In 1969, Ernest Manes proved the following remarkable result: the algebras of the ultrafilter monad on **Set** are exactly the compact Hausdorff spaces [1]. This is remarkable, because it gives a purely algebraic/category-theoretic of the otherwise purely topological notion of compact Hausdorff spaces. I will explain how this is proved, and I will give a few pointers to some extension results of the same kind.

I have talked about monads a long time ago, and about algebras of monads at about the same time, namely, in 2015. I have come back to the subject much more recently, in a 2022 post where I explained Andrea Schalk’s result that the algebras of the **Q**_{fin} monad on the category of T_{0} topological spaces are exactly the T_{0} deflationary topological semilattices with small semilattices.

## Monads, and their algebras

Let me give back the necessary definitions. 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

We then say that (**T**, η, †) is a monad, or, by abuse of language, that **T** itself is a monad. For example, there is a powerset monad **P** on **Set**, where **P***X* is the set of all subsets of *X*, η* _{X}*(

*x*)={

*x*}, and

*f*

^{†}(

*A*)=∪{

*f*(

*x*) |

*x*∈

*A*}.

I will be more interested in the *filter* monad **F** and its variants on **Set**. A filter *F* on a set *X* is a non-empty, upwards-closed collection of subsets of *X* that is closed under binary intersections. (There are some variants in that definition. For example, it is sometimes allowed to include the empty set as a member of *F*; we don’t. Also, what we call a filter on *X* here is really the special case of a filter of subsets of *X*, and more general notions of filters of elements of arbitrary lattices are possible.) The filter monad is defined as follows:

**F***X*is the set of all filters on*X*;- η
(_{X}*x*) is the*principal ultrafilter*at*x*, namely the collection of all subsets*A*of*X*that contain*x*; - for every function
*f*:*X*→**F***Y*,*f*^{†}(*F*) is the filter consisting of all the subsets*B*of*Y*such that*f*^{–1}(*B*^{#}) ∈*F*. I use the notation*B*^{#}to denote the collection of filters*G*on*Y*such that*B*∈*G*. Its inverse image*f*^{–1}(*B*^{#}) by*f*is then a subset of*X*, so*f*^{–1}(*B*^{#}) ∈*F*makes sense.

Similarly, we have the *ultrafilter* monad **U** on **Set**: simply replace “filter” by “ultrafilter” throughout in the previous definition (and be careful to redefine *B*^{#} as the the collection of *ultrafilters* that contain *B*—I will use that notation over and over with that meaning). An ultrafilter is a maximal filter (with respect to inclusion), or equivalently a filter *F* on *X* such that every subset *A* of *X* is in *F* or has its complement in *F*.

Given any monad (**T**, η, †) on **C**, **T** extends to a functor from **C** to **C**. Its action on morphisms *f* : *X* → *Y* is given by **T***f* ≝ (η* _{Y}* o

*f*)†. For example,

**P**

*f*is direct image:

**P**

*f*(

*A*) = {

*f*(

*x*) |

*x*∈

*A*}. The case of the filter monad

**F**defines

**F**

*f*(

*F*) as what I have already called the

*direct image*(as well)

*f*[

*F*] of the filter

*F*by the function

*f*, namely

*f*[

*F*] ≝ {

*B*⊆

*Y*|

*f*

^{–1}(

*B*) ∈

*F*}. The action of the ultrafilter monad is the same: for every ultrafilter

*F*on

*X*,

**U**

*f*(

*F*) =

*f*[

*F*], and that happens to be an ultrafilter.

Every monad also has a *multiplication* μ* _{X}* :

**TT**

*X*→

**T**

*X*, which is simply id

*†. For the powerset monad*

_{X}**P**, multiplication

**PP**

*X*→

**P**

*X*is a form of flattening, and maps a set

**A**of subsets of

*X*to its union ∪

**A**(namely, ∪{

*A*|

*A*∈

**A**}). For the filter and ultrafilter monads, multiplication is the so-called

*Kowalsky sum*operation, which maps every (ultra)filter

*F*of (ultra)filters on

*X*to the (ultra)filter of subsets

*A*of

*X*such that

*A*

^{#}∈

*F*.

