Alan Day [1] and Oswald Wyler [2] proved that the algebras of the filter monad on the category **Top _{0}** of T

_{0}topological spaces are exactly the continuous (complete) lattices. Martín Escardó later gave a very interesting proof of this fact [3], using a category-theoretic construction due to Anders Kock [4] which he calls

*KZ-monads*. My purpose is to talk about Escardó’s argument; but mostly, really, to put forward his notion of KZ-monad, which is a true categorical gem. I will show that, although KZ-monads are not as common as general monads, it is always useful to wonder whether a given monad is a KZ-monad, as this simplifies the study of its algebras considerably.

## The filter monad on T_{0}

The monad that Day [1] and Wyler [2] studied, the so-called *filter monad*, is defined as follows. Given any T_{0} topological space *X*, we form its *filter space* **FO***X* as follows:

- The points of
**FO***X*are the non-trivial filters of open subsets of*X*, namely the collections*F*of open subsets of*X*that are non-empty, upwards-closed, closed under binary intersections, and do not contain the empty set (i.e., are not the whole collection**O***X*of open subsets of*X*). Note that, contrarily to Part I of this series of posts, we are considering filters of open sets, not filters of subsets. - The topology of
**FO***X*is generated by the subsets ☐*U*≝ {*F*∈**FO***X*|*U*∈*F*}, where*U*ranges over the open subsets of*X*. Those subsets form a base, not just a subbase, of the topology, since ☐*U*∩ ☐*V*= ☐(*U*∩*V*), owing to the fact that filters are closed under binary intersections.

I will also consider **FO**^{⊤}*X*, the space of all filters (including the trivial filter **O***X*), with a similarly defined topology. Their specialization ordering is inclusion.

We are not requiring the filters to be Scott-open, just filters of open sets. This construction is studied in Exercise 9.3.10 of the book, where it is asked to show that *X* embeds into **FO***X* through η* _{X}* :

*x*↦ {

*U*∈

**O**

*X*|

*x*∈

*U*}, mapping each point to its filter of open neighborhoods; additionally, the image of η

*is dense in*

_{X}**FO**

*X*(with its Scott topology),

**FO**

*X*is an algebraic bc-domain, and its finite elements are the principal filters ■

*U*≝ {

*V*∈

**O**

*X*|

*U*⊆

*V*}. The upward closure of ■

*U*in

**FO**

*X*is what we wrote as ☐

*U*above, so the topology we gave to

**FO**

*X*

*is*the Scott topology. The same happens with

**FO**

^{⊤}

*X*.

In Exercise 9.3.11, it is required to show that every continuous map *f* : *X* → *Z*, where *Z* is a bc-domain (with its Scott topology), extends to a continuous map *f*† : **FO***X* → *Z*, in the sense that *f*† o η* _{X}* =

*f*. (Moreover, there is a largest such extension.) This entails that every bc-domain

*Z*is

*densely injective*, namely that every continuous map

*f*from any space

*X*to

*Z*extends to any superspace of

*X*in which

*X*embeds as a dense subspace; see Exercise 9.3.12, where it is also required to prove that every densely injective topological space must be a bc-domain.

Using **FO**^{⊤}*X* instead of **FO***X*, using a similar argument, we would obtain another proof of Scott’s theorem (Exercise 9.3.9) that the injective topological spaces are the complete continuous lattices.

Escardó showed that those notions of injectivity were strongly related to properties of the filter monads **FO** and **FO**^{⊤}. The complete details are given in [5], but I will not go to that level of generality.

## The filter monad

Let me recall that a *monad* on a category **C** is a triple (**T**, η, †) consisting of the following data:

- for each object
*X*of**C**, an object**T***X*in**C** - for each object
*X*of**C**, a morphism η:_{X}*X*→**T***X*called the*unit*of the monad - an
*extension operation*†, transforming every morphism*f*:*X*→**T***Y*into a morphism*f*^{†}:**T**X →**T**Y, so that the following equations are satisfied:- η
_{X}^{†}= id_{TX} - for every morphism
*f*:*X*→**T***Y*,*f*^{†}o η=_{X}*f* - for all morphisms
*f*:*X*→**T***Y*and*g*:*Y*→**T**Z, (*g*^{†}o*f*)^{†}=*g*^{†}o*f*^{†}.

