Notes on Condensed Mathematics

These are my notes from Dustin Clausen and Peter Scholze’s YouTube video series on Analytic Stacks, [ 1 ] . They were used in a reading seminar at Concordia University, Montreal, Canada, in April 2024. I would like to thank all the participants for their willingness to listen and for their comments and corrections.

The aim of these notes is simple: to provide a brief overview of Condensed Mathematics. They are not intended to be a comprehensive treatment, but rather a gentle introduction to the subject. The structure of the notes is as follows:

  1. Motivation

  2. Light Profinite Sets and Condensed Sets

  3. Condensed Algebra & Solids (soon...?)

  4. Affine Analytic Geometry: A Tourist Guide (in construction...)

We will try to follow the original videos as much as possible, but we will also include some additional material and comments. Whenever possible, I will try to give specific timestamps in the YouTube videos. Finally, since these are just my humble notes, if the reader finds any mistakes, please let me know. I will be happy to correct them.

1 Motivation

Condensed Mathematics seeks to provide a common framework for algebra and topology. In “classical” mathematics, we have the following simple situation:

\[ (\mathbb R,\mathrm{dis}) \xrightarrow {\ f\ } (\mathbb R,|\cdot |), \qquad x\longmapsto x. \]

This continuous map is surjective and injective: in set-theoretic terms,

\[ \operatorname {im}(f)=\mathbb R, \qquad \bigl\{ (x,y):f(x)=f(y)\bigr\} =\Delta . \]

But it is not a homeomorphism! Constructions of “classical” sets are not compatible with topology.

The idea is to consider a new category that behaves like the category of sets, but in which the underlying topology is not the protagonist, but rather a director behind the curtains. Such categories are called toposes (this is a nice interpretation by Alain Connes).

1.1 Toposes

Toposes are categories of sheaves. Given a category \(\mathcal C\) equipped with a Grothendieck topology \(\tau \), one forms the category of sheaves with respect to \(\tau \), denoted by \(\operatorname {Sh}_{\tau }(\mathcal C)\). In these notes, the details of the definition of a sheaf will not be important; only the following meta-theorem will serve as a guiding idea [ 2 , Theorem E.4.1 ] :

Meta-Theorem 1.1 Giraud

A category \(\mathcal X\) is a Grothendieck topos if and only if it has the following properties: local presentability, effective quotients, universal colimits, and disjoint coproducts.

Examples 1.1

Some basic examples of toposes:

  • The category of sets, \(\mathrm{Set}\), is a topos.

  • Let \(G\) be a group. Then the category of sets with a \(G\)-action, \(G\text{-}\mathrm{Set}\), is a topos. Here we might take:

    \[ \mathcal C=\{ \text{one object with automorphism group }G\} , \qquad \tau =\text{``trivial''}. \]

  • Let \(X\) be a topological space. Then the category of sheaves on \(X\), \(\operatorname {Sh}(X)\), is a topos. Here we might take:

    \[ \mathcal C=\{ \text{open subsets of }X\} , \qquad \tau =\text{``open covers''}. \]

  • Non-example. \(\mathrm{Ab}\) is not a topos.

2 Profinite and Condensed Sets

In this section, we review the basic definition of profinite sets. Next, we define the main objects of these notes: light condensed sets. Finally, we relate these light condensed sets to "classical" topology and see how the condensed viewpoint realizes the slogan of the previous section.

2.1 Definitions

We want to restrict ourselves to “good” topological spaces, e.g., metric ones. For this, we use light profinite sets [ 1 , Lecture 1, 00:15 ] .

Definition 2.1

Let \(\operatorname {Pro}(\mathrm{Fin})\) be the category whose objects are

\[ \lim _{i\in I}S_i, \qquad \text{each }S_i\text{ is a finite set and }I\text{ is a cofiltered system}, \]

and whose morphisms are

\[ \operatorname {Hom}\! \left(\lim _{i\in I}S_i, \lim _{j\in J}S_j\right) =\lim _{j\in J}\operatorname*{colim}_{i\in I} \operatorname {Hom}_{\mathrm{Set}}(S_i,S_j). \]
Examples 2.2

A couple of examples of profinite sets are:

  • Any finite set, \(S=S_i\).

