept this one if you want. That's not considered rude to the best of my knowledge (as long as it's not done for improper reasons of course). For what it's worth in this case I'd recommend that (independently of this answer's author's identity, incidentally - I just think this answer is meaningfully better than mine). $\endgroup$ Commented Aug 22, 2021 at 5:50
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    $\begingroup$ In the "uncondensed world," and forgetting topology, it seems the given map $\ell^1(\mathbb{N}) \to \ell^2(\mathbb{N})/\ell^1(\mathbb{N})$ is already a map of real vector spaces, no (so additivity is an interesting exercise in analysis, but it doesn't use anything about condensed sets if I've computed correctly). $\endgroup$
    – hunter
    Commented Oct 9, 2021 at 21:47
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    $\begingroup$ Of course it's also a map of underlying (ie "uncondensed") vector spaces. But it's even "continuous", ie a map of condensed abelian groups (a priori one might hope that this condition might forbid weird things). Also note that to get full points in the exercise ( ;-) ), it's not enough to check that it defines a map of condensed sets that is additive on underlying vector spaces! $\endgroup$ Commented Oct 11, 2021 at 6:15
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    $\begingroup$ Maybe a remark worth adding: In the literature, there was already a way to enlarge the category of Banach spaces into a larger abelian category so as to accomodate quotients like $\ell^2(\mathbb N)/\ell^1(\mathbb N)$, namely Waelbroeck's category of qBanach spaces. But in this category, $\ell^1(\mathbb N)$ is projective, so any map $\ell^1(\mathbb N)\to \ell^2(\mathbb N)/\ell^1(\mathbb N)$ in Waelbroeck's category lifts to a map $\ell^1(\mathbb N)\to \ell^2(\mathbb N)$, but this doesn't happen for the example given. So in this sense the given map is a "new" phenomenon. $\endgroup$ Commented Oct 13, 2021 at 8:39
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Condensed sets not being "simple low-tech" (to put it mildly) in the first place, I'm not sure to what extent this question can be answered satisfactorily. However, it is easier to address the question of why we might be unhappy with the usual category of topological spaces.

As Tyrone commented above, the introduction to Scholze's notes on condensed mathematics mentions three major issues with the category ${\bf Top}$ of toplogical spaces, from the perspective of developing a theory of "topological algebras" analogous to the rich theory we already have for purely algebraic strutures (his Question $1.1$). The second and third are rather technical (developing theories of derived categories and quasicoherent sheaves), but the first is relatively snappy: namely, the failure of the abelian category axioms. For example, the category of abelian group objects in ${\bf Sets}$ (= the category of abelian groups) is an abelian category, but the category of abelian group objects in ${\bf Top}$ is not. This means that the whole machinery of abelian categories can't be applied as we would hope.

(My original version of this answer said that the point is that ${\bf Top}$ itself is not abelian. Of course that's silly, as Connor Malin pointed out below: neither is ${\bf Sets}$!)

One important point here is that the issues here do not result from pathological objects or morphisms; we're not going to improve things by restricting to a subcategory of "nice" spaces and continuous functions. Instead, in order to get a better-behaved category we need to shift attention to a larger category, whose objects may be wilder but whose overall structure is better. This is exactly what we get in shifting from topological spaces to condensed sets, and from topological groups/rings/etc. to condensed groups/rings/et cetera (see Example $1.5$ and Proposition $1.7$ to check that this is in fact a case of passing to a larger context).

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    $\begingroup$ we're not going to improve things by restricting to a subcategory of "nice" spaces and continuous functions --- This reminds me of something I read a couple of months ago, which took me a few moments to find. It's the paragraph extending from page vii to page viii (4th paragraph of the Preface) in Beyond Topology edited by Mynard/Pearl (2009), a book I got (print version) a couple of months ago mainly for the chapter Closure by Marcel Erné (pp. 163-238), for use when I get around to incorporating revisions/additions to this 3-part answer. $\endgroup$ Commented Mar 1, 2021 at 19:22
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    $\begingroup$ (4th paragraph of the Preface I mentioned in my previous comment) "This situation can be viewed in two ways: either you consider that the class of topological spaces is too large and leaves room for too much pathology, in which case you will try to remedy the above problems by finding a subclass of topological spaces behaving better, or you realize that the class of topological spaces is too small to perform certain operations in a natural way, just like the field of real numbers is too small to factor any polynomial into linear factors." (continued) $\endgroup$ Commented Mar 1, 2021 at 19:36
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    $\begingroup$ "The former approach led among others to the theory of $k$-spaces, which became reasonably popular, notably in homotopy theory, even though this solution suffers from obvious problems, like the necessity of using a product that is different from the usual topological product." $\endgroup$ Commented Mar 1, 2021 at 19:37
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    $\begingroup$ (5th paragraph of the Preface) "The present book is about the latter, less widely known, approach. It presents to a reader with only a basic knowledge of point-set topology various generalizations of topologies, each addressing one or several particular shortcomings of topologies. Written with the graduate student in mind, this volume should also be an eye-opener for the working mathematician, the day-to-day user of topology: there is sometimes much to gain in looking beyond topology." $\endgroup$ Commented Mar 1, 2021 at 19:39
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    $\begingroup$ I think your issue should not be with $Top$ but rather with the category of abelian group objects in $Top$. Of course $Set$ does not form an abelian category, and we don't take issue with $Set$ because of it. $\endgroup$ Commented Mar 2, 2021 at 0:23

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