Examples of the difference between Topological Spaces and Condensed Sets

Question

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.

Answer 1

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!)

Answer 2

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).