  • Take \(S_i=\mathbb Z/p^i\mathbb Z\) or \(S_i=\mathbb F_p[T]/T^i\), where the transition maps are the natural projections modulo \(p^i\) or \(T^i\):

    \[ \mathbb Z/p^{i+1}\mathbb Z\longrightarrow \mathbb Z/p^i\mathbb Z, \qquad \frac{\mathbb F_p[T]}{T^{i+1}}\longrightarrow \frac{\mathbb F_p[T]}{T^i}. \]
Proposition 2.3

There are equivalences

where the equivalences are given by

\[ \begin{aligned} \text{(1)}\quad & \lim _i S_i \longmapsto \Bigl(\lim _i S_i,\ \text{inverse-limit topology}\Bigr), & \qquad \text{(2)}\quad & S\longmapsto \operatorname {Cont}(S,\mathbb F_2), \\[0.9em] \text{(3)}\quad & A\longmapsto \operatorname {Spec}(A), & \qquad \text{(4)}\quad & A\longmapsto \lim _{\scriptstyle A'\subset A,\; A'\text{ finite}} \operatorname {Hom}(A’,\mathbb F_2). \end{aligned} \]

One consequence of this proposition is the following observation: the Cantor set, \(\prod _{\mathbb N}\{ 0,1\} =\{ 0,1\} ^{\mathbb N}\) (we will usually use the exponential notation), is profinite because it is a compact Hausdorff, totally disconnected topological space, even though the product notation does not exhibit it as a cofiltered limit. In the next example, we give a description of this space in terms of cofiltered limits.

Definition 2.4 Lightness

Let \(S\in \operatorname {ProFin}\). The size of \(S\) is \(\kappa :=|S|\), and the weight is \(\lambda :=\bigl|\operatorname {Cont}(S,\mathbb F_2)\bigr|\). We say that \(S\) is light if

\[ \lambda \leq \omega , \]

where \(\omega \) is the first infinite cardinal.

We denote the subcategory of light profinite sets by

\[ \operatorname {Pro}_{\mathbb N}(\mathrm{Fin}) \hookrightarrow \operatorname {Pro}(\mathrm{Fin}). \]
Examples 2.5

Here are some examples of light profinite sets:

  • \(S=\mathbb N\cup \{ \infty \} =\lim _n\{ 0,\ldots ,n,\infty \} , \qquad \lambda =\omega ,\quad \kappa =\omega ,\) with transition map

    \[ \{ 0,\ldots ,n+1,\infty \} \longrightarrow \{ 0,\ldots ,n,\infty \} , \qquad i\longmapsto \begin{cases} i,& i\leq n,\\ \infty ,& i{\gt}n. \end{cases} \]

  • \( S=\text{Cantor set}=\{ 0,1\} ^{\mathbb N} =\lim _n\{ 0,1\} ^n, \qquad \kappa =2^\omega ,\quad \lambda =\omega . \)

  • \(S=\beta \mathbb N\), the Stone–Čech compactification of \(\mathbb N\);

    \[ \lambda =\bigl|\operatorname {Cont}(\beta \mathbb N,\mathbb F_2)\bigr| =\bigl|\operatorname {Tot}(\mathbb N,\mathbb F_2)\bigr| =\bigl|\{ \text{subsets of }\mathbb N\} \bigr| =2^\omega . \]
Remark

Light profinite sets correspond, under Proposition 2.3, to [ 1 , 00:38 ] :

\[ \{ \text{t.d. compact Hausdorff metrizable}\} \longleftrightarrow \{ \text{light profinite sets}\} \longleftrightarrow \{ \text{countable Boolean rings}\} . \]

Moreover,

\[ S\text{ light} \ \Longleftrightarrow \ \text{there exists }\{ 0,1\} ^{\mathbb N}\twoheadrightarrow S. \]

In other words, light profinite sets are obtained as quotients of the Cantor set.

