Proper maps, quasi-adjoints and the Smyth hyperspace

This month, I would like to talk about proper maps, a nifty equivalent characterization through something I call quasi-adjoints, with a few non-trivial applications. This is based on [4, Section 12].

Proper maps

There are several definitions of what a proper map is.

In classical (Hausdorff) topology, a proper map is a continuous, closed map such that the inverse of every one-point set is compact. (A map is closed if and only if the direct image of every closed set is closed.) Sometimes proper means continuous and the inverse image of a compact set is compact. The two definitions are equivalent in locally compact, Hausdorff spaces.

In non-Hausdorff topology, and following [1, paragraph after Lemma V-5.19], a proper map denotes a continuous map f : X → Y that:

  • is closed (with a modified meaning, compared to the one given above): for every closed subset F of X, ↓f[F] is closed in Y,
  • and such that f–1(Q) is compact for every compact saturated subset Q of Y.

Although that is not completely obvious, this definition generalizes the first definition of proper maps in classical topology. This is due to an equivalent characterization of the definition we just gave of proper maps (in non-Hausdorff topology): f is proper if and only if ↓f[F] is closed for every closed subset F of X and r−1(↑y) is compact for every y ∈ Y. This is [1, Lemma VI-6.21], and I will restate it and prove it as part of Theorem A below.

Now what is the use of proper maps?

There is a close notion, which I have talked about in the book. A map is perfect if and only it is continuous and satisfies the second property above (Definition 9.4.1 in the book), not necessarily the first one (being closed). That is a generalization of the first notion of proper maps we gave in classical (Hausdorff) topology.

If X is well-filtered and Y is locally compact, then every perfect map is closed, hence proper (Exercise 9.4.7 in the book), so the two notions are equivalent in that case; but they may differ otherwise.

A map between stably compact spaces is perfect (hence, equivalently, proper) if and only if it is patch-continuous and monotonic (Proposition 9.4.5 in the book), and this leads to an equivalence of categories, between the category of compact pospaces and continuous maps on the one hand and the category of stably compact spaces and perfect (equivalently, proper) maps on the other hand (Proposition 9.4.10 in the book).

As another application, let us mention that if the image f[X] of a stably compact space X by a proper map f is T0, then it is stably compact (Exercise 9.4.9 in the book).

Yet another application has to do with projective limits. For example, a projective limit of a projective system (pij : Xj → Xi)i⊑j∈I of sober spaces Xi connected by proper maps pij has projection maps pi : X → Xi that will all be proper [4, Proposition 12.12]; and this is proved by using the technology of quasi-adjoints I will describe in this post. Also, if each Xi is equipped with a tight valuation νi in a consistent way, meaning that the push-forward valuation of νj along pij is equal to νi for all i⊑j∈I, then there is a unique tight valuation ν on X of which each νi is the push-forward along pi; see Proposition 4.1 of [4], which is a reference to [5, Theorem 10.1]. I will not explain what this all means, and I am only mentioning it in order to give an idea why proper maps may be useful.

The Smyth hyperspace monad

We will see that the study of proper maps benefits from working with the Smyth hyperspace monad Q on the category Top of topological spaces and continuous maps. We have already mentioned it in the June 2021 post or the August 2020 post for example.

On objects, QX is the space of compact saturated subsets of X, for every space X, with the upper Vietoris topology: The open subsets of QX are the unions of sets of the form ☐U ≝ {Q ∈ QX | Q ⊆ U}, where U is open in X. In the August 2020 post, I stated that Q is a monad, and I gave its unit: for every x ∈ X, ηX(x) ≝ ↑x (the upward closure of x, which is in QX). It seems that I have not described its action on morphisms: for every continuous map f : X → Y, Qf maps every Q ∈ QX to ↑f[Q]. I have not described its multiplication either, apparently, and this is very simple, too: for every Q ∈ QQX, μX(Q) ≝ ∪Q; or, more explicitly, μX(Q) is the union of all the elements Q of Q.

