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Notes on condensed mathematics

Kiran S. Kedlaya

Section 3 Profinite sets

We describe the category of profinite sets, which will provide the test objects for the definition of condensed sets.
As in Section 1, we fix an infinite cardinal \(\kappa\text{.}\) In subsequent sections we will make the blanket assumption \(\kappa = \aleph_0\text{;}\) this will correspond to restricting attention to “light” condensed sets in the terminology of [8].

Reference.

This lecture is based on [8], Lectures 1 and 2. See also [6], Lecture 1.

Subsection 3.1 Pro-objects in a category

Definition 3.1.1.

A directed set is a partially ordered set \((I, \leq)\) with the property that for any \(i,j \in I\text{,}\) the set \(\{k \in I\colon k \geq i, k \geq j\}\) is nonempty. We may view a directed set as a category in which there is a single morphism \(i \to j\) if \(i \leq j\) and none otherwise.
Given a category \(\calC\text{,}\) we define an associated category \(\varprojlim_\kappa \calC\) as follows. The objects of \(\varprojlim_\kappa \calC\) (also called pro-objects of \(\calC\)) are the \(\kappa\)-small inverse systems, i.e., contravariant functors to \(\calC\) out of directed sets of cardinality \(\leq \kappa\text{.}\) The homsets are given by the formula
\begin{equation*} \Hom_{\varprojlim_\kappa \calC}\left(\varprojlim_{i \in I} X_i, \varprojlim_{j \in J} Y_j\right) = \varprojlim_{j \in J} \left( \varinjlim_{i \in I} \Hom_{\calC}(X_i, Y_j) \right)\text{.} \end{equation*}
Another way to interpret \(\varprojlim_\kappa \calC\) is as the subcategory of \(\Fun(\calC, \Set)^{\op}\) consisting of cofiltered limits of representable functors.
Note that for formal reasons, \(\varprojlim_\kappa \calC\) admits \(\kappa\)-small limits. Moreover, any functor \(\calC \to \calC'\) whose target admits \(\kappa\)-small limits induces a canonical inverse limit functor \(\varprojlim_\kappa \calC \to \calC'\text{.}\) For instance this applies when \(\calC\) admits a forgetful functor to \(\Set\text{.}\)

Example 3.1.2.

In the category \(\varprojlim_\kappa \Ab\text{,}\) consider the inverse system of \(\ZZ/p^n \ZZ\) over all positive integers \(n\text{,}\) which has inverse limit \(\ZZ_p\text{.}\) According to the definition, the family of reduction maps \(\ZZ/p^{2n} \ZZ \to \ZZ/p^n \ZZ\) constitutes a morphism \(\varprojlim_\kappa \ZZ/p^n \ZZ \to \varprojlim_\kappa \ZZ/p^n \ZZ\text{.}\) However, this is the same morphism in \(\varprojlim_\kappa \Ab\) as the one constituted by the family of identity maps \(\ZZ/p^{n} \ZZ \to \ZZ/p^n \ZZ\text{,}\) which is consistent with the fact that they both induce the identity map on the inverse limits in \(\Ab\text{.}\)
Another way to say this is that if we take the inverse system of \(\ZZ/p^{2n}\ZZ\) over all even positive integers \(n\text{,}\) the resulting pro-object is isomorphic to the one we started with. More generally, this implies if we start with any index set \(I\) and replace it with a cofinal subset \(I'\text{,}\) meaning one such that for any \(i \in I\) there exists \(i' \in I'\) with \(i \leq i'\text{.}\)

Remark 3.1.3.

Building on Example 3.1.2, keep in mind that pro-objects indexed by different partially ordered sets \(I\) can nonetheless be isomorphic; that is, the index set is not an intrinsic datum of a pro-object but only an artifact used to give a concrete description.
This is closely related (via Stone duality; see Remark 3.2.4) to the fact that it is useful to describe a topology on a set in terms of a neighborhood basis, but the neighborhood basis is not an intrinsic datum of the resulting topological space.
Another side effect is that it can be confusing to think about properties of a morphism of pro-objects such as injectivity or surjectivity. It is better to view these as properties of the corresponding morphisms of topological spaces (again see Remark 3.2.4).

Remark 3.1.4.

As an example of Remark 3.1.3, consider the category of pro-systems of abelian groups, for which the inverse limit functor is left exact and hence admits right derived functors. For sequential pro-objects the derived functors beyond \(R^1 \varprojlim\) all vanish ([28], tag 07KW); by Remark 3.1.3 the same applies to any countable pro-object in abelian groups. By contrast, for a larger pro-object it can happen that \(R^i \varprojlim\) is nonzero for \(i \gt 1\text{.}\)

Subsection 3.2 The category of profinite sets

Definition 3.2.1.

The category of profinite sets is defined to be \(\Prof := \varprojlim_\kappa \Set^{\fin}\) where \(\Set^{\fin}\) denotes the full subcategory of \(\Set\) consisting of finite sets.
We introduce two key examples, one “maximal” and one “minimal”.