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
= id_{X}_{X} - α o μ
= α o_{X}**T**α.

I will simply say that α : **T***X* → *X* itself is an algebra of the monad **T**, or a **T**-algebra.

This is a fundamental construction. Let me simply say that algebras of monads on **Set** generalize the notion of varieties of algebras defined by operators and equations; and that the algebras of a monad **T** on a category **C** form a category **C ^{T}** (the

*Eilenberg-Moore category*of

**T**), such that the forgetful functor from

**C**to

^{T}**C**has a left adjoint, and such that the resulting adjunction has exactly

**T**as associated monad. (A morphism from α :

**T**

*X*→

*X*to β :

**T**

*Y*→

*Y*in

**C**is a morphism

^{T}*f*:

*X*→

*Y*in

**C**such that

*f*o α = β o

**T**

*f*.)

It is a standard routine in mathematics to attempt and determine the algebras of various naturally occurring monads. For example, the algebras of the **P** monad on **Set** are exactly the complete (sup-)semilattices: namely, a **P**-algebra α : **P***X* → *X* equips *X* with an ordering ≤ (defined by *x*≤*y* if and only if α({*x*,*y*})=*y*) such that every subset *A* of *X* has a supremum (which turns out to be α(*A*)), and morphisms of **P**-algebras are those functions that preserve such suprema.

In general, it is hard to give an explicit description of **T**-algebras for arbitrary **T** (except for so-called finitary monads on **Set** and a few exceptional monads such as **P**). It is therefore pretty remarkable that the algebras of the ultrafilter monad **U** were completely characterized [1]. A nice exposition of the matter is given in Marius Stekelenburg’s Bachelor thesis [2], and I will shamelessly borrow a lot from it.

## The algebras of the ultrafilter monad on **Set**

Given an algebra α : **U***X* → *X* of the ultrafilter monad **U**, one may rightly guess that α will map every ultrafilter on *X* to its limit. For that, every ultrafilter needs to have a limit, and that forces *X* to be compact. It is at least believable that α will pick the unique limit of the ultrafilter given as its argument, forcing *X* to be Hausdorff. The deep part of the theorem is that the equations defining the algebra will force *X* to be a *topological* convergence space, that is, one whose convergence is entirely determined by a topology, in the sense that an ultrafilter converges to a point if and only if every open neighborhood of that point is in the ultrafilter.

All right. Before we start dealing with that argument, let us check that a compact Hausdorff topological space *X* gives rise to a **U**-algebra. In other words, let us check the easy part first.

### All compact T_{2} spaces are algebras of the ultrafilter monad on **Set**

So let us assume that *X* is a compact Hausdorff topological space.

- Every ultrafilter
*F*has at least one limit. This is a well-known characterization of compactness, but let me spell that out.

A*limit*of a filter*F*is a point*x*such that every open neighborhood*U*of*x*is in*F*.

In general, we show that, on a compact space, every filter has at least one*cluster point*, namely a point*x*such that every open neighborhood*U*of*x*intersects every set*A*in*F*. This is easy: consider the family*F’*of all closed sets that lie in*F*.*F’*is a filtered family of non-empty closed subsets of*X*. Since*X*is compact,*F’*has a non-empty intersection (see Proposition 4.4.9 in the book, for example). Let*x*be a point in that intersection. For every*A*∈*F*, the closure cl(*A*) is also in*F*, hence contains*x*; then every open neighborhood*U*of*x*intersects cl(*A*) (at*x*), hence also intersects*A*.