Showing that ∪Q is compact saturated in X is easy, although the result may seem surprising at first. We consider any directed open cover (Ui)i ∈ I of ∪Q, and we claim that ∪Q is included in one of the open sets Ui (see Proposition 4.4.7 in the book to see why this shows compactness). For every Q ∈ Q, Q is included in ∪Q, hence in ∪i ∈ I Ui. Since Q is compact, and by directedness, Q is included in some Ui; namely, Q ∈ ☐Ui. This shows that (☐Ui)i ∈ I is an open cover of Q. It is easy to see that this open cover is directed, since the ☐ operator is monotonic. Using the fact that Q is compact, Q is then included in some ☐Ui. But then every element Q of Q is in ☐Ui, hence is included in Ui. Therefore ∪Q is included in Ui.

We see that, for every open subset U of X, μX–1(☐U) = ☐☐U, so μX is continuous. Similarly, ηX–1(☐U) = U, so ηX is continuous. For every continuous map f : X → Y, for every open subset V of Y, (Qf)–1(☐V) = ☐f–1 (V), so Qf is continuous, too.

We will need to observe that the specialization (pre)ordering on QX is reverse inclusion ⊇; in particular, QX is T0; and we will repeatedly use the following:

Inverse image trick. Any two continuous maps f and g with a common domain and a common T0 codomain that have the same inverse images of open sets must be equal. It even suffices to check that f and g have the same inverse images of subbasic open sets, for any given subbase of the topology on the codomain.

Proof. For every point x in the common domain, it suffices to show that for every subbasic open subset V of the (T0) codomain, f(x) ∈ V if and only if g(x) ∈ V; equivalently, that x ∈ f–1 (V) if and only if x ∈ g–1 (V), and that is obvious since f–1 (V)=g–1 (V) by assumption. ☐

The monad equations are checked as follows.

  • μX o ηQX = idQX: using the inverse image trick, the inverse image of ☐V by the left-hand side is ηQX–1 (μX–1 (☐V)) = ηQX–1 (☐☐V) = ☐V.
  • μX o QηX = idQX: using the inverse image trick, the inverse image of ☐V by the left-hand side is (QηX)–1 (μX–1 (☐V)) = (QηX)–1 (☐☐V) = ☐ (ηX–1 (☐V)) = ☐V.
  • μX o QμX = μX o μQX: the inverse image of every basic open subset ☐V of QX by both sides of the equation is ☐☐☐V, then we rely on the inverse image trick (again).

We should not forget to verify that η and μ are natural transformations. For η, this means that for every continuous map f : X → Y, Qf o ηX = ηY o f; indeed, every point x of X is mapped by the left-hand side to ↑f[↑x] = ↑f(x) (since f, being continuous, is monotonic). For μ, this means verifying that for every continuous map f : X → Y, Qf o μX = μY o QQf. To this end, we use the same inverse image trick as above: for every open subset V of Y, the inverse image of ☐V under Qf o μX is μX–1 ((Qf)–1 (☐V)) = μX–1 (☐ f–1 (V)) = ☐☐ f–1 (V), while the inverse image of ☐V under μY o QQf is (QQf)–1 (μY–1 (☐V)) = (QQf)–1 (☐☐V) = ☐☐ f–1 (V).

Quasi-adjoints

Let us call a quasi-adjoint to a continuous map r : X → Y a continuous map ς : Y → QX such that:

  1. ηY ≤ Qr o ς;
  2. ς o r ≤ ηX.

I had introduced a close notion in [3, Definition 4.1], which I had called a quasi-retraction: r : X → Y is a quasi-retraction if and only if it has an associated quasi-section, namely a continuous map ς : Y → QX such that ηY = Qr o ς. I will mention quasi-retractions just once more below.

The categorical reading

If you are familiar with Kleisli categories, you should see a pattern in the definition of quasi-adjoints. (If you are ill at ease with categories, please proceed to the next subsection.)

A morphism from Y to QX such as ς is by definition the same thing as a morphism from Y to X in the Kleisli category of the monad Q. The identity at an object X in that Kleisli category is ηX. Composition g • f of g and f in the Kleisli category is μ o Qg o f. If the Kleisli category is poset-enriched (namely, its homsets are equipped with an ordering in such a way that Kleisli composition is monotonic, which is the case here), then a (right) adjoint to ς : Y → X (in the Kleisli category, namely ς : Y → QX in the original category) is a morphism ϱ : X → Y (in the Kleisli category, hence ϱ : X → QY in the original category) such that ηY ≤ ϱ • ς and ς • ϱ ≤ ηX. Equivalently, we say that ς is a left adjoint to ϱ.

