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Notes on condensed mathematics

Kiran S. Kedlaya

Section 4 Condensed sets

We describe the category of condensed sets. This is formally similar to the construction of solid modules, but using the richer category of profinite sets as the test objects. As a result, it is able to absorb both the theory of solid modules (see Section 7) and some nontrivial amount of topology (see Section 8). However, if you are only interested in solid modules, you can skip forward to Section 9 where we continue the discussion of solid modules without the language of condensed sets.
From now on, we make the blanket assumption \(\kappa = \aleph_0\) in the definition of the category \(\Prof\text{;}\) this will correspond to restricting attention to “light” condensed sets in the terminology of [8].
Note: by convention, whenever we write \(\RR\) we mean the corresponding object in \(\Top\) with the usual archimedean topology. In case we mean the underlying set with the discrete topology, we instead write \(\RR_{\disc}\text{.}\)
Another convention: I will take \(\NN\) (natural numbers) to be the nonnegative integers. If I really mean the positive integers (probably because I want to use the multiplicative structure) I will write \(\NN^\times\) instead. See Definition 5.3.1 for the reason I chose this convention.

Reference.

This lecture is based on [8], Lectures 1 and 2. See also [6], Lecture 1.
Although we do not use this perspective here, we also wish to draw attention to the alternate interpretation available in the language of pyknotic sets of Barwick–Haine [2]. Roughly speaking, this amounts to using two separate cardinality cutoffs.
There is also a form of Stone duality (Remark 3.2.8) that expresses condensed sets in terms of fpqc sheaves on the category of Boolean algebras of limited cardinality. See [12], Theorem 4.4 (and also Definition 19.3.3 for a closely related construction).

Subsection 4.1 The category of condensed sets

We now now imitate the passage from modules to solid modules over a ring to pass from sets to condensed sets. One important difference is that we do impose certain structure on the functors; this will correspond to enforcing the sheaf axiom with respect to a certain topology.

Definition 4.1.1.

The category of condensed sets is the full subcategory \(\CSet\) of \(\Fun(\Prof^{\op}, \Set)\) consisting of those functors \(X\) such that:
  • \(X\) carries finite disjoint unions to finite products (in particular \(X(\emptyset)\) is a singleton set);
  • for every surjection \(T \to S\) in \(\Prof\text{,}\) the diagram
    \begin{equation*} X(S) \to X(T) \rightrightarrows X(T \times_S T) \end{equation*}
    is an equalizer.
For any \(S \in \Prof\text{,}\) the representable functor \(h_S\) belongs to \(\CSet\) (we will prove a stronger statement in Lemma 4.1.5). By Yoneda’s lemma, this yields a fully faithful functor \(\Prof \to \CSet\text{.}\) We will see a bit later that \(\CSet\) also contains some other familiar categories as full subcategories, including \(\Set\) (Corollary 4.1.6), \(\CHaus\) (Proposition 4.2.4), and even a sizable subcategory of \(\Top\) (Proposition 4.4.2).

Remark 4.1.2.

The compatibilities in the definition of \(\CSet\) can be reformulated as the statement that we are considering presheaves in sets on \(\Prof\) which are sheaves for the following Grothendieck topology: a family \(\{U_i \to S\}_{i \in I}\) of morphisms is a covering if and only if there is a finite subset \(J\) of \(I\) such that \(\bigsqcup_{i \in J} U_i \to S\) is surjective as a morphism in \(\Top\text{.}\) (The point is that this topology is generated by finite disjoint unions and arbitrary surjections.)
The fact that we are considering sheaves has a consequence which is standard but worth recalling here: while a monomorphism in \(\CSet\) is precisely the same thing as a pointwise injection, an epimorphism in \(\CSet\) is not necessarily a pointwise surjection. Rather, a morphism \(Y \to X\) in \(\CSet\) is an epimorphism if and only if the following weaker condition holds: for every \(S \in \Prof\) and every \(x \in X(S)\text{,}\) we can find a surjection \(T \to S\) in \(\Prof\) such that \(x\) lifts to some \(y \in Y(T)\text{,}\) which need not equalize the maps \(Y(T) \rightrightarrows Y(T \times_S T)\text{.}\) In particular, while the objects of \(\Prof\) do formally form a set of compact generators of \(\CSet\text{,}\) they are not projective objects of \(\CSet\text{.}\) (Compare this to Proposition 1.3.4, where there is no sheafification involved.)
That said, at some point I will start referring to monomorphisms and epimorphisms in \(\CSet\) as “injections” and “surjections”. Do not read too much into this!

