What intuitive notion is formalized by condensed mathematics?

Question

Preface: I ask this question from the position of a curious layperson who is excited about new conceptual advances in mathematics.

It is often said that the notion of "topology" formalises qualitative aspects of the notion of "nearness", or "nearness without distance". In this popular MO answer, Dan Piponi writes that topology can also be thought of as the "art of reasoning about imprecise measurements".

Is there a similarly intuitive notion that condensed mathematics can be considered a formalisation of? I am aware that it might be unreasonable (or premature at best) to ask for this kind of intuition.

I have read that in condensed mathematics, it is possible to speak of distinct points that are "infinitely" near each other — is this the notion that is being formalised?

Answer 1

Condensed sets axiomatize the notion of convergence rather than the notion of neighborhoods. Unlike topological spaces, they allow a sequence to converge "for multiple different reasons". This is similar in spirit to how homotopy theory allows to objects to be equal "for multiple different reasons".

$ \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\~}{\tilde} \newcommand{\O}{\mathcal O} \newcommand{\Set}{\mathrm{Set}} \newcommand{\Top}{\mathrm{Top}} \newcommand{\iso}{\xrightarrow\sim} \newcommand{\Prof}{\mathrm{Prof}} \newcommand{\op}{\mathrm{op}} \newcommand{\epi}{\twoheadrightarrow} \newcommand{\light}{\mathrm{light}} \newcommand{\Cvg}{\operatorname{Cvg}} \newcommand{\im}{\operatorname{im}} \newcommand{\tail}{\operatorname{tail}} \newcommand{\xra}{\xrightarrow} \newcommand{\red}[1]{{\color{red}{#1}}} \newcommand{\blue}[1]{{\color{blue}{#1}}} \require{AMScd} $We can get some intuition by restricting to the profinite sets $S=*$ and $S=\N\cup\{\infty\}$. I will call such a thing a partial condensed set to emphasize that I'm not looking at the full definition.

First let me introduce some nonstandard terminology. Let $$\N^+=\N\cup\{\infty\}$$ be the one-point compactification of $\N$. For a topological space $X$, I define a convergent sequence in $X$ to be a continuous map $$x_\bullet\colon\N^+\to X,$$ while a convergeable sequence is a map $\N\to X$ (where $\N$ has the discrete topology) which admits a continuous extension to $\N^+$. That is, for me a convergent sequence is not just a sequence $x_1,x_2,\dotsc$, it includes the limit point $x_\infty$. When $X$ is non-Hausdorff, there may be many convergent sequences with the same underlying convergeable sequence. So you have to get out of the habit of saying "is this sequence convergent?" and maybe instead say "how does this sequence converge?"

Given a topological space $X$, write $|X|$ for the underlying set of $X$, and $\O(X)$ for the set of open subsets of $X$, so that $X$ is the pair $X=(|X|,\O(X))$. Note that $$|X|=C(*,X) \quad\text{and}\quad \O(X)=C(X,\Omega),$$ where $\Omega$ is the Sierpiński space $\Omega=\{0,1\}$ with topology $\{\emptyset,\{1\},\Omega\}$ (and $C(X,Y)$ denotes the set of continuous maps $X\to Y$). So the topological space $X$ is completely determined by how it interacts with the topological spaces $*$ and $\Omega$. Condensed sets index spaces by how they interact with $\{*,\N^+\}$ (and more general profinite sets), rather than with $\{*,\Omega\}$.

Definition. A (partial) condensed set $S$ consists of

• an underlying set $|S|$
• a set of convergent sequences $\Cvg(S)$

subject to axioms introduced in my other answer. We think of $|S|$ as $C(*,S)$ and $\Cvg(S)$ as $C(\N^+,S)$, and sometimes write them as $S(*)$ and $S(\N^+)$.

Crucially, an element $\alpha\in\Cvg(S)$ has terms $\alpha_n\in S$ for $n\in\N\cup\{\infty\}$, but unlike topological spaces, $\alpha$ is not necessarily determined by its terms. Because $\Cvg(S)$ is not so closely tied to $S$, condensed sets are able to mix convergence information better than topological spaces are. Here is an example.

Let

• $A=\mathbb Z[\frac12]$ with the discrete topology
• $B=\mathbb Z[\frac12]$ with the topology inherited from $\mathbb R$
• $C=\mathbb Z_3$ (3-adic integers) with the 3-adic topology

Then $B\oplus_A C$ is a condensed abelian group with underlying set $\mathbb Z_3$, but an exotic convergence structure. The elements \begin{align*} \left(\alpha\colon \left(\tfrac34\right)^n\to 0\right) &\in\Cvg(B)\\ \left(\beta\colon \left(\tfrac34\right)^n\to 0\right) &\in\Cvg(C) \end{align*} define two different elements $(\alpha,0)$ and $(0,\beta)$ of $\Cvg(B\oplus_A C)$. We could also do \begin{align*} \left(\alpha\colon \sum_{n=0}^n \left(\tfrac34\right)^n \to 4\right) &\in\Cvg(B)\\ \left(\beta\colon \sum_{n=0}^n \left(\tfrac34\right)^n \to 4\right) &\in\Cvg(C) \end{align*} to get two different reasons that the series $\sum_{n=0}^\infty \left(\frac34\right)^n$ converges to $4$ in $B\oplus_A C$. An even better mixture is $B\otimes_A C$, although $\Cvg$ of this might be harder to describe explicitly.

