# The Vietoris monad and
weak distributive laws

Richard Garner  Centre of Australian Category Theory, Macquarie University,
NSW 2109, Australia<EMAIL_ADDRESS>
(Date: 28th August 2024)

###### Abstract.

The Vietoris monad on the category of compact Hausdorff spaces is a
topological analogue of the power-set monad on the category of sets.
Exploiting Manes’ characterisation of the compact Hausdorff spaces as algebras
for the ultrafilter monad on sets, we give precise form to the above analogy
by exhibiting the Vietoris monad as induced by a weak distributive law, in the
sense of Böhm, of the power-set monad over the ultrafilter monad.

The support of Australian Research Council grants DP160101519 and FT160100393
is gratefully acknowledged.

##  1\. Introduction

In his 1922 paper [27], Vietoris described how the set of closed subspaces of
a compact Hausdorff space $X$ can itself be made into a compact Hausdorff
space, now often referred to as the _Vietoris hyperspace_ $VX$. The Vietoris
construction is important not just in topology, but also in theoretical
computer science, where its various generalisations provide different notions
of _power domain_ [22], and in general algebra, where its restriction to zero-
dimensional spaces links up under Stone duality with the theory of _Boolean
algebras with operators_ [14].

The assignation $X\mapsto VX$ in fact underlies a monad $\mathbb{V}$ on the
category $\mathrm{\mathcal{K}{\mathcal{H}}aus}$ of compact Hausdorff spaces.
This monad structure was sketched briefly by Manes in [19, Exercise I.5.23],
but received its first detailed treatment by Wyler in [28]; in particular,
Wyler identified the $\mathbb{V}$-algebras as Scott’s _continuous lattices_
[21].

Clearly, the Vietoris monad is related to the _power-set_ monad $\mathbb{P}$
on the category of sets. In both cases, the monad unit and multiplication are
given by inclusion of singletons and by set-theoretic union; and both
underlying functors are “power-object” constructions—differing in the
distinction between _closed_ subspaces and _arbitrary_ subsets, and in the
need to topologise in the former case.

In this article, we give a new account of the Vietoris monad on
$\mathrm{\mathcal{K}{\mathcal{H}}aus}$ which explains its similarities with
the power-set monad on $\mathrm{\mathcal{S}et}$ by deriving it from it in a
canonical way; for good measure, this account also renders the slightly
delicate topological aspects of the Vietoris construction entirely automatic.

The starting point is Manes’ result [18] identifying compact Hausdorff spaces
as the algebras for the ultrafilter monad $\bbbeta$ on
$\mathrm{\mathcal{S}et}$. In light of this, we recognise our situation as the
following one: we have a monad—namely, the power-set monad $\mathbb{P}$ on
sets—which we would like to “lift” appropriately to the category of algebras
for another monad on the same base—namely, the ultrafilter monad $\bbbeta$.

At this point, the categorically-minded reader will doubtless think of Beck’s
theory [2] of distributive laws. For monads $\mathbb{S}$, $\mathbb{T}$ on
${\mathcal{C}}$, a _distributive law_ of $\mathbb{S}$ over $\mathbb{T}$ is a
natural transformation $\delta\colon TS\Rightarrow ST$ satisfying four axioms
expressing compatibility with the monad structures of $\mathbb{S}$ and
$\mathbb{T}$. As we will recall in Section 3.1 below, distributive laws
correspond to _liftings_ of $\mathbb{S}$ to a monad on the category of
$\mathbb{T}$-algebras, and also to _extensions_ of $\mathbb{T}$ to a monad on
the Kleisli category of $\mathbb{S}$.

In particular, we can ask: is there a distributive law of $\mathbb{P}$ over
$\bbbeta$ for which the associated lifting of $\mathbb{P}$ to
$\mathrm{\mathcal{K}{\mathcal{H}}aus}$, the category of $\bbbeta$-algebras, is
the Vietoris monad? Unfortunately, the answer to this question is _no_ , since
the kind of lifting mandated by the theory of distributive laws is too strong;
if the Vietoris monad did lift the power-set monad in this sense, then the
underlying set of $VX$ would comprise the full power-set of $X$, rather than
just the closed subsets.