Remark 4.1.3.

Evaluation at a fixed singleton in \(\Prof\) defines a functor \(\CSet \to \Set\text{,}\) which we will denote by \(X \mapsto X(*)\text{.}\) We will interpret this as an “underlying set” construction, although this should be interpreted carefully as the functor is not faithful; that is, one cannot view condensed sets as “sets with extra structure”.

Remark 4.1.4.

The functor \(\Prof \to \CSet\) does not preserve colimits. For example, any object of \(\Prof\) can be viewed as a colimit over its closed subspaces of cardinality \(\leq \kappa\text{,}\) and this remains valid in \(\Top\text{,}\) but this description no longer applies in \(\CSet\text{.}\)
Before proceeding, we first enrich our collection of examples of condensed sets.

Proof.

The only issue is to check the compatibilities in the definition of \(\CSet\text{.}\) Compatibility with disjoint unions is clear because disjoint unions in \(\Top\) are coproducts. For compatibility with surjections, note that this is obvious at the level of sets, and we can use Lemma 3.2.7 to promote the conculsion to \(\Top\text{.}\)

Proof.

The composition \(\Set \to \Top \to \CSet\) carries \(X\) to the functor taking \(S \in \Prof\) to \(\Hom_{\Top}(S, X)\text{,}\) which is the set of closed-open partitions of \(S\) indexed by finite subsets of \(X\text{.}\) Hence for any \(X,Y \in \Set\text{,}\) any natural transformation between the functors \(S \mapsto \Hom_{\Top}(S, X)\) and \(S \mapsto \Hom_{\Top}(S, Y)\) on \(\Prof\) can be reconstructed from its effect on all singletons, or equivalently on one singleton.

Remark 4.1.7.

It is worth noting that there is no cardinality cutoff in Corollary 4.1.6. That is, \(\CSet\) contains the entire category \(\Set\) as a full subcategory; the effect of the cardinality cutoff is only felt in “topological” ways.
For instance, in light of Example 3.2.3, we have the object \(\NN_\infty \in \Prof\text{,}\) at which we can then evaluate any condensed set. For an object \(X \in \Top\text{,}\) this gives the collection of “limited” convergent sequences in \(X\) (i.e., if \(X\) is not Hausdorff we must specify a choice of the limit); we thus obtain an arrow-theoretic interpretation of sequential convergence. In particular, this test object can distinguish an object of \(\Top\) from its underlying set viewed as a discrete topological space.
By contrast, since we are now assuming \(\kappa = \aleph_0\) we will have no comparable construction to study convergence of larger nets. This will then limit the extent to which topological spaces embed into \(\CSet\text{,}\) but not to an extent that will impede our work (Proposition 4.4.2).
Here is a key example of a condensed set not in the essential image of \(\Top\text{.}\)

Proof.

By hypothesis we have a finite covering of \(T \times_S T\) such that \(f_1-f_2\) is constant on each term. Since the topology on \(T \times_S T\) is generated by products \(U_1 \times_S U_2\) where \(U_j \subseteq T\) is closed-open, we may assume our covering is given by closed-open subsets of this form. We can then choose a finite partition \(\{V_i\}_{i \in I}\) of \(T\) such that every \(U_j\) is a finite union of some of the \(V_i\text{.}\) We can further refine this partition to ensure that the images of the \(V_i\) in \(S\) form a covering by closed-open subsets which is then refined by some finite partition; since our desired conclusion may be checked locally on \(S\text{,}\) we may reduce to the case where each \(V_i\) surjects onto \(S\text{.}\)
At this point, for any \(i,j \in I\text{,}\) \(V_i \times_S V_j\) is nonempty and \(f_1 - f_2\) is constant on \(V_i \times_S V_j\text{;}\) let \(c_{ij} \in X\) denote this constant value. By the same token, for any \(i,j,k \in I\text{,}\) \(V_i \times_S V_j \times_S V_k\) is nonempty and so
\begin{equation} c_{ik} = c_{ij} + c_{jk}\text{.}\tag{4.1} \end{equation}
Now choose an index \(i_0 \in I\) and define a locally constant function \(g \colon T \to X\) by specifying that \(g(t_i) = c_{ii_0}\) for all \(t_i \in V_i\text{.}\) By construction, for any \((t_i, t_j) \in V_i \times_S V_j\text{,}\) using (4.1) we find that
\begin{equation*} g(t_i) - g(t_j) = c_{ii_0} - c_{ji_0} = c_{ij} = f(t_i) - f(t_j)\text{.} \end{equation*}
Consequently, \(h := f - g\) factors through \(S\text{.}\)