If*F*is not just a filter, but an ultrafilter, any cluster point*x*of*F*must actually be a limit of*F*: for every open neighborhood*U*of*x*, either*U*is in*F*, and we are done, or its complement*X*–*U*is; but in that case*U*must intersect*X*–*U*by definition of*x*as a cluster point, and that is plainly impossible. - Every ultrafilter has at most one limit; in fact every filter has at most one limit. Now that is a characterization of Hausdorffness. Let us assume that a filter
*F*has two limits*x*and*y*. If*X*is Hausdorff (a.k.a. T_{2}), then*x*has an open neighborhood*U*, and*y*has a disjoint open neighborhood*V*. Since*x*is a limit of*F*,*U*is in*F*. Similarly,*V*is in*F*. This forces*U*∩*V*to be in*F*. But*U*∩*V*is empty, and the empty set belongs to no filter. - We can therefore define a map α :
**U***X*→*X*that sends every ultrafilter to its unique limit. Let us check the algebra equations:- α o η
= id_{X}_{X}means that for every*x*∈*X*, the limit of the principal filter at*x*is*x*. That is straightforward. - α o μ
= α o_{X}**U**α: this one is more intimidating. We consider any ultrafilter of ultrafilters*F*∈**UU***X*, and we must show that the unique limit of the Kowalsky sum μ(_{X}*F*) of*F*is equal to the unique limit of**U**α(*F*).

A point*x*is the limit of μ(_{X}*F*) if and only if:

— every open neighborhood*U*of*x*is in μ(_{X}*F*),

— namely if and only if every open neighborhood*U*of*x*is such that*U*^{#}∈*F*.

A point*x*is the limit of**U**α(*F*) if and only if:

— every open neighborhood*U*of*x*is in**U**α(*F*),

— i.e., every open neighborhood*U*of*x*is such that α^{–1}(*U*) ∈*F*.

Now we recall that*U*^{#}is the collection of ultrafilters*G*on*X*such that*U*∈*G*, while α^{–1}(*U*) is the collection of ultrafilters*G*on*X*whose limit α(*G*) is in*U*. For any open subset*U*, if the limit α(*G*) of*G*is in*U*then*U*is in*G*. Therefore α^{–1}(*U*) ⊆*U*^{#}.

This immediately implies that every limit of**U**α(*F*) is a limit of μ(_{X}*F*). But limits exist and are unique, so the unique limit α(**U**α(*F*)) of the former is equal to the unique limit α(μ(_{X}*F*)) of the latter.

- α o η

### Building a compact Hausdorff space from a **U**-algebra

In the converse direction, we start from a **U**-algebra α : **U***X* → *X*, and we need to show several things: that *X* can be given a topology that is compact and Hausdorff, and such that α maps every ultrafilter *F* to its unique limit.

If we can nurture any hope of achieving such a result, then for every open subset *U* of *X* (in the topology that is yet to be defined), we must have that for every ultrafilter *F* on *X*, α(*F*) ∈ *U* implies *U* ∈ *F*. Let us simply take that as a definition:

A subset *U* of *X* is called *open* (relatively to α)

if and only if for every ultrafilter *F* on *X* such that α(*F*) ∈ *U*, *U* is in *F*.

That is not an arbitrary choice: a topology is uniquely determined by its notion of convergence. Hence, if *X* can be equipped with a topology such that α characterizes its notion of convergence, the above choice is the unique possible choice.

We claim that this is a topology:

- Let (
*U*)_{i}_{i ∈ I}be any family of open subsets of*X*(under that definition), and*U*be their union. For every ultrafilter*F*on*X*such that α(*F*) ∈*U*, α(*F*) is in some*U*, so_{i}*U*is in_{i}*F*, using the fact that*U*is open. Since_{i}*F*is upwards-closed,*U*is also in*F*. This shows that*U*is open as per our definition. - Let (
*U*)_{i}_{i ∈ I}be any finite family of open subsets of*X*, and*U*be their intersection. For every ultrafilter*F*on*X*such that α(*F*) ∈*U*, α(*F*) is in every*U*, so_{i}*U*is in_{i}*F*, for every*i*in*I*. Since*F*is closed under finite intersections,*U*is also in*F*. Therefore*U*is open.