Every morphism r : X → Y in the original category gives rise to a morphism ηY o r from X to Y in the Kleisli category.

Then a quasi-adjoint to r : X → Y (in the original category, Top) is simply a left adjoint to ηY o r in the Kleisli category of Q. Indeed, let ϱ ≝ ηY o r. Then:

  • ϱ • ς = μY o QηY o Qr o ς = Qr o ς, so ηY ≤ ϱ • ς is equivalent to condition 1: ηY ≤ Qr o ς.
  • ς • ϱ = μX o Qς o ηY o r = μX o ηX o ς o r (by naturality of η) = ς o r, so ς • ϱ ≤ ηX is equivalent to condition 2: ς o r ≤ ηX.

The concrete reading

Condition 1 means that for every y ∈ Y, ↑y should contain (remember, the specialization ordering is reverse inclusion) Qr (ς (y)) = ↑r[ς (y)], or equivalently that:

1′. for every x ∈ ς (y), y ≤ r(x).

Condition 2 means that for every x ∈ X, ς (r (x)) should contain ↑x, or equivalently that:

2′. x ∈ ς (r (x)).

Proper = has a quasi-adjoint

At last we arrive at the point where we can connect all the dots. The important part is the equivalence between 1 and 3 in the theorem below. The equivalence between 1 and 2 is well-known [1, Lemma VI-6.21], and we have already mentioned it, but it is worth stating here.

Theorem A. For a continuous map r : X → Y, the following are equivalent:

  1. r is a proper map;
  2. ↓r[F] is closed for every closed subset F of X and r−1(↑y) is compact for every y ∈ Y;
  3. r has a quasi-adjoint.

The quasi-adjoint ς, if it exists, is uniquely determined by ς(y)=r−1(↑y) for every y ∈ Y.

Proof. 2 ⇒ 1. Let Q be a compact saturated subset of Y. We must show that r−1(Q) is compact. To this end, let (Ui)i ∈ I be a directed open cover of r−1(Q). For every y ∈ Q, r−1(↑y) is compact by assumption, and included in r−1(Q), so it is included in some Ui, say Uiy. Let Fiy be the complement of Uiy: r−1(↑y) does not intersect Fiy , so ↓r[Fiy] does not contain y. This means that y is in the set Y–↓r[Fiy], which is open by assumption. This holds for every y ∈ Q, so, since Q is compact, a finite number of such sets cover Q. In other words, there is a finite subset E of Q such that Q ⊆ ∪y∈E (Y–↓r[Fiy]). For every x ∈ r−1(Q), there is a y ∈ E such that x ∈ r−1(Y–↓r[Fiy]); we claim that x must then be in Uiy. Indeed, otherwise, x would be in Fiy, so r(x) would be in ↓r[Fiy], leading to a contradiction. Therefore (Uiy)y∈E is a finite subcover of r−1(Q).

In order to prove the other implications, we make the following observation:

(∗) for every map ς : Y → QX such that ς(y)=r−1(↑y) for every y ∈ Y, for every open subset U of X, the complement of ς−1(U) in Y is equal to ↓r[F], where F is the complement of U.

Indeed, for every y ∈ Y, y ∉ ς−1(U) if and only if ς(y)=r−1(↑y) is not included in U, if and only if there is an x ∈ F such that y ≤ r(x).

3 ⇒ 2. Let ς be a quasi-adjoint of r. We claim that ς(y)=r−1(↑y), which will also show the final uniqueness result of the theorem.

For every x ∈ ς(y), y ≤ r(x) by condition 1′ in the concrete definition of quasi-adjoints, so x ∈ r−1(↑y). Conversely, for every x ∈ r−1(↑y), we have y ≤ r(x). Since ς is continuous hence monotonic, ς(y) ⊇ ς(r(x)). By condition 2′, x is in ς(r(x)), hence in ς(y).

It follows that, since ς(y) ∈ QX by assumption, r−1(↑y) is compact. For every closed subset F of X, we consider its complement U. Since ς is continuous, ς−1(U) is open. But its complement is precisely ↓r[F] by (*), so ↓r[F] is closed.