Example 3.2.2.

By the (abstract) Cantor set, we will mean the object \(\{0,1\}^{\NN}\) of \(\Prof\) indexed by \(\NN\text{,}\) where \(n\) corresponds to \(\{0,1\}^n\) and the projection \(\{0,1\}^{n+1} \to \{0,1\}^n\) is the map that forgets the last coordinate. This example is in a certain sense maximal when \(\kappa = \aleph_0\text{;}\) see Proposition 3.3.6.

Example 3.2.3.

Let \(\NN_\infty\) be the object of \(\Prof\) indexed by \(\NN\text{,}\) where \(n\) corresponds to \(\{0,\dots,n,\infty\}\) and the projection \(\{0,\dots,n+1,\infty\} \to \{0,\dots,n,\infty\}\text{.}\) This example plays an important role in the relationship between topological spaces and condensed sets; see Remark 4.1.7.
An interesting example of a morphism \(S \to \NN_\infty\) is given by the \(p-\)-adic valuation map \(v_p \colon \ZZ_p \to \NN_\infty\text{.}\)
We will freely pass between the given definition of profinite sets and the following alternate interpretation.

Remark 3.2.4. Stone duality, part 1.

Since \(\Set\) admits limits, we obtain a natural functor \(\Prof \to \Set\) taking an inverse system to its inverse limit. Alternatively, by viewing every finite set as a topological space for the discrete topology, we may factor \(\Prof \to \Set\) through the category \(\Top\) of topological spaces.
Now observe that the inverse limit of a family of Hausdorff topological spaces is a closed subspace of the full product over the same family: the compatibility with morphisms is a collection of conditions involving two factors at a time, and hence a closed condition for the product topology. By Tykhonoff’s theorem, any product of compact Hausdorff spaces is compact, so any inverse limit of compact Hausdorff spaces is compact. As a result, the functor \(\Prof \to \Top\) factors through the full subcategory \(\CHaus\) of \(\Top\) consisting of compact Hausdorff spaces admitting a neighborhood basis of cardinality \(\leq \kappa\text{.}\) (When \(\kappa = \aleph_0\text{,}\) Urysohn’s theorem says that these are just the metrizable compact Hausdorff spaces.)
Conversely, an object of \(\CHaus\) belongs to the essential image of \(\Prof \to \Top\) if and only if it is totally disconnected, meaning that any two distinct points can be separated by a partition of the space into two closed-open subsets. Namely, we can write down a corresponding inverse system with terms corresponding to the finite closed-open coverings; see Remark 4.2.2 for the detailed construction in a more general context. One side effect of the construction is that we can represent any object of \(\Prof\) as an inverse system with surjective transition maps, which we will almost always do in practice.

Remark 3.2.5.

Note that the cardinality cutoff \(\kappa\) applies to a neighborhood basis of a totally disconnected compact Hausdorff space, not to the underlying set of the topological space. For example, for \(\kappa = \aleph_0\text{,}\) the Cantor set \(\{0,1\}^{\NN}\) belongs to \(\Prof\) but its underlying set \(\prod_{n=1}^\infty \{0,1\}\) has cardinality \(2^\kappa\text{.}\)

Remark 3.2.6.

The object of \(\CHaus\) corresponding to \(\NN_\infty \in \Prof\) is the one-point compactification of \(\NN\text{.}\) That is, if we interpret \(\NN\) as an object of \(\Top\) for the discrete topology, then \(\NN_\infty\) corresponds to the topological space with underlying set \(\NN \cup \{\infty\}\) in which an open subset of \(\NN\) is either an arbitrary subset of \(\NN\) or \(\infty\) plus a cofinite subset of \(\NN\text{.}\)
In connection with Stone duality, we will make frequent use of the following fact about compact Hausdorff topological spaces, which in a sense makes them quite similar to objects of more algebraic categories.

Proof.

The quotient map property says that a subset \(V\) of \(X\) is closed if its preimage \(f^{-1}(V)\) is closed (the “only if” statement being the property that \(f\) is continuous). The surjectivity of \(f\) means that \(f(f^{-1}(V)) = V\text{;}\) since any closed subset of \(Y\) is again compact, \(V\) is the image of a quasicompact topological space and hence is also quasicompact (we can test whether a finite subcollection of a covering is a covering by checking the inverse images). Since \(X\) is Hausdorff, for any \(x \in X \setminus V\text{,}\) for each \(y \in V\) we can find open subsets separating \(x\) and \(y\text{.}\) We can choose finitely many \(y\) so that the opens covering those \(y\) cover all of \(V\text{;}\) intersecting the corresponding opens covering \(x\) produces an open neighborhood of \(x\) within \(X \setminus V\text{.}\) Since \(x \in X \setminus V\) was arbitrary, we conclude that \(X \setminus V\) is open and hence \(V\) is closed.
We will make occasional use of a third interpretation of profinite sets.

Remark 3.2.8. Stone duality, part 2.