For further reference, let us call this the *α-topology*. Its open subsets the *α-open* subsets of *X*, its closed subsets are the *α-closed* subsets of *X*, and so on.

Let us practice a bit and express some related notions.

A subset *C* of *X* is α-closed if and only if it is the complement of an α-open subset, namely if and only if for every ultrafilter *F* on *X* such that α(*F*) ∉ *C*, the complement of *C* is in *F*. The complement of *C* is in *F* if and only if *C* is not in *F*, since *F* is an ultrafilter. By contraposition, *C* is α-closed if and only if for every ultrafilter *F* on *X* that contains *C*, α(*F*) is in *C*. We recognize the familiar idea that a set *C* is closed if and only if it contains every limit of every ultrafilter containing *C*, with α used instead of forming limits.

We remember that *C*^{#} is exactly the collection of ultrafilters *F* that contain *F*. Therefore *C* is α-closed if and only if:

- α sends every
*F*in*C*^{#}to an element of*C*, - or equivalently if
*C*^{#}⊆ α^{–1}(*C*), - or equivalently α(
*C*^{#}) ⊆*C*, where α(*C*^{#}) is the (direct) image of*C*^{#}by α, namely {α(*G*) |*G*∈*C*^{#}}.

**Lemma A.** The closure of any subset *A* of *X* in the α-topology is exactly the direct image α(*A*^{#}) of *A*^{#} by α.

*Proof.* Let *C* be that image. We first observe that *A* ⊆ *C*. In order to show this, we pick any point *x* in *A*, and we wish to show that *x* is in *C*, namely that there is an ultrafilter *G* in *A*^{#} such that α(*G*)=*x*. We define *G* as η* _{X}*(

*x*). We verify that

*G*is in

*A*

^{#}, namely that

*A*is in

*G*, since that is equivalent to

*x*being in

*A*. And α(

*G*)=α(η

*(*

_{X}*x*))=

*x*, by the first algebra law α o η

*= id*

_{X}_{X}.

We claim that *C* is α-closed. This is a lot harder! We must show that for every *F* ∈ *C*^{#}, α(*F*) is in *C*. Henceforth, let us fix *F* ∈ *C*^{#}. Showing that namely that α(*F*) is in *C* means finding an ultrafilter *G* ∈ *A*^{#} such that α(*F*)=α(*G*). One might think of simply defining *G* as *F*, as then α(*F*)=α(*G*) would be obvious; but there is no reason why *F* would be in *A*^{#}. Instead, we will build a well-designed element *F* of **UU***X*, and apply the two sides of the second algebra law α o μ* _{X}* = α o

**U**α to it.

- We start by defining
*E*≝ {*A*^{#}} ∪ {α^{–1}(*B*) |*B*∈*F*}; this is a family of sets of ultrafilters on*X*, namely a subset of**PU***X*. We will show that*E*is included in some ultrafilter*F*∈**UU***X*. In order to this, it suffices to verify that it has the*finite intersection property*, namely that any finite intersection of elements of*E*is non-empty: then the family ↑*E*of all subsets of**PU***X*that contain some element of*E*will be a filter; by Zorn’s Lemma, that filter will be included in a maximal filter, which will be the desired ultrafilter*F*. *E*consists of sets of two kinds: the set*A*^{#}, and the sets of the form α^{–1}(*B*),*B*∈*F*. In order to check the finite intersection property for*E*, we first observe that any finite intersection of sets of the second kind, namely any intersection of α^{–1}(*B*_{1}), …, α^{–1}(*B*) where_{n}*B*_{1}, …,*B*_{n}∈*F*, is equal to α^{–1}(*B*_{1}∩ … ∩*B*_{n}), and that*B*_{1}∩ … ∩*B*_{n}is in*F*, since*F*is a filter. In other words, any finite intersection of sets of the second kind is a set of the second kind.

Hence, in order to show that every finite intersection of elements of*E*is non-empty, it suffices to show that the unique set of the first kind,*A*^{#}, intersects every set of the second kind, namely every set α^{–1}(*B*), for any*B*∈*F*.