- η

In that case, **T** gives rise to a functor from **C** to **C**, the various units assemble as a natural transformation, and there is also a *multiplication* natural transformation μ : **TT** → **T**, defined on each object *X* by μ* _{X}* ≝ id

_{T}

_{X}^{†}. This satisfies a few

*monad laws*, among which μ

*o η*

_{X}_{T}

*= μ*

_{X}*o*

_{X}**T**η

*= id*

_{X}_{TX}. (There is a third one, which I will not make any use of here.)

**FO** is a monad on **Top** (and on **Top _{0}**). We have already defined the space

**FO**

*X*, the map η

*:*

_{X}*x*↦ {

*U*∈

**O**

*X*|

*x*∈

*U*} will serve as unit, and for every continuous map

*f*:

*X*→

**FO**

*Y*, we will use the largest extension

*f*† of

*f*to

**FO**

*X*introduced earlier. The action of

**FO**on morphisms

*f*:

*X*→

*Y*is given by

**FO**(

*f*)(

*F*) =

*f*[

*F*], the “image filter” map, namely

*f*[

*F*] = {

*V*∈

**O**

*Y*|

*f*

^{–1}(

*V*) ∈

*F*}. Similarly,

**FO**

^{⊤}is a monad.

For future reference, we note that η_{X}^{–1}(☐*U*) = *U*.

## KZ-monads

Here “KZ” is an abbreviation for “of Kock-Zöberlein type”. Technically, KZ-monads are monads on *2-categories* [4], but, just like Escardó [3], we will be content with the more restricted setting of *poset-enriched* categories. This will relieve us of much of the complications of general 2-categories.

A *poset-enriched category* is a category **C**, equipped with orderings ≤ on each homset Hom_{C}(*X*, *Y*), in such a way that composition o is monotonic in each of its arguments. **Top _{0}** is poset-enriched: we simply write

*f*≤

*g*, for every pair of maps

*f*,

*g*∈ Hom

_{C}(

*X*,

*Y*), if and only if

*f*(

*x*) ≤

_{Y}*g*(

*x*) for every

*x*∈

*X*, where ≤

*is the specialization ordering of*

_{Y}*Y*; the fact that composition is monotonic in its arguments is due to the fact that every continuous map is monotonic with respect to the specialization orderings of its domain and of its codomain.

A functor *F* between poset-enriched categories is *poset-enriched* if and only if *F* is monotone on every homset, namely if and only if for all objects *X* and *Y* in the source category, for all morphisms *f*, *g* : *X* → *Y*, if *f*≤*g* then *F*(*f*)≤*F*(*g*).

A *right KZ-monad* (**T**, η, †) on a poset-enriched category **C** is one such that **T** is a poset-enriched functor, and such that η_{T}* _{X}* ≤

**T**η

*for every object*

_{X}*X*[3,5].

Now, look: **FO** *is* a right KZ-monad on **TOP _{0}**. (Does that surprise you?)

This is proved as follows. First, let us consider two continuous maps *f*, *g* : *X* → *Y* such that *f*≤*g*. Then **FO***f*(*F*) = *f*[*F*] = {*V* ∈ **O***Y* | *f*^{–1}(*V*) ∈ *F*} is included in **FO***g*(*F*) = *g*[*F*] = {*V* ∈ **O***Y* | *g*^{–1}(*V*) ∈ *F*}, since if *f*^{–1}(*V*) ∈ *F*, then *g*^{–1}(*V*) contains *f*^{–1}(*V*) (since *f*≤*g* and *V* is upwards-closed), hence *g*^{–1}(*V*) is in *F*, since *F* is upwards-closed.

Second, for every filter *F* ∈ **FO***X*, η_{FO}* _{X}*(

*F*) is the collection of open neighborhoods of

*F*in

**FO**

*X*, while

**FO**η

*(*

_{X}*F*) = η

*[*

_{X}*F*] = {

*∈*

**U****OFO**

*Y*| η

_{X}^{–1}(

*) ∈*

**U***F*}. In order to show that η

_{FO}*(*

_{X}*F*) ⊆

**FO**η

*(*

_{X}*F*), we consider any element of η

_{FO}*(*

_{X}*F*), namely any open neighborhood

**of**

*U**F*in

**FO**

*X*. By definition,

**contains a basic open subset ☐**

*U**U*such that

*F*∈ ☐

*U*, namely such that

*U*∈

*F*. Then η

_{X}^{–1}(

*) contains η*

**U**

_{X}^{–1}(☐

*U*) =

*U*, which is in

*F*; by definition of

**FO**η

*(*

_{X}*F*),

**is therefore in**

*U***FO**η

*(*

_{X}*F*), which completes the argument.