Example 4.1.9.

Let \(F \in \Fun(\Prof^{\op}, \Set)\) be the functor taking \(S\) to the quotient of the group of continuous maps \(S \to \RR\) by the subgroup of locally constant maps. It is clear that this is compatible with finite disjoint unions. To check compatibility with a surjection \(T \to \RR\text{,}\) we must check that if \(f\colon T \to \RR\) is continuous and the two maps \(f_1, f_2\colon T \times_S T \to \RR\) differ by a locally constant map, then \(f\) can be written as a locally constant map plus a map that factors through \(S\) (which will then necessarily be continuous); this follows from Lemma 4.1.8.
When \(S\) is a singleton, we have \(F(S) = \{0\}\text{,}\) so if \(F\) were in the essential image of \(\Top\) it would arise from a singleton space. However, we can compute some other evaluations to see that this is far from what is going on.
  • For \(S = \NN_\infty\text{,}\) \(F(S)\) consists of convergent sequences modulo eventually constant sequences, or equivalently, null sequences modulo sequences of finite support.
  • For \(S = \{0,1\}^\NN\text{,}\) we can produce a nontrivial element of \(F(S)\) using the standard “middle third” representation in \(\RR\text{:}\)
    \begin{equation*} (s_1, s_2, \dots) \mapsto \sum_{n=1}^\infty (2s_n) 3^{-n}\text{.} \end{equation*}
    This defines a continuous map \(S \to \RR\) which is not locally constant.

Subsection 4.2 Free resolutions of topological spaces

While by definition one can only evaluate condensed sets on test objects in the category of profinite sets, we can use the functor \(\Top \to \CSet\) to evaluate condensed sets on arbitrary topological spaces. To quantify the extent to which this retains information from \(\Top\text{,}\) we use the following concept.

Definition 4.2.1.

By a free resolution of an object \(S \in \CHaus\text{,}\) we will mean a coequalizer diagram of the form
\begin{equation*} S_2 \rightrightarrows S_1 \to S \end{equation*}
in which \(S_1, S_2 \in \Prof\text{.}\)
Given such a diagram, for any \(X \in \CSet\) we can define the evaluation \(X(S)\) as an equalizer:
\begin{equation*} X(S) \to X(S_1) \rightrightarrows X(S_2)\text{.} \end{equation*}
By construction, this reproduces the previous definition when \(X \in \Prof\text{.}\)

Remark 4.2.2.

For any \(S \in \CHaus\text{,}\) we can form a surjection \(S_1 \to S\) with \(S_1 \in \Prof\) by taking an inverse limit over finite coverings by compact neighborhoods in some countable basis. More precisely, for any two finite coverings \(\{U_i\}_{i \in I}\) and \(\{V_j\}_{j \in J}\text{,}\) if the second covering refines the first, then we make one choice of a function \(f\colon J \to I\) such that \(V_j \subseteq U_{f(j)}\) for all \(j\) (the final construction will depend on these choices).
Now note that \(S_1\) also belongs to \(\CHaus\) (it again has a countable neighborhood basis), as then does the fiber product \(S_1 \times_S S_1\) (as a closed subset of the absolute product). We can thus repeat the construction to obtain a surjection \(S_2 \to S_1 \times_S S_1\) and hence a free resolution of \(S\text{.}\)
In connection with Remark 3.2.4, note that if \(S\) is totally disconnected, then \(S_1 \to S\) is an isomorphism. That is, this construction completes the identification of \(\Prof\) with the full subcategory of totally disconnected objects of \(\CHaus\text{.}\)

Example 4.2.3.