Let therefore*B*be any element of*F*. We must find an ultrafilter*G*in*A*^{#}such that α(*G*) is in*B*. Since*F*∈*C*^{#},*C*is in*F*, so*B*∩*C*is also in*F*. Since*F*does not contain the empty set,*B*∩*C*is non-empty, and we can therefore pick a point*x*in*B*∩*C*. Since*x*is in*C*, by definition of*C*, we can write*x*as α(*G*) for some*G*in*A*^{#}. This is the desired ultrafilter*G*:*G*is in*A*^{#}, and α(*G*)=*x*is in*B*∩*C*, hence in*B*.- Now that we have shown that
*E*has the finite intersection property, we obtain an ultrafilter*F*∈**UU***X*that contains*E*by Zorn’s Lemma, as previously stated. We now use the second algebra law α o μ= α o_{X}**U**α as follows.- μ
(_{X}*F*) is the collection of sets*B*such that*B*^{#}is in*F*, by definition. In particular, it contains the collection of sets*B*such that*B*^{#}is in*E*. One such set*B*is*A*. (Remember that*A*^{#}is the set of the first kind in*E*.) Therefore*A*is in μ(_{X}*F*), meaning that*G*≝ μ(_{X}*F*) is in*A*^{#}. **U**α(*F*) is the collection of sets*B*such that α^{–1}(*B*) is in*F*, by definition. In particular, it contains the collection of sets*B*such that α^{–1}(*B*) is in*E*. Remember the shape of the sets of the second kind in*E*: exactly the sets α^{–1}(*B*) with*B*in*F*. Therefore**U**α(*F*) contains every element*B*of*F*. This means that*F*⊆**U**α(*F*).

Since ultrafilters are maximal filters, any two ultrafilters such that one is included in the other one must be equal. Hence*F*=**U**α(*F*).

It follows that α(**U**α(*F*)) = α(*F*).- The second algebra law says that α(μ
(_{X}*F*))=α(**U**α(*F*)). Hence α(*G*)=α(*F*). We recall that*G*is in*A*^{#}. This completes our task of finding an ultrafilter*G*∈*A*^{#}such that α(*F*) is equal to α(*G*); hence, as promised, we have shown that α(*F*) is in*C*. This holds for every*F*∈*C*^{#}, so*C*is α-closed.

- μ

Let us summarize: *C* is an α-closed set containing *A*. It remains to show that every α-closed set *C’* containing *A* must contain *C*. For any such α-closed set *C’* (containing *A*), *C’*^{#} contains *A*^{#}: every ultrafilter *G* that contains *A* must contain its superset *C’*. Hence the image of *C’*^{#} by α contains the image of *A*^{#} by α. The former is included in *C’* since *C’* is α-closed, and the latter is *C*, by definition. ☐

The non-trivial technique used in the proof of Lemma A proceeds as follows: we build a family *E* as the union of two kinds of sets *E*_{1} and *E*_{2}, we verify that *E* has the finite intersection property, so that *E* is included in some ultrafilter *F* ∈ **UU***X*; we use the second algebra law in the form of the equality α(μ* _{X}*(

*F*))=α(

**U**α(

*F*)); we use the fact that

*F*contains

*E*

_{1}to deduce something about α(μ

*(*

_{X}*F*)), and we use the fact that

*F*contains

*E*

_{2}to deduce something about α(

**U**α(

*F*)), and then we compare what we have gotten in the two cases. We will use that technique once more below (in Lemma C).

**Lemma B.** In any topological space, the limits of an ultrafilter *F* are exactly the points that lie in the closure of every element *A* of *F*.

*Proof.* That one is much easier (whew!); and I have already spelled out part of the argument earlier in this post. Let *x* be any limit of *F*, and let *A* ∈ *F*. If *x* is not in the closure of *A*, then it lies in its complement *U*, which is open. Since *x* is a limit of *F*, *U* must be in *F*, and since *F* is a filter, *U* ∩ *A* must be in *F*, too. This is impossible, since *U* ∩ *A* is empty.