Similarly, **FO**^{⊤} is also a right KZ-monad on **TOP _{0}**.

## The algebras of a right KZ-monad

In a poset-enriched category, there is a notion of *adjoint maps*. (No, not adjunctions—a different notion, but in the same spirit; in fact, in a 2-categorical perspective, they are really adjoint 1-cells, as defined with respect to the 2-cell structure.) We say that *f* : *X* → *Y* is *left-adjoint* to *g* : *Y* → *X* if and only if *f* o *g* ≤ id* _{Y}* and id

*≤*

_{X}*g*o

*f*. Then we write

*f*⊣

*g*, and we also say that

*g*is

*right-adjoint*to

*f*.

As in the usual case of poset adjunctions, right-adjoints are unique if they exist. (Similarly with left-adjoints.) Indeed, let us assume that we have two right-adjoints *g*, *g’* to *f* : *X* → *Y*. Then *g’* o *f* o *g* ≤ *g’* since *f* ⊣ *g*, and *g* ≤ *g’* o *f* o *g* since *f* ⊣ *g*‘, so *g* ≤ *g’*; symmetrically, *g’* ≤ *g*, so *g*=*g’*.

Let me remind you that an *algebra* of a monad (**T**, η, †) is an object *X* together with a morphism α : **T***X* → *X* (the *structure map* of the algebra) such that: α o η* _{X}* = id

_{X}and α o μ

*= α o*

_{X}**T**α. I will simply call α itself the

**T**-algebra.

The following lemma, in this form, is due to Escardó [5, Lemma 4.1.1], and is a restatement of a result of Kock, who proved it in a more general 2-categorical context [4]. This yields a rather unexpected characterization of the algebras of a right KZ-monad **T**: they are exactly certain right-adjoints to the unit η* _{X}*. In particular, since right-adjoints are unique if they exist, there is

*at most one*structure of

**T**-algebra on every object

*X*, which is a pretty amazing feat.

**Lemma (Kock-Escardó).** Let (**T**, η, †) be a monad on a poset-enriched category **C**, where **T** is a poset-enriched functor. The following are equivalent:

- η
_{T}≤_{X}**T**ηfor every object_{X}*X*(namely,**T**is a right KZ-monad); - η
_{T}⊣ μ_{X}for every object_{X}*X*; - μ
⊣_{X}**T**ηfor every object_{X}*X*; - a morphism α :
**T***X*→*X*is a**T**-algebra if and only if η⊣ α and α o η_{X}= id_{X}(i.e., iff it is a_{X}*coreflective right-adjoint to the unit*η);_{X}

*Proof.* We follow Escardó’s proof. We decompose claim 4 into its ‘if’ part and its ‘only if’ part, and we will call them [4⇐] and [4⇒], respectively.

1 ⇒ [4⇒]. The equality α o η* _{X}* = id

*holds for every*

_{X}**T**-algebra α, by definition. We show that η

*o α ≤ id*

_{X}

_{T}*as follows: by naturality of η, η*

_{X}*o α =*

_{X}**T**α o η

*; that is ≤*

_{X}**T**α o

**T**η

*by 1 (and the fact that*

_{X}**T**is poset-enriched); then

**T**α o

**T**η

*=*

_{X}**T**(α o η

*) = id*

_{X}

_{T}*, since α o η*

_{X}*= id*

_{X}*.*

_{X}[4⇒] ⇒ 2. This is because μ* _{X}* : μ :

**TT**

*X*→

**T**

*X*is a

**T**-algebra (the

*free*

**T**-algebra) on

**T**

*X*, and is therefore right-adjoint to η

_{T}*.*

_{X}2 ⇒ [4⇐]. This is the clever part. Let us assume η_{T}* _{X}* ⊣ μ

*, and let us consider a coreflective right-adjoint α to the unit η*

_{X}*; namely, η*

_{X}*⊣ α and α o η*

_{X}*= id*

_{X}*. The latter is one of the two conditions we need to ensure that is a*

_{X}**T**-algebra. The other one is α o μ

*= α o*

_{X}**T**α, and we will prove it by showing that the two sides of the equality are right-adjoint to the same morphism.