We can also describe \(\Prof\) in terms of Boolean algebras: the space \(X\) corresponds to \(\Hom_{\Top}(X, \FF_2)\) (i.e., the partitions of \(X\) into two closed-open subsets labeled \(0\) and \(1\)). and we recover \(X\) by taking \(\MaxSpec\) (the set of maximal ideals with the Zariski topology). From this point of view, the cardinality cutoff is that \(|\Hom_{\Top}(X, \FF_2)| \leq \kappa\text{.}\)

Subsection 3.3 Effects of the cardinality cutoff

For the remainder of this section, let us assume that \(\kappa = \aleph_0\text{.}\) We spell out some effects of this condition, starting with an important special case of Remark 3.1.3.

Remark 3.3.1.

Now that we are assuming \(\kappa = \aleph_0\text{,}\) every pro-object is isomorphic to an inverse system on the index set \(\NN\text{.}\) In \(\Prof\text{,}\) we can further ensure (by Remark 3.2.4) that the transition maps are surjective.
We will use this repeatedly in the following discussion. However, when describing pro-objects in practice it is sometimes more natural to use an alternate index set, e.g., when forming products the index set \(\NN \times \NN\) occurs naturally.

Proof.

Per Remark 3.3.1, we may write \(S\) as a sequential inverse limit of finite sets with surjective transition maps: \(S = \varprojlim_{n \in \NN} S_n\text{.}\) For each \(n \in \NN, x \in S_n\text{,}\) let \(U_{n,x}\) be the preimage of \(x\) in \(S\text{;}\) this is an open (and closed) subset of \(S\text{,}\) and \(U\) is the union of \(U_{n,x}\) over all pairs \((n,x)\) such that \(U_{n,x} \subseteq U\text{.}\)
Now consider the set of pairs \((n,x)\) such that \(U_{n,x} \subseteq U\) and there is no \(n' \lt n\) for which the image \(x'\) of \(x\) in \(X_{n'}\) satisfies \(U_{n',x'} \subseteq U\) (that is, this pair is “irredundant”). For these pairs, the sets \(U_{n,x}\) form a countable disjoint union of objects of \(\Prof\) with union \(U\text{.}\)

Proof.

Per Remark 3.3.1, we may write \(S\) as a sequential inverse limit of finite sets with surjective transition maps: \(S = \varprojlim_{n \in \NN} S_n\text{.}\) We are given a coherent sequence of maps \(Y \to S_n\) which we wish to factor through \(Z\text{.}\) It will suffice to show that given any maps \(Y \to S_{n+1}\text{,}\) \(Z \to S_n\) which yield the same composition \(Y \to S_n\text{,}\) we can fill in a map \(Z \to S_{n+1}\text{;}\) that is, we can fit a dashed arrow into the following diagram.
Figure 3.3.4.
By partitioning \(Z\) into closed-open subsets, we may reduce to the case where \(S_n\) is a singleton (which means we can ignore the bottom right corner of the diagram).
At this point, we may choose representations \(Y = \varprojlim_i Y_i\text{,}\) \(Z = \varprojlim_i Z_i\) such that \(Y \to Z\) is represented by a sequence of compatible injective morphisms \(Y_i \to Z_i\text{.}\) (More precisely, start with any representation of \(Z\) and let \(Y_i\) be the image of \(Y \to Z \to Z_i\text{;}\) the injectivity of \(Y \to Z\) ensures that \(\varprojlim_i Y_i = Y\text{.}\)) The morphism \(Y \to S_{n+1}\) factors through some \(Y_i\text{;}\) but now it is straightforward to factor \(Y_i \to S_{n+1}\) through \(Z_i\) since all of the sets involved are finite.

Remark 3.3.5.

By contrast, objects of \(\Prof\) are in general not projective unless they are finite. For example, for \(S = \NN_\infty\text{,}\) we have a covering of \(S\) by a disjoint union of two copies of \(\NN_\infty\text{,}\) one mapping via \(n \mapsto 2n\) and the other via \(n \mapsto 2n+1\text{.}\) This covering cannot be split: there are only two set-theoretic splittings, corresponding to choosing one point in the fiber over \(\infty\text{,}\) and in both cases we end up with a sequence in \(\NN_\infty\) mapping to the wrong limit point.
In a similar vein, we have the following. Again, this is not particularly useful in practice but is convenient for some theoretical considerations; it means we can view an object in \(\CSet\) as its value on the Cantor set plus some descent data.

Proof.

Per Remark 3.1.3, we may write \(S\) as a sequential inverse limit of finite sets with finite transition maps: \(S = \varprojlim_{n \in \NN} S_n\text{.}\) It will suffice to show that any surjection \(\{0,1\}^{n_1} \to S_n\) for some \(n_1 \in \NN\) can be lifted to a surjection \(\{0,1\}^{n_2} \to S_{n+1}\) for some \(n_2 \geq n_1\text{;}\) this is obvious because all of the sets involved are finite.
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