We are now ready to define light condensed sets [ 1 , Lecture 1, 01:00 ] .

Definition 2.6

A light condensed set is a functor

\[ X:\operatorname {Pro}_{\mathbb N}(\mathrm{Fin})^{\mathrm{op}} \longrightarrow \mathrm{Set} \]

such that

  1. \(X(\varnothing )=\{ *\} \).

  2. The restriction map is an isomorphism

    \[ X(U_1\sqcup U_2)\xrightarrow {\sim }X(U_1)\times X(U_2). \]

  3. If \(T\twoheadrightarrow S\), then

    \[ X(S)\xrightarrow {\sim } \operatorname {eq}\! \left( X(T)\mathrel {\substack {\xrightarrow {\ p_1^*\ }\\[-0.6ex] \xrightarrow [\ p_2^*\ ]{}}} X(T\times _S T) \right) =\{ s\in X(T):p_1^*(s)=p_2^*(s)\} , \]

    where \(p_1,p_2:T\times _ST\rightrightarrows T\) are the projections.

Examples 2.7

Let \(X\) be a topological space; set

\[ \underline X(S):=C(S,X)=\{ \text{continuous functions }S\to X\} . \]

Then \(\underline X\) is a condensed set. In particular, \(\operatorname {Hom}(-,S)\) is a condensed set for all \(S\). A key feature of the profinite world is that surjections \(S\twoheadrightarrow S'\) are quotient maps.

Remark

Since every light profinite set is a quotient of the Cantor set, we have the following description (think of the analogy with Galois theory, regarding the Cantor set as “an algebraic closure of \(\mathbb Q\)”): \(X\in \mathrm{Cond}^{\mathrm{light}}\) is completely determined by

\[ X(\text{Cantor set})\circlearrowleft \operatorname {End}(\text{Cantor set}). \]

However, describing \(\operatorname {End}(\text{Cantor set})\) and its action might be too complicated to track down.

Note that a light condensed set is a sheaf on \(\operatorname {Pro}_{\mathbb N}(\mathrm{Fin})\) with respect to the Grothendieck topology generated by finite disjoint unions and all surjective maps. Thus, light condensed sets form a topos within the framework presented in Section 1, i.e., a category of sheaves. We call this category \(\mathrm{Cond}^{\mathrm{light}}\).

2.2 Light condensed sets and topology

Let us return to the example of the map \(f\) at the beginning of these notes and see how condensed sets help us understand the situation. We have the following diagram:

\[ (\mathbb R,\mathrm{disc})\longrightarrow (\mathbb R,|\cdot |), \qquad \underline{\mathbb R}^{\mathrm{disc}} \xrightarrow {\ f\ }\underline{\mathbb R}. \]

\(f\) is not an isomorphism of sheaves (i.e., light condensed sets)!

\[ f(*):\underline{\mathbb R}^{\mathrm{disc}}(*)=\mathbb R \longrightarrow \mathbb R=\underline{\mathbb R}(*), \]

whereas

\[ f(\mathrm{Cant}): \{ g:\{ 0,1\} ^{\mathbb N}\to \mathbb R:\ g\text{ locally constant}\} \not\simeq \{ g:\{ 0,1\} ^{\mathbb N}\to \mathbb R:\ g\text{ continuous}\} . \]

We want to emphasize that, in the context of condensed mathematics, condensed sets are just “sets”, so there is no underlying topology. The “topology” is encoded internally, in their functoriality. Thus, in some sense, condensed sets are sets whose topology is “behind the curtain”. Let us see how this works in more detail [ 1 , Lecture 2, 01:15 ] .