- By composing the adjunctions η
_{T}⊣ μ_{X}with η_{X}⊣ α, we obtain η_{X}_{T}o η_{X}⊣ α o μ_{X}. Explicitly, (η_{X}_{T}o η_{X}) o (α o μ_{X}) = η_{X}_{T}o (η_{X}o α) o μ_{X}≤ η_{X}_{T}o μ_{X}(because η_{X}⊣ α, so η_{X}o α ≤ id_{X}_{T}) ≤ id_{X}_{TT}(because η_{X}_{T}⊣ μ_{X}); and (α o μ_{X}) o (η_{X}_{T}o η_{X}) = α o (μ_{X}o η_{X}_{T}) o η_{X}(because η_{X}_{T}⊣ μ_{X}) ≥ α o η_{X}= id_{X}(really, μ_{X}o η_{X}_{T}= id_{X}_{TX}, by the monad laws, so we have an equality here). - Hence it remains to show that η
_{T}o η_{X}⊣ α o_{X}**T**α. Since η is natural, we have ηo_{Y}*f*=**T***f*o ηfor every morphism_{X}*f*:*X*→*Y*. In particular, if we take*f*≝ η_{T}(and_{X}*Y*≝**T***X*), we obtain that η_{T}o η_{X}=_{X}**T**ηo η_{X}. Hence, instead of showing that η_{X}_{T}o η_{X}⊣ α o_{X}**T**α, we will show that equivalent claim that**T**ηo η_{X}⊣ α o_{X}**T**α. - In order to show that
**T**ηo η_{X}⊣ α o_{X}**T**α, we compute: (**T**ηo η_{X}) o (α o_{X}**T**α) =**T**ηo (η_{X}o α) o_{X}**T**α ≤**T**ηo_{X}**T**α =**T**(ηo α) ≤ id_{X}_{TT}; and (α o_{X}**T**α) o (**T**ηo η_{X}) = α o (_{X}**T**α o**T**η) o η_{X}= α o η_{X}= id_{X}._{X}

Hence, as promised, both α o μ* _{X}* and α o

**T**α are right-adjoint to the same morphism, namely η

_{T}*o η*

_{X}*=*

_{X}**T**η

*o η*

_{X}*; so they are equal.*

_{X}At this point, we have the chain of implications 1 ⇒ [4⇒] ⇒ 2 ⇒ [4⇐]. In particular, 1 implies both [4⇒] and [4⇐], hence implies 4; and 4, which implies [4⇐], therefore implies 2. In summary, we have proved 1 ⇒ 4 ⇒ 2.

2 ⇒ 1. Assumption 2 is that η_{T}* _{X}* ⊣ μ

*. We wish to show that η*

_{X}

_{T}*≤*

_{X}**T**η

*. This is immediate: since η*

_{X}

_{T}*⊣ μ*

_{X}*, (η*

_{X}

_{T}*o μ*

_{X}*) ≤ id*

_{X}_{TT}

*; we compose with*

_{X}**T**η

*, so that η*

_{X}

_{T}*o μ*

_{X}*o*

_{X}**T**η

*≤*

_{X}**T**η

*. By one of the monad laws, μ*

_{X}*o*

_{X}**T**η

*= id*

_{X}

_{T}*, whence η*

_{X}

_{T}*≤*

_{X}**T**η

*.*

_{X}Together with 1 ⇒ 4 ⇒ 2, we obtain that 1, 2, and 4 are equivalent.