However, we are clearly very close to having a lifting of $\mathbb{P}$ to
$\bbbeta$-algebras; and, in fact, we are also very close to having a
distributive law of $\mathbb{P}$ over $\bbbeta$. For indeed, such a
distributive law would be the same as an extension of $\bbbeta$ to the Kleisli
category of $\mathbb{P}$, which is the category $\mathrm{\mathcal{R}el}$ of
sets and relations; and the extension of structure from
$\mathrm{\mathcal{S}et}$ to $\mathrm{\mathcal{R}el}$ was analysed in detail by
Barr [1]. As observed in [26, §2.11], it is a direct consequence of Barr’s
results that:

  * •

A functor $F\colon\mathrm{\mathcal{S}et}\rightarrow\mathrm{\mathcal{S}et}$ has
_at most_ one extension to a locally monotone functor
$\tilde{F}\colon\mathrm{\mathcal{R}el}\rightarrow\mathrm{\mathcal{R}el}$,
which exists just when $F$ is weakly cartesian;

  * •

If $F,G$ are weakly cartesian, then $\alpha\colon F\Rightarrow G$ has _at
most_ one extension to a natural transformation
$\tilde{\alpha}\colon\tilde{F}\Rightarrow\tilde{G}$, existing just when
$\alpha$ is weakly cartesian.

(The definition of weak cartesianness is recalled in Section 4.1 below.) In
the case of the ultrafilter monad $\bbbeta$ on $\mathrm{\mathcal{S}et}$, it is
well known that the underlying endofunctor and the monad multiplication are
weakly cartesian, and so extend; while the unit is not, and so does not. This
not-quite extension of $\bbbeta$ to $\mathrm{\mathcal{R}el}$ turns out to
correspond to a not-quite distributive law $\delta\colon\beta P\Rightarrow
P\beta$, which is compatible with both monad multiplications and the unit of
$\mathbb{P}$, but not with the unit of $\bbbeta$.

One perspective on this situation can be found in [5, 10, 25, 24]. As was
already essentially observed in [1], the not-quite extension of $\bbbeta$ to
$\mathrm{\mathcal{R}el}$ is an example of a _lax_ monad extension in the sense
of [5]. It was noted in [10, Exercise 1.I], and confirmed in [25], that such
lax monad extensions correspond to suitably-defined lax distributive laws, and
further explained in [24] that these correspond, in turn, to suitable lax
liftings. These facts are important for the area of _monoidal topology_ ; see,
for example, [11].

However, for the ends we wish to pursue here, a different point of view is
relevant. In 2009, with motivation from quantum algebra, Street [23] and Böhm
[3] introduced various notions of _weak distributive law_ of a monad
$\mathbb{S}$ over a monad $\mathbb{T}$, involving a natural transformation
$\delta\colon TS\Rightarrow ST$ satisfying Beck’s original axioms relating to
the monad multiplications, but weakening in different ways those relating to
the monad units. Each of these kinds of weak distributive law of $\mathbb{S}$
over $\mathbb{T}$ was shown to correspond to a kind of “weak lifting” of
$\mathbb{S}$ to $\mathbb{T}$-algebras.

In particular, one of the kinds of weak distributive law involves simply
dropping from Beck’s original notion the axiom relating to the unit of
$\mathbb{T}$. Thus, the not-quite distributive law $\delta\colon\beta
P\Rightarrow P\beta$ we described above is a weak distributive law, in this
sense, of $\mathbb{P}$ over $\bbbeta$; and so there is a corresponding weak
lifting of $\mathbb{P}$ to $\bbbeta$-algebras. Our main result identifies this
weak lifting by proving:

######  Theorem.

The Vietoris monad on the category of compact Hausdorff spaces is the weak
lifting of the power-set monad associated to the canonical weak distributive
law of the power-set monad over the ultrafilter monad.

As an application of this result, we obtain a simple new proof of Wyler’s
characterisation of the $\mathbb{V}$-algebras as the continuous lattices; and
we conclude the paper with remarks on possible variations and generalisations
of our main result.