1 ⇒ 3. Let ς(y) ≝ r−1(↑y) for every y ∈ Y. This is compact saturated since r is proper, hence perfect. Hence ς is a map from Y to QX.

We check that ς is continuous. For every open subset U of X, ς−1(U) is the complement of ↓r[F], where F is the complement of U in X, by (*). Since r is proper, ↓r[F] is closed, so ς−1(U) is open.

Let us check condition 1′ of the concrete definition of quasi-adjoints. For every y ∈ Y, (Qr o ς)(y) = ↑r[ς(y)]= ↑r[r−1(↑y)]. Every element y’ of that set is such that y’ ≥ r(x) for some x ∈ X such that r(x) ≥ y, so y’ ∈ ↑y.

Let us check condition 2′. For every x ∈ X, we need to check that x ∈ ς(r(x)) = r−1(↑r(x)), or equivalently that r(x) ≥ r(x), which is obvious. ☐

The equivalence between items 1 and 3 in Theorem A may seem to indicate that, in order to show that a continuous map r has a quasi-adjoint, you do not need to imagine what the quasi-adjoint may be, and you just have to check that r is proper. We will see that, perhaps rather curiously, it is sometimes easier to find the quasi-adjoint directly, and then deduce that r is proper.

Before that, let me come back briefly to quasi-retractions. Lemma 4.11 of [3] states that given a continuous map r : X → Y where Y is T0, r is proper and surjective if and only if it is a quasi-retraction satisfying property 2′. Let me restate this as follows.

Lemma A(bis). For a continuous map r : X → Y where Y is T0, the following are equivalent:

  1. r is proper and surjective;
  2. there is a continuous map ς : Y → QX such ηY = Qr o ς (instead of just ≤ for quasi-adjoints) and ς o r ≤ ηX.

Proof. 2 ⇒ 1. Assuming 2, ς is in particular a quasi-adjoint to r, so r is proper by Theorem A. For every y ∈ Y, ηY(y) = ↑y is equal to (Qr o ς) (y) = ↑r[ς(y)] = ↑r[r−1(↑y)], because ηY = Qr o ς and since ς(y)=r−1(↑y), as stated at the end of Theorem A. In particular, y, which is in ↑y, is in ↑r[r−1(↑y)], hence is larger than or equal to r(x) for some x ∈ X such that r(x)≥y. Since Y is T0, y≥r(x)≥y implies y=r(x). Therefore r is surjective.

1 ⇒ 2. By Theorem A, since r is proper it has a quasi-adjoint ς. Hence ηY ≤ Qr o ς and ς o r ≤ ηX. We reinforce ηY ≤ Qr o ς to an equality. To this end, we need to show that ηY ≥ Qr o ς, namely that for every y ∈ Y, ηY(y) = ↑y is included in (remember, the ordering on Smyth hyperspaces is reverse inclusion) (Qr o ς) (y) = ↑r[ς(y)] = ↑r[r−1(↑y)]. Since the latter is upwards-closed, it suffices to show that y ∈ ↑r[r−1(↑y)]. Profiting from the fact that r is surjective, we write y as r(x), for some x ∈ X. Then x ∈ r−1(↑y), and its image under r is certainly less than or equal to y (being, in fact, equal to y). ☐

Applications

Our first application is the following.

Proposition B. The functor Q preserves proper maps: for every proper map r : X → Y, Qr : QX → QY is proper.

Proof. Relying on the equivalence between items 1 and 3 of Theorem A, we know that r has a quasi-adjoint ς : Y → QX, and we will simply find a quasi-adjoint to Qr. We claim that this quasi-adjoint is ς’ ≝ ηQX o μX o Qς. Note that ς’ is already a morphism, namely a continuous map from QY to QQX, by definition. (Oh, why not take Qς for ς’? Well, simply because that would not work. Let us note that since ηQX o μX o Qς will turn out to the the quasi-adjoint, and quasi-adjoints are unique, no other candidate can work. But I agree that may seem mysterious for now.)