1 ⇒ 3. Assumption 1 entails that η_{T}* _{X}* ≤

**T**η

*, and also that η*

_{X}

_{TT}*≤*

_{X}**T**η

_{T}*. We wish to show that μ*

_{X}*⊣*

_{X}**T**η

*. We have (μ*

_{X}*o*

_{X}**T**η

*) = id*

_{X}

_{T}*by one of the monad laws. In order to show that*

_{X}**T**η

*o μ*

_{X}*≥ id*

_{X}_{TT}

*, we first use the naturality of μ in order to derive*

_{X}**T**η

*o μ*

_{X}*= μ*

_{X}

_{T}*o*

_{X}**TT**η

*. Since*

_{X}**T**η

*≥ η*

_{X}

_{T}*, this is larger than or equal to μ*

_{X}

_{T}*o*

_{X}**T**η

_{T}

*. Since*

_{X}**T**η

_{T}*≥ η*

_{X}

_{TT}*, this is in turn larger than or equal to μ*

_{X}

_{T}*o η*

_{X}_{TT}

*= id*

_{X}_{TT}

*(using a monad law).*

_{X}3 ⇒ 1. Assumption 3 is that μ* _{X}* ⊣

**T**η

*, so (*

_{X}**T**η

*o μ*

_{X}*) ≥ id*

_{X}

_{T}*. We compose with η*

_{X}

_{T}*, so*

_{X}**T**η

*o μ*

_{X}*o η*

_{X}

_{T}*≥ η*

_{X}

_{T}*. By one of the monad laws, μ*

_{X}*o η*

_{X}

_{T}*= id*

_{X}

_{T}*, so*

_{X}**T**η

*≥ η*

_{X}

_{T}*. ☐*

_{X}## The algebras of the **FO** monad

Since the **FO** monad on **TOP _{0}** is a right KZ-monad, the above Lemma immediately tells us what its algebras are. Those are exactly the coreflective right-adjoints to the unit, namely the continuous maps α :

**FO**

*X*→

*X*such that η

*o α ≤ id*

_{X}_{FOX}and α o η

*= id*

_{X}*.*

_{X}In particular, if *X* has such an **FO**-algebra structure, then *X* must be a retract of **FO***X*. Since **FO***X* is a bc-domain and since bc-domain are closed under retracts, *X* must be a bc-domain.

Conversely, for every bc-domain *X*, there is a unique largest continuous extension of the identity map on *X* to the larger space **FO***X*, as we have seen earlier: this is because bc-domains are densely injective spaces, and because *X* embeds as a dense subspace of **FO***X* through η* _{X}*. Let us write α for this map. The fact that α extends id

*means that α o η*

_{X}*= id*

_{X}*. Exercise 9.3.11 of the book gives us an explicit formula for α: for every*

_{X}*F*∈

**FO**

*X*, α(

*F*) = sup

_{U ∈ F}inf

*U*. (The notation inf U denotes the infimum of the non-empty collection of elements of

*U*. That always exists in any bc-domain.) Since

*F*is a filter, and inf

*U*depends antitonically on

*U*, the family of points inf

*U*, when

*U*ranges over

*F*, is directed. Since the topology of

*X*is the Scott topology, it follows that every open neighborhood

*V*of α(

*F*) contains inf

*U*for some

*U*∈

*F*; if so,

*V*contains the whole of

*U*, since

*V*is upwards-closed, and therefore

*V*itself is in

*F*. This shows that η

*(α(*

_{X}*F*)) is included in

*F*. Therefore η

*o α ≤ id*

_{X}_{FOX}.

By the above Lemma of right KZ-monads, this is enough to show that α is the structure map of an **FO**-algebra on *X*. Additionally, this is the *only* possible **FO**-algebra on *X*.

Similarly, the **FO**^{⊤}-algebras are exactly the continuous (complete) lattices, in a unique way, and we have retrieved the result we started this post with, due to Day and Wyler.

## Are KZ-monads frequent?

That is hard to say. My impression is that KZ-monads are relatively rare, but that it is always a good idea to check, given any monad you might be interested in, whether it is KZ. If so, determining its algebras is really a lot simpler.

A first reason is that you need to work in a poset-enriched category. On **Set**, there is only one way to obtain a poset-enriched structure, and that is to equip each homset with the equality relation as ordering. A right KZ-monad is then one such that η_{T}* _{X}* =

**T**η

*for every every object*

_{X}*X*, and the Kock-Escardó Lemma then says that it is equivalent to require that η

_{T}*and μ*

_{X}*are*

_{X}*inverses*for every object

*X*, or that μ

*and*

_{X}**T**η

*are inverses; or that, finally, there is a (unique)*

_{X}**T**-algebra on a set

*X*if and only if η

*is an isomorphism, and the structure map must then be its inverse. Those are exactly the*

_{X}*idempotent monads*on

**Set**.

For example, the powerset monad is not idempotent on **Set**, hence not right KZ.

On general poset-enriched category, any idempotent monad, namely any monad in which η_{T}* _{X}* =

**T**η

*for every every object*

_{X}*X*, is a right KZ-monad, whatever poset enrichment we choose. For example, the sobrification monad

**S**on

**Top**is idempotent, hence right KZ. (Its algebras are the sober spaces.)

_{0}The Hoare powerspace monad **H** on **Top _{0}** is not idempotent. For every closed subset

*C*of

*X*, η

_{H}*(*

_{X}*C*) is the collection of closed subsets of

*C*, while

**H**η

*(*

_{X}*C*) is the closure cl {↓

*x*|

*x*∈

*C*}. Hence, for example, taking

*X*≝ {

*a*,

*b*} with the discrete topology, and

*C*≝

*X*, η

_{H}*(*

_{X}*C*) is the collection of all subsets of

*X*, while

**H**η

*(*

_{X}*C*) only contains the empty set, {

*a*}, and {

*b*}.