Answer 2

There is an implicit notion in this question that condensed mathematics was specifically constructed to formalise some intuition about how to work with 'topology-like' structures. This is historically inaccurate: the usefulness of condensed mathematics was in some sense discovered by accident, as a 'trivial' case of some other construction (the pro-étale topology).

While it might still be possible to give a post hoc justification, let me focus instead on the historical motivation. This is not what you ask, and I apologise if you know all of this already. However, the pro-étale topology is something that is intuitive and tries to formalise a specific idea.

The starting point is, roughly speaking, building universal covers in algebraic geometry. If $X$ is a smooth complex algebraic variety, then its universal cover $\widetilde X \to X$ need not be a map of algebraic varieties: for instance, the universal cover of $\mathbf C^\times$ is the exponential map $\mathbf C \to \mathbf C^\times$ (in this case, $\widetilde X$ happens to be an algebraic variety, but that also fails in general). However, a theorem first proven in SGA$4_{\text{III}}$ (as far as I can tell) says that every finite covering space of a complex algebraic variety is again a complex algebraic variety (and the map is algebraic). This is called Grauert–Remmert in SGA, and in later writings is often informally called the Riemann existence theorem. For instance, the unique connected degree $n$ covering of $\mathbf C^\times$ is the $n$-th power map $\mathbf C^\times \to \mathbf C^\times$.

For a long time, people were content with this: this observation (plus a lot of theory) was enough to define étale cohomology: a way to construct (something isomorphic to) singular cohomology $H^i(X,\mathbf Z/n\mathbf Z)$ defined purely by algebraic geometry (in particular, it can be done for varieties in positive characteristic as well). With a limit procedure, you can extract $\ell$-adic cohomology $H^i(X,\mathbf Z_\ell) = \lim_n H^i(X,\mathbf Z/\ell^n\mathbf Z)$ if you want cohomology with characteristic $0$ coefficients.

If you want a universal cover instead of just finite étale covers, you can take the limit, which is now a profinite cover. In algebraic terms, if the degree $n$ cover of $k[x,x^{-1}]$ (the coordinate ring of $\mathbf A^1_k \setminus \{0\}$) is given by $k[x^{1/n},x^{-1/n}]$, then the pro-étale universal cover is given by $k[x^{1/\mathbf N},x^{-1/\mathbf N}]$, the union of all of these. The map $\widetilde{\mathbf C^\times} \to \mathbf C^\times$ is then Galois with group $\hat{\mathbf Z}$ (where $\widetilde{\mathbf C^\times}$ is a scheme but not of finite type over $\mathbf C$, so not an algebraic variety). Likewise, the pro-étale universal cover of $\operatorname{Spec} k$ is the spectrum of a separable algebraic closure $\bar k$ of $k$, so this recovers classical profinite Galois theory. (All of this could have been done a long time ago, but it was Scholze's work in $p$-adic geometry that brought this to the attention: you're somehow forced to think about pro-étale covers because a $p$-adic analytic variety is only pro-étale-locally perfectoid, not étale-locally.)

Then it turns out that taking the pro-étale topology fixes some of the technical issues of working with $\ell$-adic étale cohomology: you can now really take the cohomology of the sheaf $\mathbf Z_\ell = \lim_n \mathbf Z/\ell^n\mathbf Z$, whereas in the étale topology you have to take the limit after taking cohomology, which leads to all sorts of complications.

Finally, condensed mathematics is the pro-étale topology on the spectrum of an algebraically closed field; in other words, a point. So all the algebraic geometry disappears and you're "just" doing topos theory. I think it was a well-known principle that internal abelian group objects (or group objects, groupoid objects, category objects) in a topos is better behaved than internal group objects (...) in an arbitrary category. So from this point of view, the usefulness of condensed mathematics is explained by the fact that it is a topos in which topological spaces embed fully faithfully. (Insert set-theoretic caveat; in practice it is usually enough to work with $\kappa$-condensed for some cardinal $\kappa$, so that you really get a topos.)