Conversely, let *x* be a point that lies in the closure of every *A* ∈ *F*. For every open neighborhood *U* of *x*, *U* intersects the closure of *A* (at *x*), hence it also intersects *A*. This implies that the collection *G* of subsets *B* of *X* that contain *U* ∩ *A* for some *A* ∈ *F* is a filter. Trivially, *F* is included in *G*. Since *F* is an ultrafilter, hence a maximal filter, we must therefore have *G*=*F*. In particular, *U*, which is trivially in *G*, must be in *F*. This shows that *x* is a limit of *F*. ☐

We can now prove the following important result.

**Lemma C.** Let α : **U***X* → *X* be a **U**-algebra. For every ultrafilter *F* on *X*, a point *x* is a limit of *F* in the α-topology if and only if α(*F*)=*x*. In particular, every ultrafilter has exactly one limit.

*Proof.* In one direction, we claim that α(*F*) is a limit of *F*: we take any open neighborhood *U* of α(*F*); by our definition of the α-topology, since α(*F*) ∈ *U*, we obtain that *U* is in *F*.

In the other direction, we consider any limit *x* of *F*, and we claim that α(*F*)=*x*. This is the harder part. By Lemma B, *x* is in the closure of every *A* ∈ *F*. By Lemma A, that closure is just the direct image α(*A*^{#}). Hence *x* is in α(*A*^{#}) for every *A* ∈ *F*.

- We consider the family
*E*≝ {*A*^{#}|*A*∈*F*} ∪ {α^{–1}({*x*})}; this is a family of sets of ultrafilters on*X*, namely a subset of**PU***X*. We will show that*E*is included in some ultrafilter*F*∈**UU***X***.**In order to this, it suffices to verify that*E*has the finite intersection property, as before: then the family ↑*E*of all subsets of**PU***X*that contain some element of*E*will be a filter; by Zorn’s Lemma, that filter will be included in a maximal filter, which will be the desired ultrafilter*F*. - Given any two elements
*A*^{#}and*B*^{#}, their intersection*A*^{#}∩*B*^{#}is the collection of ultrafilters on*X*that contains both*A*and*B*, equivalently*A*∩*B*; hence*A*^{#}∩*B*^{#}= (*A*∩*B*)^{#}. In particular, if both*A*and*B*are in*F*, then so is*A*∩*B*. In general, the intersection of any finite family of elements*A*_{1}^{#}, …,*A*_{n}^{#}where*A*_{1}, …,*A*_{n}are in*F*is equal to (*A*_{1}∩ … ∩*A*_{n})^{#}, and*A*_{1}∩ … ∩*A*_{n}is in*F*.

Hence it remains to show that for every*A*in*F*,*A*^{#}intersects α^{–1}({*x*}). In other words, we must show that*x*is in the direct image α(*A*^{#}) of*A*^{#}for every*A*∈*F*. We have noticed that this is indeed the case, as a consequence of Lemma A and Lemma B. - As announced, we form an ultrafilter
*F*∈**UU***X*containing*E*. We now use the second algebra law α o μ= α o_{X}**U**α as follows.- μ
(_{X}*F*) is the collection of sets*B*such that*B*^{#}is in*F*, by definition. In particular, it contains the collection of sets*B*such that*B*^{#}is in*E*, and therefore every*A*in*F*. Therefore*F*is included in μ(_{X}*F*), and by maximality, is equal to it. It follows that α(μ(_{X}*F*))=α(*F*). **U**α(*F*) is the collection of sets*B*such that α^{–1}(*B*) is in*F*, by definition. In particular, it contains the collection of sets*B*such that α^{–1}(*B*) is in*E*. One such set is {*x*}, so {*x*} is in the ultrafilter**U**α(*F*). Since**U**α(*F*) is upwards-closed,**U**α(*F*) must contain every superset of*x*, so the principal ultrafilter η(_{X}*x*) is included in**U**α(*F*). By the maximality of ultrafilters,**U**α(*F*)=η(_{X}*x*). By the first algebra law, α(η(_{X}*x*))=*x*, so α(**U**α(*F*))=*x*.- From α(μ
(_{X}*F*))=α(**U**α(*F*)), it follows that α(*F*)=*x*. We are done! ☐