Let \(X\in \mathrm{Cond}^{\mathrm{light}}\). There is an “evaluation” map

\[ \coprod _{\beta \in X(\{ 0,1\} ^{\mathbb N})}\{ 0,1\} ^{\mathbb N} \longrightarrow X(*), \qquad (\beta ,x)\longmapsto \operatorname {Res}_x(\beta ), \]

where the point \(x\) determines the map \((\{ *\} \longrightarrow \{ 0,1\} ^{\mathbb N},\ *\longmapsto x)\). Using this map, we equip \(X(*)\) with the quotient topology. To emphasize that this set carries a topology, we sometimes also denote it by \(X(*)_{\mathrm{top}}\). This construction allows us to pass from condensed sets to topological spaces:

Proposition 2.8

The construction above defines a functor

\[ \mathrm{Cond}^{\mathrm{light}} \to {\rm Top}, \qquad X\longmapsto X(*)_{\mathrm{top}}, \]

which is right adjoint to

\[ {\rm Top} \to \mathrm{Cond}^{\mathrm{light}}, \qquad T\longmapsto \underline T. \]

Furthermore, we have a fully faithful functor

\[ \{ \text{``Metrizably compactly generated'' Top. spaces}\} \hookrightarrow \{ \text{light condensed sets}\} . \]
Examples 2.9

We can see how this construction works in the case of the Cantor set and the interval \([0,1]\):

\[ \underline{[0,1]}(*)_{\mathrm{top}}=[0,1], \qquad \{ 0,1\} ^{\mathbb N}\twoheadrightarrow [0,1], \qquad (a_i)\longmapsto \sum _i a_i\left(\frac12\right)^i. \]

2.3 Gluing

To study these new objects, we return to a basic principle of topos theory: objects can be constructed by gluing. In the case of condensed sets, they can be constructed by gluing light profinite sets. One can also observe this phenomenon in simplicial sets; see @Infinito categorías.

The Yoneda functor

\[ \operatorname {Pro}(\mathrm{Fin})\hookrightarrow \mathrm{Cond}^{\mathrm{light}}, \qquad S\longmapsto \underline S=\operatorname {Hom}(-,S), \]

is fully faithful. It follows from abstract nonsense that \(\mathrm{Cond}^{\mathrm{light}}\) has arbitrary colimits. Moreover, every \(X\in \mathrm{Cond}^{\mathrm{light}}\) can be written as

\[ X=\operatorname*{colim}_{i\in I}\underline{T_i}, \qquad T_i\in \operatorname {Pro}_{\mathbb N}(\mathrm{Fin})! \]
Examples 2.10

Some basic examples are:

  • Equivalence relations (recall that coequalizers are colimits):

    \[ R\subset X\times X;\qquad \operatorname {Coeq} \bigl(R\rightrightarrows T\bigr)=X/R. \]

    In particular, for a quotient map \(S\twoheadrightarrow X\), we have

    \[ X=\operatorname {Coeq}(S\times _X S\rightrightarrows S). \]

    For a more concrete example, see the Cantor-set construction in Example 2.9.

  • Consider the real line \(\mathbb R\) as the union of intervals \([-n,n]\). Then we have

    \[ \mathbb R=\bigcup _n[-n,n] \longleftarrow \bigcup _n\{ 0,1\} ^{\mathbb N}. \]
Remark

We conclude with an important warning about condensed sets: colimits of topological spaces do not necessarily coincide with colimits in \(\mathrm{Cond}^{\mathrm{light}}\). However, they do in a “countable setting”. The reason is that the topology \(X(*)_{\mathrm{top}}\) can only capture “metrizable compactly generated” topological spaces. For example, we have [ 1 , Lecture 3, 01:55 ]

\[ \{ 0,1\} ^{\mathbb N} \neq \operatorname*{colim}_{\substack {Z\subseteq \{ 0,1\} ^{\mathbb N}\\ Z\text{ is countable}}} Z \qquad \bigl(\mathrm{Top}:\checkmark , \ \mathrm{Cond}^{\mathrm{light}}:\times \bigr), \]

but

\[ \mathbb R=\bigcup _n[-n,n] \qquad \bigl(\mathrm{Top}\text{ and Cond.}\bigr). \]

References

1

D. Clausen, P. Scholze, Analytic Stacks, YouTube videos, available at https://www.youtube.com/playlist?list=PLx5f8IelFRgGmu6gmL-Kf_Rl_6Mm7juZO (2024).

2

Emily Riehl, Category Theory in Context, available at https://math.jhu.edu/~eriehl/161/context.pdf.