Using binary representations we get a surjection from the abstract Cantor set \(S_1 := \{0,1\}^n\) to the closed interval \(S := [0,1]\text{.}\) Explicitly, this is the map \((s_1,s_2,\dots) \mapsto \sum_{n=1}^\infty s_n 2^{-n}\text{.}\) This map is bijective over all points except dyadic rationals in \((0,1)\text{,}\) whose fibers consist of two points. We can thus write \(S_1 \times_S S_1\) as a copy of the diagonal \(S_1\) plus two copies of \((0,1) \cap \ZZ[\tfrac{1}{2}]\text{,}\) topologized in a certain way that makes the whole thing compact. The result can in turn be covered by one copy of \(S_1\) (to cover the diagonal) and two copies of \([0,1]\) (to cover the remaining points together with their accumulation points on the diagonal), or in turn by three copies of \(S_1\text{.}\)

Proof.

We already know formally that the restriction of this functor to \(\Prof\) is fully faithful. Using the existence of a free resolution, we may extend this to \(\CHaus\text{.}\)

Remark 4.2.5.

Another way to interpret Proposition 4.2.4 is to say that we can also interpret \(\CSet\) as the category of sheaves on \(\CHaus\) for the Grothendieck topology generated by finite coverings. When working in the context of nonarchimedean analysis/geometry the interpretation in terms of \(\Prof\) is perhaps more natural; however, one of the points of condensed mathematics is to prove access to archimedean analysis/geometry on an equal footing, and in the archimedean setting one certainly would like access to test objects like the closed unit interval.
Going further, one even dreams of doing some sort of global analysis/geometry that bridges between archimedean and nonarchimedean, and in that context one might like access to even more exotic test objects like the idele class group of a number field.

Remark 4.2.6.

The forgetful functor \(\CHaus \to \Set\) admits a left adjoint \(\beta\colon \Set \to \CHaus\) known as Stone-Čech compactification. Concretely, this is the functor
\begin{equation*} S \mapsto \MaxSpec \FF_2[S] \subset \Spec \FF_2[S] \end{equation*}
for the usual Zariski topology on \(\Spec \FF_2[S]\text{;}\) maximal ideals of \(\FF_2[S]\) are classically known as ultrafilters on the sets \(S\text{.}\)
If the underlying set of \(S\) has cardinality \(\leq \kappa\text{,}\) this gives a canonical covering of \(S\) by an object of \(\Prof\text{.}\) However, in general even if \(S \in \Prof\) its underlying set may be as large as \(2^\kappa\) (Remark 3.2.5).

Subsection 4.3 Condensed sets and topological spaces

Definition 4.3.1.

We factor the forgetful functor \(\CSet \to \Set\) through \(\Top\) as follows. Pick any singleton set \(*\) and write the forgetful functor \(\CSet \to \Set\) as \(X \mapsto X(*)\text{.}\) Now equip \(X(*)\) with the finest topology (i.e., the one with the most open sets) such that for any \(S \in \Prof\) and any element of \(X(S)\text{,}\) the resulting map \(S \cong S(*) \to X(*)\) is continuous as a map of topological spaces.
For \(X \in \Prof\) we formally recover the correct topology on the underlying set. Using Remark 4.2.2 we may upgrade this conclusion to cover \(X \in \CHaus\text{.}\)
This construction can be viewed as starting with the embedding \(\Prof \to \Top\) and then making a left Kan extension to \(\CSet\text{.}\) One thing that makes this interesting is colimits in \(\Top\) and \(\CSet\) may disagree (Remark 4.1.4). A consequence of this will be that \(\Top \to \CSet\) is not fully faithful.
However, \(\Top \to \CSet\) will become fully faithful if we restrict to “small enough” topological spaces (Proposition 4.4.2). Among other things, this will provide a mechanism by which condensed sets detect some sophisticated invariants of algebraic topology; see Section 8.

Proof.