- μ

### Morphisms of **U**-algebras = continuous maps

Given two **U**-algebras α : **U***X* → *X* and β : **U***Y* → *Y*, a morphism *f* from α to β in the Eilenberg-Moore category **Set ^{U}** is a function

*f*:

*X*→

*Y*such that

*f*o α = β o

**U**

*f*, namely a function such that, for every ultrafilter

*F*on

*X*,

*f*applied to the α-limit of

*F*is equal to the β-limit of

*f*[

*F*].

That is exactly the definition of a continuous map between convergence spaces (provided convergence spaces are defined in terms of ultrafilters). I will therefore skip the proof that such maps are exactly the continuous maps from *X* (with the α-topology) to *Y* (with the β-topology).

Let us recapitulate what we have obtained, in a formal way.

- There is a functor
**K**that maps every**U**-algebra α :**U***X*→*X*to*X*, seen as a topological space, when equipped with the α-topology. Then*X*is compact Hausdorff, so**K**is a functor from the Eilenberg-Moore category**Set**to^{U}**CompHaus**, the category of compact Hausdorff spaces and continuous maps. - In the opposite direction, there is a functor
**A**from**CompHaus**to**Set**. We have already given its action on objects: every compact Hausdorff space^{U}*X*gives rise to a**U**-algebra α :**U***X*→*X*, where α maps every ultrafilter to its unique limit in*X*. On morphisms, we simply observe that every continuous map preserves limits, hence defines a morphism of**U**-algebras. - The functors
**K**and**A**are mutually inverse (on the nose, as category theorists say!). If we start from a compact Hausdorff space*X*, then**KA***X*has the same points as*X*, and its topology is the unique topology that induces the same notion of convergence as*X*; that must be the topology of*X*. If we start from a**U**-algebra α :**U***X*→*X*instead, then Lemma C tells us that α must map every ultrafilter on*X*to its limit in the α-topology, hence that α=**AK**(α).

Category theorists are less interested in isomorphisms of categories (which is a rarely occurring event) than in equivalence of categories, and will therefore usually sum up all that by saying that **CompHaus** and **Set ^{U}** are equivalent categories. But they really are isomorphic categories. This is the contents of Manes’ theorem [1].

## Beyond Manes’ theorem

Quite some work has been done in the wake of Manes’ theorem, which after all is already 53 years old. I have already said that determining the algebras of monads is a difficult task, and we only have a sparse collection of results characterizing the algebras of various filter-related monads. For example, and unless I have overlooked something, we have no explicit description of the algebras of the *filter* monad on **Set**, or of the *prime filter* monad on **Set**. (Oops. We *do* have an explicit description of the algebras of the prime filter monad on **Set**, because prime filters are just ultrafilters there; so Manes’ theorem applies and the algebras of the prime filter monad on **Set** are the compact Hausdorff topological spaces, too. Found by Vincent Moreau, September 17th, 2024.)

Symmetrically, we have a limited number of characterizations of notions of topological spaces (or of convergence spaces) as algebras of monads. For example, I don’t know whether there is any characterization of stably compact spaces as algebras of some monad (apart from the trivial identity monad on the category of stably compact spaces!).

Still, let me cite some results akin to Manes’ theorem in the literature.