##  2\. The monads

###  2.1. The power-set monads

We begin by recalling the various monads of interest and their categories of
algebras. Most straightforwardly, we have:

######  Definition 1.

The power-set monad $\mathbb{P}$ on $\mathrm{\mathcal{S}et}$ has $PX$ given by
the set of all subsets of $X$, and $Pf\colon PX\rightarrow PY$ given by direct
image. The unit $\eta_{X}\colon X\rightarrow PX$ and multiplication
$\mu_{X}\colon PPX\rightarrow PX$ are given by $\eta_{X}(x)=\\{x\\}$ and
$\mu_{X}({\mathcal{A}})=\bigcup{\mathcal{A}}$.

The $\mathbb{P}$-algebras can be identified as complete lattices in two
different ways, depending on whether we view the $\mathbb{P}$-algebra
structure as providing the sup operation or the inf operation; the maps of the
category of $\mathbb{P}$-algebras are then respectively the sup-preserving
maps and the inf-preserving maps.

###  2.2. The ultrafilter monad

Recall that a _filter_ on a set $X$ is a non-empty subset
${\mathcal{F}}\subseteq PX$ such that, for all $A,B\subseteq X$, we have
$A,B\in{\mathcal{F}}$ iff $A\cap B\in{\mathcal{F}}$. A filter is an
_ultrafilter_ if it contains exactly one of $A$ and $X\setminus A$ for each
$A\subseteq X$.

######  Definition 2.

The ultrafilter monad $\bbbeta$ on $\mathrm{\mathcal{S}et}$ has $\beta X$
given by the set of all ultrafilters on $X$, and $\beta f\colon\beta
X\rightarrow\beta Y$ the function taking _pushforward_ along $f$:

${\mathcal{F}}\mapsto f_{!}({\mathcal{F}})=\\{B\subseteq
Y:f^{-1}(B)\in{\mathcal{F}}\\}=\\{B\subseteq Y:f(A)\subseteq B\text{ for some
}A\in{\mathcal{F}}\\}\hbox to0.0pt{ .\hss}$

The unit $\eta_{X}\colon X\rightarrow\beta X$ and multiplication
$\mu_{X}\colon\beta\beta X\rightarrow\beta X$ are defined by
$\eta_{X}(x)=\\{A\subseteq X:x\in A\\}$ and $\mu_{X}(\mathbf{F})=\\{A\subseteq
X:A^{\\#}\in\mathbf{F}\\}$, where for any $A\subseteq X$ we define
$A^{\\#}=\\{{\mathcal{F}}\in\beta X:A\in{\mathcal{F}}\\}$.

The algebras for the ultrafilter monad were identified by Manes [18] as the
compact Hausdorff spaces. Recall that, for a topological space $X$, an
ultrafilter ${\mathcal{F}}\in\beta X$ is said to _converge_ to $x\in X$ if
each neighbourhood of $x$ is in ${\mathcal{F}}$; and that, when $X$ is compact
Hausdorff, each ${\mathcal{F}}\in\beta X$ converges to a unique point
$\xi({\mathcal{F}})$. Manes showed that the function $\xi\colon\beta
X\rightarrow X$ so determined endows the compact Hausdorff $X$ with
$\bbbeta$-algebra structure, and that _every_ $\bbbeta$-algebra arises thus.
Under this identification, the $\bbbeta$-algebra maps are the continuous ones.

###  2.3. The Vietoris monad

The _Vietoris hyperspace_ [27] $VX$ of a compact Hausdorff space $X$ is the
set of all closed subspaces of $X$, endowed with the topology (sometimes
called the “hit-and-miss” topology) generated by the following subbasic open
sets for each $C\in VX$:

$C^{+}=\\{A\in VX:A\cap C=\emptyset\\}\qquad\text{and}\qquad C^{-}=\\{A\in
VX:A\nsubseteq C\\}\hbox to0.0pt{ .\hss}$

######  Definition 3.

[28] The Vietoris monad $\mathbb{V}$ on $\mathrm{\mathcal{K}{\mathcal{H}}aus}$
has $VX$ given as above, and $Vf\colon VX\rightarrow VY$ given by direct
image. The unit $\eta_{X}\colon X\rightarrow VX$ and multiplication
$\mu_{X}\colon VVX\rightarrow VX$ are given by $\eta_{X}(x)=\\{x\\}$ and
$\mu_{X}({\mathcal{A}})=\bigcup{\mathcal{A}}$.