Denote the functors as \(F\colon \CSet \to \Top\) and \(G\colon \Top \to \CSet\text{.}\) The assertion is that for \(S \in \CSet, X \in \Top\text{,}\) there is a natural bijection \(\Hom_{\Top}(F(S), X) \to \Hom_{\CSet}(S, G(X))\) specified by a pair of natural transformations \(\id \to G \circ F\text{,}\) \(F \circ G \to \id\text{.}\)
For the first composition, starting with \(X \in \CSet\text{,}\) for each \(S \in \Prof\) we must map \(X(S)\) to \((G \circ F(X))(S) = \Hom_{\Top}(S, F(X))\text{.}\) Given an element of \(X(S)\text{,}\) restriction along inclusions of singletons defines a set-theoretic map \(S \to F(X)\text{;}\) the definition of \(F\) ensures that this map is continuous.
For the second composition, starting with \(X \in \Top\text{,}\) we have a natural bijection \((F \circ G)(X) \to X\) which we must check is continuous. By the definition of \(F\text{,}\) it suffices to check that precomposing this map with any map out of an object of \(\Prof\) still yields a continuous map, which follows from the definition of \(G\text{.}\)

Remark 4.3.3.

Note that the adjunction in Proposition 4.3.2 is reversed from what might naively expect. A related point is that the functor \(\CSet \to \Top\) does not commute with products; this will come back to haunt us when we start considering algebraic structures on condensed sets, as in Section 5.

Subsection 4.4 Sequential topological spaces

We next identify the topological spaces which we can view faithfully as condensed sets.

Definition 4.4.1.

A space \(X \in \Top\) is first countable if every point has a countable neighborhood basis. By contrast, \(X\) is second countable if \(X\) itself has a countable neighborhood basis.
The space \(X\) is sequential if every subset of \(X\) which is sequentially closed (i.e., contains the limit of any convergent sequence with terms in the subset) is actually closed. An equivalent condition is that \(X\) can be written as a quotient of a first countable space (although in practice all reasonable examples are already first countable).
Let \(\Top^{\seq}\) denote the full subcategory of \(\Top\) consisting of sequential spaces. This contains \(\CHaus\) as a full subcategory by the definition of the latter.

Proof.

The composition \(\Top \to \CSet \to \Top\) induces a canonical equivalence at the level of underlying sets, so it suffices to check that the topology does not change when we start with a sequential space. This follows from the fact that a map between sequential topological spaces is continuous if and only if it is sequentially continuous.

Proof.

As per Remark 4.1.2, we need to know that for any \(S\in \Prof\) and any \(y \in Y(S)\text{,}\) we can find a surjection \(T \to S\) in \(\Prof\) such that \(y\) lifts to some \(x \in Y(T)\text{.}\) To begin with, \(X_1 \to Y\) is an epimorphism and so we can find a surjection \(T_1 \to S\) in \(\Prof\) such that \(y\) lifts to some \(x_1 \in X_1(T_1)\text{.}\) We can then find a surjection \(T_2 \to T_2\) such that \(x_1\) lifts to some \(x_2 \in X_2(T_2)\text{.}\) Repeating, we obtain an inverse system \(\varprojlim_{n \in \NN} T_n\) in \(\Prof\) such that \(y\) admits a compatible sequence of lifts to \(X_n(T_n)\) for each \(n\text{.}\) This inverse system formally admits a limit \(T \in \Prof\) such that \(y\) lifts to some \(x \in X(T)\text{.}\)

Proof.

Apply Proposition 4.4.3 to the inverse system
\begin{equation*} \prod_{n' \lt n+1} X_{n'} \times \prod_{n' \geq n+1} Y_{n'} \to \prod_{n' \lt n} X_{n'} \times \prod_{n' \geq n} Y_{n'} \qquad (n=1,2,\dots) \end{equation*}
and the identity map out of the last term.

Remark 4.4.5.

Proposition 4.4.3 cannot be weakened to omit the condition that the transition maps are epimorphisms. This can be seen using an example based on discrete sets; for instance, let \(Y\) be a singleton set; set \(X_n = \{n, n+1, \dots\}\text{;}\) and form the inverse system \(\varprojlim_{n \in \NN} X_n\) using the inclusions \(X_{n+1} \to X_n\) (which has empty inverse limit).
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