Michael Barr showed the following variant [3]. Instead of **Set**, consider the category **Rel** on sets are binary relations between them as morphisms. The idea is that, outside the realm of compact Hausdorff spaces, taking limits of ultrafilters no longer defines a function, but a mere binary relation between ultrafilters and points. There is still a way of defining an ultrafilter functor on **Rel**. This no longer defines a monad, rather a weaker form one may call an *oplax* monad; in it, the unit η and the multiplication are no longer natural, and rather satisfy so-called oplax naturality conditions η* _{Y}* o

*R*⊆

**U**

*R*o η

*and μ*

_{X}*o*

_{Y}*R*⊆

**UU**

*R*o μ

*(for every relation*

_{X}*R*between

*X*and

*Y*, and where ⊆ refers to inclusion of relations, seen as sets of pairs). Correspondingly, the right notion is not that of an algebra, but of a

*lax*algebra, where we only require α o η

*⊇ id*

_{X}_{X}and α o μ

*⊇ α o*

_{X}**T**α. Barr then showed that the lax

**U**-algebras on

**Rel**are exactly the

*topological spaces*, without any other restriction such as being compact or T

_{2}. The most readable introduction to this is probably Stekelenburg’s bachelor thesis [2, Section 4]. He also notes that the proof is essentially the same as the proof given above of Manes’ theorem, replacing functions by relations.

In another direction, Bob Flagg replaces **Set** by the category **Poset** of posets and monotone maps, and ultrafilters on *X* by *prime* filters of upwards-closed subsets of a poset *X* [4]. Note that, if *X* has equality as ordering, then a prime filter of upwards-closed subsets of *X* is just a prime filter of subsets of *X*, namely an ultrafilter. There is a monad **U’** on **Poset** which maps every poset to its set of prime filters of upwards-closed subsets, itself ordered by inclusion. Flagg shows that the **U’**-algebras are exactly the *compact pospaces*, and that the morphisms of **U’**-algebras are exactly the monotone, continuous maps between them. The proof is necessarily different from the proof I have given of Manes’ theorem, since in that proof, we have repeatedly used the fact that ultrafilters are maximal; something that does not hold for prime filters.

Since compact pospaces and stably compact spaces are essentially the same thing (see Chapter 9 in the book), one may think that the **U’**-algebras are the stably compact spaces. While this holds some truth at the level of objects, the difficulty lies in morphisms: that would be true if we decided that the morphisms of stably compact spaces had to be the perfect maps. It is not clear whether the category of stably compact spaces and *continuous* maps is equivalent to the category of algebras of any non-trivial monad.

I will conclude by mentioning a result obtained by Day [5] and Wyler [6] independently. Instead of considering the ultrafilter monad, look at the filter monad. Not on **Set**, though, rather on the category **Top**_{0} of T_{0} topological spaces. In other words, for every T_{0} space *X*, we let **F***X* be the collection of filters of open subsets of *X* (instead of filters of arbitrary subsets), with the topology generated by the subsets ☐*U* ≝ {*F* ∈ **F***X* | *U* ∈ *F*}, where *U* ranges over the open subsets of *X*. Then **F** defines a monad on **Top**_{0}, and Day and Wyler showed that its algebras are exactly the continuous lattices, with their Scott topologies. I am tempted to explain Martín Escardó’s elegant proof of that result [7] in a future post, and that is the main reason why this post’s title has a “I.” in it. Escardó’s proof rests on the amazing notion of a *KZ-monad*, a notion, originally due to Anders Kock, and which deserves to be better known.

- Ernest Manes.
*A triple theoretic construction of compact algebras*. In B. Eckmann and M. Tierney, editors, Seminar on Triples and Categorical Homology Theory, volume 80 of Lecture Notes in Mathematics. Springer, 1969. Also Reprints in Theory and Applications of Categories 18 (2008), 73–94. - Marius Stekelenburg.
*Ultrafilters and topology*. Bachelor project in mathematics, Leiden University, 2014. - Michael Barr.
*Relational algebras*. Reports of the Midwest Category Seminar, no. 137, pages 39–55, Springer, 1970. - Bob Flagg.
*Algebraic theories of compact pospaces*. Topology and its Applications 77(3), pages 277–290. - 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 Escardó.
*Injective spaces via the filter monad*. Topology Proceedings 22(2), 1997.

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