We will reason categorically. Q is a poset functor, meaning that for all morphisms f and g witht the same domain and the same codomain, if f≤g then Qf≤Qg. Indeed, for every compact saturated subset Q of the domain, Qf(Q) = ↑f[Q] contains Qg(Q) = ↑g[Q], and we remember that the specialization ordering on Smyth hyperspaces is reverse inclusion. (In fact, Q is even a right KZ-monad, see the August 2022 post.) We check conditions 1 and 2 of the definition of a quasi-adjoint:

  1. By condition 1 of the definition of quasi-adjoints, ηY ≤ Qr o ς. Hence QQr o ς’ = QQr o ηQX o μX o Qς = ηQY o μY o QQr o QQς (by naturality of η and μ) ≥ ηQY o μY o QηY = ηQY (by one of the monad laws).
  2. By condition 2, ς o r ≤ ηX. Hence ς’ o Qr = ηQX o μX o Qς o Qr ≤ ηQX o μX o QηX = ηQX (by one of the monad laws). ☐

Proposition B has a converse, so we have the following.

Proposition C. For every continuous map r : X → Y, Qr : QX → QY is proper if and only if r is proper.

Proof. We assume Qr proper. Hence it has a quasi-adjoint ς’ : QY → QQX. Let ς ≝ μX o ς’ o ηY. Then:

  1. Qr o ς = Qr o μX o ς’ o ηY = μY o QQr o ς’ o ηY (by naturality of μ) ≥ μY o ηQY o ηY (by condition 1 defining ς’ as a quasi-adjoint) = ηY (by one of the monad laws).
  2. ς o r = μX o ς’ o ηY o r = μX o ς’ o Qr o ηX (by naturality of η) ≤ μX o ηQX o ηX (by condition 2 for ς’) = ηX (by one of the monad laws).

Hence ς is a quasi-adjoint to r, and we conclude by using Theorem A. ☐

We play a similar game with the Hoare hyperspace monad H. We will only look at its functor part: for every space X, HX is the collection of closed subsets of X, with the lower Vietoris topology, generated by the subbasic open sets ♢U ≝ {C ∈ HX | C intersects U}, where U ranges over the open subsets of X; for every continuous map f : X → Y, Hf maps every C ∈ HX to cl(f[C]). We note that HX is T0 (in fact sober), with inclusion as specialization ordering; also, for every open subset V of Y, (Hf)–1 (♢V) = ♢f–1 (V), an equality that we will use in conjunction with the inverse image trick.

Proposition D. The functor H preserves proper maps: for every proper map r : X → Y, Hr : HX → HY is proper.

Proof. We will use the map σX : HQX → QHX that is at the core of the de Brecht-Kawai theorems on hyperspace commutation (see the October 2022 post): σX is defined as mapping every C ∈ HQX to the collection of closed subsets C of X that intersect every Q ∈ C. That collection is compact saturated, and σX is not just continuous, but even a topological embedding, and for every open subset U of X, σX−1(☐♢U) = ♢☐U, as we have seen in the aforementioned post. The sets ☐♢U with U open in X form a subbase of the topology on QHX; this is a consequence of the fact that ☐ commutes with directed unions and finite intersections, and that ♢ commutes with (finite) unions.

We claim that:

  • σ is a natural transformation; namely, for every morphism f : A → B, σB o HQf = QHf o σA. We use the inverse image trick: (σB o HQf)−1 (☐♢V) = (HQf)−1 (σB−1 (☐♢V)) = (HQf)−1 (♢☐V) = ♢☐f−1 (V), while (QHf o σA)−1 (☐♢V) = σA−1 ((QHf)−1 (☐♢V)) = σA−1 (☐♢f−1 (V)) = ♢☐f−1 (V).
  • Compatibility with the Smyth unit: for every space A, σA o HηA = ηHA. We again use the inverse image trick: the inverse image of ☐♢V by the left-hand side is (HηA)−1 (σA−1 (☐♢V)) = (HηA)−1 (♢☐V) = ♢ηA−1 (☐V) = ♢V.
  • H is a poset functor, namely f≤g implies Hf≤Hg. In fact, we have already seen that it is a left KZ-monad, see the August 2022 post; otherwise, I’ll let you verify it.