If that is your point of view, you could also take any Grothendieck topology on topological spaces for which representable presheaves are sheaves (such a topology is called subcanonical) and use that as the basis for your theory. This is not so far from the truth: one way to obtain condensed sets is by endowing compact Hausdorff spaces with the coherent topology: coverings are finite families of jointly surjective maps of compact Hausdorff spaces $\coprod_{i=1}^n X_i \twoheadrightarrow X$. To connect this with the profinite theory, observe that every compact Hausdorff space is a quotient of a profinite space (for instance, the Stone–Čech compactification of the underlying discrete space), so profinite spaces (or even extremally disconnected spaces) form a basis for this Grothendieck topology.

(Actually, I have no idea what happens if you put a different Grothendieck topology on a suitable subcategory of topological spaces; in principle there could be many different theories here, but I don't know how to tell a priori which ones will tell you something meaningful.)

Answer 3

$ \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\~}{\tilde} \newcommand{\O}{\mathcal O} \newcommand{\Set}{\mathrm{Set}} \newcommand{\Top}{\mathrm{Top}} \newcommand{\iso}{\xrightarrow\sim} \newcommand{\Prof}{\mathrm{Prof}} \newcommand{\op}{\mathrm{op}} \newcommand{\epi}{\twoheadrightarrow} \newcommand{\light}{\mathrm{light}} \newcommand{\Cvg}{\operatorname{Cvg}} \newcommand{\im}{\operatorname{im}} \newcommand{\tail}{\operatorname{tail}} \newcommand{\xra}{\xrightarrow} \newcommand{\red}[1]{{\color{red}{#1}}} \newcommand{\blue}[1]{{\color{blue}{#1}}} \require{AMScd} $By requst, split off from my other answer about intuition. Here I will unwind the definition of "condensed set" on the profinite sets $S=*$ and $S=\N^+$ to get more explicit axioms that look more like the definitions you'd see in an undergrad topology course.

Axioms

1. a convergent sequence should have terms, so we require there to be maps $$ (-)_n\colon \Cvg(S) \to |S| $$ for all $n\in\N\cup\{\infty\}$. In particular, this means there is a map $$ \Cvg(S) \to |S|^{\N\cup\{\infty\}} $$ where the RHS is just functions between sets.
   If $X$ is a Hausdorff space, then the map $\Cvg(X) \to |X|^\N $ is injective: every convergeable sequence is convergent in exactly one way. In a non-Hausdorff space, this is not true, but still $ \Cvg(X) \to |X|^{\N\cup\{\infty\}} $ is injective, since maps of topological spaces are just functions between the underlying set satisfying a condition.
   The radical departure of condensed sets from topological spaces is that $$ \Cvg(S) \to |S|^{\N\cup\{\infty\}} $$ need not be injective; that is, a sequence may converge "for multiple different reasons".
   So we can write an element $\alpha\in\Cvg(S)$ as $$ \alpha\colon \Big(\alpha_n \to \alpha_\infty\Big) \quad\text{or}\quad \Big(\alpha_n \xra\alpha \alpha_\infty\Big) $$ and think of it as "a reason that $(\alpha_n)$ converges to $\alpha_\infty$"

2. every constant sequence should be convergent, so we require a map $$ \Delta\colon |S| \to \Cvg(S) $$ such that $\Delta(x)_n=x$ for all $n\in\N\cup\{\infty\}$.

3. every subsequence of a convergent sequence should be convergent. If you think about it, we can implement this by a map $$ f^* \colon \Cvg(S) \to \Cvg(S) $$ for every continuous map $f\colon \N^+\to\N^+$, such that $f^*(\alpha)_n=\alpha_{f(n)}$ for all $n\in\N\cup\{\infty\}$. This also lets us scramble the first $n$ terms around, or send it to an eventually constant sequence.

You can see that these axioms are asking for a map $S(f)\colon S(Y)\to S(X)$ for every continuous map $f\colon X\to Y$, where $X,Y\in\{*,\N^+\}$. In other words, a contravariant functor from the subcategory $\{*,\N^+\}\subset\Top$ to $\Set$.

The official definition of a (light) condensed set is not just any functor $ S\colon (\Prof^\light)^\op \to \Set, $ but a sheaf for the topology of finite collections of jointly surjective maps. This can be split into two more concrete axioms:

4. for every $X,Y\in\Prof^\light$, we have $$ S(X\amalg Y)=S(X)\times S(Y). $$ (More precisely, the natural map $S(X\amalg Y) \to S(X) \times S(Y)$ is a bijection). This makes sense geometrically: if $S(X\amalg Y)$ is supposed to be $C(X\amalg Y, S)$, we should always have $$C(X\amalg Y, S)=C(X, S)\times C(Y, S). $$ First, this tells us that the extension of $S$ from $\{*,\N^+\}$ to $\Prof^\light$ doesn't get us anything new on finite discrete sets: if $X$ is a discrete finite set with $n$ elements, then $S(X)=|S|^X$.
   More interestingly, this allows us to implement the head and tail of a sequence. The isomorphism of topological spaces \begin{align*} * \amalg \N^+ &\iso \N^+\\ * &\mapsto 0\\ n &\mapsto n+1 \end{align*} gives an isomorphism $$ S(\N^+) \iso S(*\amalg \N^+) = S(*) \times S(\N^+) $$ In other words, there is a bijection \begin{align*} \Cvg(S) &\iso |S| \times \Cvg(S)\\ \alpha &\mapsto (\alpha_0, \tail(\alpha)) \end{align*} where $\tail(\alpha)_n=\alpha_{n+1}$. This implements "a sequence is convergent if and only if its tail is convergent" in this more refined setting of "convergence".

5. given a surjection $p\colon Y\epi X$ of profinite sets, we want to be able to construct a map $\~f\colon X \to S$ by constructing a map $f\colon Y\to S$ and then requiring that $f$ is constant on the fibers of $p$. \begin{CD} Y @>f>> S\\ @VVV @|\\ X @>>\exists\,\~f> S \end{CD} The categorical way to say this is: let $p\colon Y\epi X$ be a surjection of profinite sets, then form the pullback \begin{CD} Y\times_X Y @>\pi_2>>Y\\ @V\pi_1VV \lrcorner @VV p V\\ Y @>>p> X \end{CD} Explicitly, $Y\times_X Y = \{(y_1,y_2) \in Y^2\mid \pi(y_1) = \pi(y_2)\}$. Then we have maps $$ S(X) \xrightarrow{p^*} S(Y) \xrightarrow[\pi_2^*]{\pi_1^*} S(Y\times_X Y) $$ so we have an inclusion $$ \im(p^*) \subset \{f \in S(Y) \mid \pi_1^*(f) = \pi_2^*(f) \} $$ and the requirement is that this is an equality.
   How to think about this in terms of sequences? I dunno, let's take the example $Y=\red{\N^+}\cup\blue{\N^+}$ and $X=\N^+$, with the map $p$ sending $p(\red a) = 2\red a$ and $p(\blue b)=2\blue b+1$. In this case $Y\times_X Y=\{(\red\infty,\blue\infty\})$. So in this case it's saying that two sequences \begin{align*} \Big(x_1,x_3,x_5,\dotsc \xra\alpha L\Big) &\in \Cvg(S)\\ \Big(x_2,x_4,x_6,\dotsc \xra\beta L\Big) &\in \Cvg(S) \end{align*} with the same limit can be interleaved to a sequence $$ \Big(x_1,x_2,x_3,x_4,\dotsc \xra{p^*(\alpha,\beta)} L \Big) \in \Cvg(S) $$ and this is a bijection $$ \{(\alpha,\beta)\in\Cvg(S) \mid \alpha_\infty = \beta_\infty\} \iso \Cvg(S). $$

This is just one example, there are SO MANY things you can do with convergent sequences.

Answer 4

I like the earlier answers but I want to give more naive and intuitive justifications for why condensed sets are sheaves and more importantly why sheaves on the category of totally disconnected compact Hausdorff spaces. Hence I try to present a (informal) way of arriving at the formal notion of condensed sets from the vague intuition that they should be spaces with convergence data with the additional criteria of adequacy that they are well-behaved with respect to additional algebraic structure.

Yuri Sulyma already illustrated how sheaves appear naturally. In general sheaves on a category can be interpreted as generalized spaces (see nlab for more details). We also know that a category of sheaves is a topos and hence well-behaved with respect to additional structure (for example the category of abelian group objects in a topos is always an abelian category). Therefore taking the approach of constructing condensed sets as sheaves on a suitable site is adequate.

But why look at sheaves on the category of totally disconnected compact Hausdorff spaces? We have to construct a category of spaces such that maps out of these spaces capture convergence data. As Yuri Sulyma explained $\mathbb N \cup \infty$ is a clear example of such a space. This space should definitely be in our category but it can also guide us in finding other similar spaces (I think it's even better to look at the Cantor set, in fact for every light profinite set $X$ there is a surjection from the Cantor set to $X$).

1. Every space should be compact because we want to capture the limit of any convergence sequence (the space $\mathbb N$ should not be in our category).
2. Every space should be Hausdorff because all limits should be unique (a space like $\mathbb N \cup \infty _1 \cup \infty _2$ with two copies of $\infty$ with the same open neighborhoods should not be in our category).
3. Converging sequences allow us to cut of initial segments without changing the convergence data. In terms of topological spaces this is realized by saying that the topology has a base consisting of clopen (closed and open) sets (think of these clopens as final/initial segments). Together with Hausdorff this implies that our spaces have to be totally disconnected.

Sadly I don't have a justification in the same spirit for the choice of covering families.