However, **H**η* _{X}*(

*C*) is included in η

_{H}*(*

_{X}*C*), because every set of the form ↓

*x*with

*x*∈

*C*is a closed subset of

*C*. This show that

**H**is a

*left KZ-monad*… not a right KZ-monad, but I am sure you had wondered whether those things existed from the very start (if there are right KZ-monads, then surely there must also be left KZ-monads).

A left KZ-monad is defined exactly as right KZ-monads, except with ≤ replaced by ≥. (In other words, left KZ-monads are right KZ-monads on the same category, except with the orderings on each homset reversed.) Explicitly, a left KZ-monad is a monad **T** such that η_{T}* _{X}* ≥

**T**η

*for every object*

_{X}*X*. The Kock-Escardó lemma for left KZ-monads reads as follows.

**Lemma (Kock-Escardó).** Let (**T**, η, †) be a monad on a poset-enriched category **C**, where **T** is a poset-enriched functor. The following are equivalent:

- η
_{T}≥_{X}**T**ηfor every object_{X}*X*(namely,**T**is a*left*KZ-monad); - μ
⊣ η_{X}_{T}for every object_{X}*X*; **T**η⊣ μ_{X}for every object_{X}*X*;- a morphism α :
**T***X*→*X*is a**T**-algebra if and only if α ⊣ ηand α o η_{X}= id_{X}(i.e., iff it is a_{X}*reflective left-adjoint to the unit*η);_{X}

There is also a so-called *Smyth powerspace monad* **Q** on **Top _{0}**, where

**Q**

*X*is the space of compact saturated subsets of

*X*, with the upper Vietoris topology, see this page for example. Apparently I have not yet stated here that this yields a monad. It does, and the unit maps every point

*x*in a space

*X*to the compact saturated set ↑

*x*. The specialization ordering is

*reverse*inclusion. In that case,

**Q**η

*(*

_{X}*Q*) (the set of compact saturated subsets

*Q’*of

*X*that are above, namely included in,

*Q*)

*contains*η

_{Q}*(*

_{X}*C*) (the set of compact saturated subsets

*Q’*of

*X*that are included in ↑

*x*for some

*x*in

*Q*). This looks like the opposite relation as with

**H**, but remember that the specialization ordering is

*reverse*inclusion. Hence

**Q**is also a

*left*KZ-monad.

It follows immediately that the **Q**-algebras, just like the **H**-algebras, are determined as the reflective left-adjoints to the unit, and are unique if they exist. That is all the nicer as we never had to look at the multiplication of the monad, thus saving us some complications! Indeed, the equivalence between 1 and 4 in the Kock-Escardó lemma does not mention multiplication at all.

The continuous valuation monad, which I have briefly talked about, for example here, is neither a left nor a right KZ-monad, though. The Plotkin powerspace monad, which is implicit here, is neither.

However, and to conclude with a positive note, the formal ball monad **B** on the category of quasi-metric spaces and 1-Lipschitz continuous maps *is* a left KZ-monad [6]. That allowed me to swiftly characterize the **B**-algebras α : **B***X* → *X* as the 1-Lipschitz maps that send every formal ball (*x*, *r*) to some point in the closed ball centered at *x* of radius *r*, and which send (*x*, 0) to *x* [6, Proposition 3.4]. This result extends to the category of standard quasi-metric spaces and 1-Lipschitz continuous maps, but that is a more technical result.

- Alan Day. Filter monads, continuous lattices and closure systems. Canadian Journal of Mathematics, XXVII(1):50–59, 1975.
- Oswald Wyler. Algebraic theories for continuous semilattices. Archive for Rational Mechanics and Analysis, 90(2):99–113, 1985.
- Martín Hötzel Escardó. Injective spaces via the filter monad. Topology Proceedings 22(2), 1997.
- Anders Kock. Monads for which structures are adjoint to units (version 3). Journal of Pure and Applied Algebra, 104:41–59, 1995.
- Martín Hötzel Escardó. Properly injective spaces and function spaces. Topology and its Applications 89:75–120, 1998.
- Jean Goubault-Larrecq. Formal ball monads. Topology and its Applications 263:372–391, 2019.

— Jean Goubault-Larrecq (August 20th, 2022)