By Theorem A, r has a quasi-adjoint ς. Let ς’ ≝ σX o Hς : HY → QHX. We verify that ς’ is a quasi-adjoint to Hr:

  1. We have QHr o ς’ = QHr o σX o Hς = σY o HQr o Hς (by naturality of σ) ≥ σY o HηY (by condition 1 of quasi-adjoints, namely ηY ≤ Qr o ς, plus the fact that H is a poset functor) = ηHY (by compatibility with the Smyth unit).
  2. ς’ o Hr = σX o Hς o Hr ≤ σX o HηX (by condition 2 of quasi-adjoints, ς o r ≤ ηX, plus the fact that H is a poset functor) = ηHX (by compatibility with the Smyth unit). ☐

The essence of the above proof abstracts to the following. We can replace Top by any full (necessarily poset-enriched) subcategory that is closed under the Q functor, namely on which Q defines a monad.

Theorem E. Let F be any poset functor on a full subcategory of Top that is closed under Q. If there is a natural transformation σ : FQ → QF such that (compatibility with the Smyth unit:) for every object A of the subcategory, σA o FηA = ηFA, then F preserves proper maps.

This is in fact what we did in Proposition B, too, where we took σA ≝ ηQA o μA : QQA → QQA. The compatibility property with the Smyth unit reads ηQA o μA o QηA = ηQA, which is a trivial consequence of the monad laws.

That applies to various other functors. For example, there is a poset functor V that maps every topological space to its space of continuous valuations, with the weak topology, generated by subbasic open sets [U > r] ≝ {ν ∈ VX | ν(U)>r}, where U ranges over the open subsets of X and r over the non-negative real numbers. Continuous valuations are a notion that is very close to measures, but they behave better with respect to topology. I have talked a bit about continuous valuations here. The action of V on morphisms is given by: for every continuous map f : X → Y, for every ν ∈ VX, for every open subset V of Y, Vf(ν) = ν(f–1(V)). The specialization ordering on VX is given by μ≤ν if and only if μ(U)≤ν(U) for every open subset U of X.

It is a pretty non-trivial fact that there is a natural transformation σ : VQ → QV, which maps every continuous valuation ξ on QX (for every space X) to the (compact saturated) collection of continuous valuations ν on X such that ν(U) ≥ ξ(☐U) for every open subset U of X. This stems from various papers of mine, and I may talk about all that another time. (In fact σ is even a weak distributive law of Q over V, and again, I will refrain from saying what this is; you may have a look at [2, Section 4] if you are brave.) The point is that the equality σA o VηA = ηVA holds. Indeed, for every μ ∈ VA, ξ ≝ VηA(μ) is the continuous valuation on QA such that ξ(V) = μ(ηA–1(V)) for every open subset V of QA. Writing V as a union ∪i ∈ I ☐Ui where each Ui is open in A, ξ(V) = μ(∪i ∈ I ηA–1(☐Ui)) = μ(∪i ∈ I Ui). In particular, when V is of the form ☐U, ξ(☐U) = μ(U), and therefore (σA o VηA) (μ) is the collection of continuous valuations ν such that ν(U) ≥ μ(U) for every open subset U of X; in other words, the upward closure of μ in VA, namely ηVA(μ).

Using Theorem E, it follows that [4, Theorem 12.11]:

Proposition F. V preserves proper maps on Top.

And that will be enough for this time! I will be on holidays during most of the month of August, and therefore there will probably be no post next month, or only a short one, maybe, who knows.

  1. Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. Continuous Lattices and Domains. Number 93 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003.
  2. Jean Goubault-Larrecq. Weak Distributive Laws between Monads of Continuous Valuations and of Non-Deterministic Choice. https://arxiv.org/abs/2408.15977, August 2024.
  3. Jean Goubault-Larrecq.  QRB-Domains and the Probabilistic Powerdomain. Logical Methods in Computer Science, February 29, 2012, Volume 8, Issue 1. https://doi.org/10.2168/LMCS-8(1:14)2012
  4. Jean Goubault-Larrecq. On the Preservation of Projective Limits by Functors of Non-Deterministic, Probabilistic, and Mixed Choice. https://arxiv.org/abs/2407.10235, July 2024.
  5. Jean Goubault-Larrecq. Products and projective limits of continuous valuations on T0 spaces. Mathematical Structures in Computer Science 31(2):234–254, 2021.
jgl-2011

— Jean Goubault-Larrecq (July 20th, 2025)