It was shown in [28] that the $\mathbb{V}$-algebras are the _continuous
lattices_ of [21]. Recall that, for elements $x,y$ of a poset $L$, we write
$x\ll y$ if, whenever $D\subseteq L$ is directed and $y\leqslant\sup D$, there
exists some $d\in D$ with $x\leqslant d$. A _continuous lattice_ is a complete
lattice $L$ such that every $x\in L$ satisfies $x=\sup\\{s:s\ll x\\}$. Under
Wyler’s identification, a continuous lattice $L$ becomes a compact Hausdorff
space under its _Lawson topology_ , which is generated by the subbasic open
sets

$s^{+}=\\{x\in L:s\ll x\\}\qquad\text{and}\qquad s^{-}=\\{x\in L:s\nleqslant
x\\}\qquad\text{for $s\in L$}\hbox to0.0pt{ ,\hss}$

and a $\mathbb{V}$-algebra via the function $VL\rightarrow L$ taking infima of
closed sets.

##  3\. Distributive laws and weak distributive laws

###  3.1. Distributive laws

We now recall Beck’s classical theory [2] of _distributive laws_ and their
associated liftings and extensions, and the generalisation of this theory to
_weak distributive laws_ [3] which will be necessary for our main result. We
begin with Beck’s original notion.

######  Definition 4.

Let $\mathbb{S}=(S,\nu,\omega)$ and $\mathbb{T}=(T,\eta,\mu)$ be monads on a
category ${\mathcal{C}}$. A _distributive law of $\mathbb{S}$ over
$\mathbb{T}$_ is a natural transformation $\delta\colon TS\Rightarrow ST$
rendering commutative the four diagrams:

$\textstyle{TSS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta
S}$$\scriptstyle{T\omega}$$\textstyle{STS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\delta}$$\textstyle{SST\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega
T}$$\textstyle{TTS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\delta}$$\scriptstyle{\mu
S}$$\textstyle{TST\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta
T}$$\textstyle{STT\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\mu}$$\textstyle{{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\nu}$$\scriptstyle{\nu
T}$$\textstyle{{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta
S}$$\scriptstyle{S\eta}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{ST}$$\textstyle{{TS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{ST}}$$\textstyle{{TS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{ST}}$$\textstyle{{TS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{ST}\hbox
to0.0pt{.\hss}}$

The basic result about distributive laws is that they correspond both to
liftings and to extensions, in the sense of the following definition.

######  Definition 5.

Let $\mathbb{S}=(S,\nu,\omega)$ and $\mathbb{T}=(T,\eta,\mu)$ be monads on a
category ${\mathcal{C}}$. If we write
$U^{\mathbb{T}}\colon{\mathcal{C}}^{\mathbb{T}}\rightarrow{\mathcal{C}}$ for
the forgetful functor from the category of $\mathbb{T}$-algebras, then a
_lifting_ of $\mathbb{S}$ to ${\mathcal{C}}^{\mathbb{T}}$ is a monad
$\tilde{\mathbb{S}}$ on ${\mathcal{C}}^{\mathbb{T}}$ such that

$U^{\mathbb{T}}\circ\tilde{S}=S\circ U^{\mathbb{T}}\qquad
U^{\mathbb{T}}\circ\tilde{\nu}=\nu\circ U^{\mathbb{T}}\quad\text{and}\quad
U^{\mathbb{T}}\circ\tilde{\omega}=\omega\circ U^{\mathbb{T}}\hbox to0.0pt{
.\hss}$

On the other hand, if we write
$F_{\mathbb{S}}\colon{\mathcal{C}}\rightarrow{\mathcal{C}}_{\mathbb{S}}$ for
the free functor into the Kleisli category of $\mathbb{S}$, then an
_extension_ of $\mathbb{T}$ to ${\mathcal{C}}_{\mathbb{S}}$ is a monad
$\tilde{\mathbb{T}}$ on ${\mathcal{C}}_{\mathbb{S}}$ such that

$\tilde{T}\circ F_{\mathbb{S}}=F_{\mathbb{S}}\circ T\qquad\tilde{\eta}\circ
F_{\mathbb{S}}=F_{\mathbb{S}}\circ\eta\quad\text{and}\quad\tilde{\mu}\circ
F_{\mathbb{S}}=F_{\mathbb{S}}\circ\mu\hbox to0.0pt{ .\hss}$

######  Proposition 6.

[2, §1], [20, Theorem 2.5]. For monads $\mathbb{S}$, $\mathbb{T}$ on
${\mathcal{C}}$, there are bijections between distributive laws of
$\mathbb{S}$ over $\mathbb{T}$, liftings of $\mathbb{S}$ to
${\mathcal{C}}^{\mathbb{T}}$ and extensions of $\mathbb{T}$ to
${\mathcal{C}}_{\mathbb{S}}$.

###### Proof.

Given a distributive law $\delta\colon TS\Rightarrow ST$, we define the
corresponding lifting of $\mathbb{S}$ to $\mathbb{T}$-algebras to have action
on objects given by

$\tilde{S}(X,\,TX\ext@arrow
01{20}0\rightarrowfill@{}{x}X)\qquad=\qquad(SX,\,TSX\ext@arrow
01{20}0\rightarrowfill@{}{\delta_{X}}STX\ext@arrow
01{20}0\rightarrowfill@{}{Sx}SX)\hbox to0.0pt{ .\hss}$

and remaining data inherited from $\mathbb{S}$: thus $\tilde{S}(f)=Sf$,
${\tilde{\nu}}_{X}=\nu_{X}$ and ${\tilde{\omega}}_{X}=\omega_{X}$. Conversely,
for a lifting of $\mathbb{S}$ to $\mathbb{T}$-algebras with action
$\tilde{S}(X,x)=(SX,\sigma_{X,x})$, the corresponding distributive law
$\delta\colon TS\Rightarrow ST$ is given by:

(3.1) $\delta_{X}=TSX\ext@arrow
01{20}0\rightarrowfill@{}{TS\eta_{X}}TSTX\ext@arrow
01{20}0\rightarrowfill@{}{\sigma_{F^{\mathbb{T}}\\!X}}STX\hbox to0.0pt{
.\hss}$

Next, for a distributive law $\delta\colon TS\Rightarrow ST$, the
corresponding extension of $\mathbb{T}$ to ${\mathcal{C}}_{\mathbb{S}}$ is
given on objects by $\tilde{T}X=TX$ and on a Kleisli map from $X$ to $Y$ by

$\tilde{T}(X\ext@arrow 01{20}0\rightarrowfill@{}{f}SY)=TX\ext@arrow
01{20}0\rightarrowfill@{}{Tf}TSY\ext@arrow
01{20}0\rightarrowfill@{}{\delta_{Y}}STY\hbox to0.0pt{, \hss}$

while the unit and multiplication have components

$X\ext@arrow 01{20}0\rightarrowfill@{}{\eta_{X}}TX\ext@arrow
01{20}0\rightarrowfill@{}{\nu_{TX}}STX\qquad\text{and}\qquad TTX\ext@arrow
01{20}0\rightarrowfill@{}{\mu_{X}}TX\ext@arrow
01{20}0\rightarrowfill@{}{\nu_{TX}}STX\hbox to0.0pt{ .\hss}$

Conversely, given an extension $\tilde{\mathbb{T}}$ of $\mathbb{T}$, we may
view each map $1_{SX}\colon SX\rightarrow SX$ as a Kleisli map from $SX$ to
$X$, and applying $\tilde{T}$ yields a Kleisli map from $TSX$ to $TX$, which
provides the $X$-component of the corresponding distributive law:

$\tilde{T}(SX\ext@arrow 01{20}0\rightarrowfill@{}{1_{SX}}SX)=TSX\ext@arrow
01{20}0\rightarrowfill@{}{\delta_{X}}STX\hbox to0.0pt{ .\hss}\qed$

We can describe the algebras for the lifted monad $\tilde{\mathbb{S}}$
associated to a distributive law in various other ways. One is in terms of the
_composite monad_ $\mathbb{ST}$ on ${\mathcal{C}}$, which is the monad induced
by the composite adjunction
$\smash{({\mathcal{C}}^{\mathbb{T}})^{\tilde{\mathbb{S}}}\leftrightarrows{\mathcal{C}}^{\mathbb{T}}\leftrightarrows{\mathcal{C}}}$;
its underlying endofunctor is $ST$, its unit is $\nu\eta\colon 1\Rightarrow
ST$ and its multiplication is $\omega\mu\circ S\delta T\colon STST\Rightarrow
ST$. Another is in terms of “$\delta$-algebras”:

######  Definition 7.

Let $\delta\colon TS\Rightarrow ST$ be a distributive law of $\mathbb{S}$ over
$\mathbb{T}$. A _$\delta$ -algebra_ is an object $X\in{\mathcal{C}}$ endowed
with $\mathbb{T}$-algebra structure $t\colon TX\rightarrow X$ and
$\mathbb{S}$-algebra structure $s\colon SX\rightarrow X$ and rendering
commutative the diagram below. The category ${\mathcal{C}}^{\delta}$ of
$\delta$-algebras is the full subcategory of
${\mathcal{C}}^{\mathbb{S}}\times_{\mathcal{C}}{\mathcal{C}}^{\mathbb{T}}$ on
the $\delta$-algebras.

(3.2)
$\textstyle{TSX\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{X}}$$\scriptstyle{Ts}$$\textstyle{STX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{St}$$\textstyle{SX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{TX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{X}$

The basic result relating these notions is the following; for the proof, see
[2].

######  Lemma 8.

For any distributive law $\delta\colon TS\Rightarrow ST$ of $\mathbb{S}$ over
$\mathbb{T}$, there are canonical isomorphisms between the category of
$\tilde{\mathbb{S}}$-algebras in ${\mathcal{C}}^{\mathbb{T}}$, the category of
$\mathbb{ST}$-algebras in ${\mathcal{C}}$, and the category of
$\delta$-algebras in ${\mathcal{C}}$.

###  3.2. Weak distributive laws

As explained in the introduction, weak distributive laws generalise
distributive laws by relaxing the axioms relating to the monad units. There
are various ways of doing this, studied in Street [23] and Böhm [3], but we
will need only one, which we henceforth refer to with the unadorned name “weak
distributive law”. In the terminology of [3], our notion is that of a monad in
$\mathrm{\mathcal{E}{\mathcal{M}}}^{w}(\mathrm{\mathcal{C}at})$ whose $2$-cell
data satisfy the conditions of Lemma 1.2(3) of _ibid_.

######  Definition 9.

Let $\mathbb{S}=(S,\nu,\omega)$ and $\mathbb{T}=(T,\eta,\mu)$ be monads on a
category ${\mathcal{C}}$. A _weak distributive law of $\mathbb{S}$ over
$\mathbb{T}$_ is a natural transformation $\delta\colon TS\Rightarrow ST$
rendering commutative the three diagrams:

$\textstyle{TSS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta
S}$$\scriptstyle{T\omega}$$\textstyle{STS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\delta}$$\textstyle{SST\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega
T}$$\textstyle{TTS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\delta}$$\scriptstyle{\mu
S}$$\textstyle{TST\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta
T}$$\textstyle{STT\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\mu}$$\textstyle{{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\nu}$$\scriptstyle{\nu
T}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{ST}$$\textstyle{{TS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{ST}}$$\textstyle{{TS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{ST}\hbox
to0.0pt{ .\hss}}$

Thus, a weak distributive law in our sense simply drops from Beck’s definition
the axiom relating to the unit of $\mathbb{T}$. Such weak distributive laws
correspond to _weak_ liftings and to _weak_ extensions, where the definitions
of these are a bit more subtle.

######  Definition 10.

Let $\mathbb{S}=(S,\nu,\omega)$ and $\mathbb{T}=(T,\eta,\mu)$ be monads on a
category ${\mathcal{C}}$. A _weak lifting_ of $\mathbb{S}$ to
${\mathcal{C}}^{\mathbb{T}}$ comprises a monad $\tilde{\mathbb{S}}$ on
${\mathcal{C}}^{\mathbb{T}}$ and natural transformations

(3.3) $U^{\mathbb{T}}\tilde{S}\ext@arrow 04{22}0{\Rightarrowfill@}{}{\ \
\iota\ \ }SU^{\mathbb{T}}\ext@arrow 04{22}0{\Rightarrowfill@}{}{\ \ \pi\ \
}U^{\mathbb{T}}\tilde{S}$

such that $\pi\iota=1$, and such that each of the following diagrams commutes:

(3.4)
$\textstyle{U^{\mathbb{T}}\tilde{S}\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota\tilde{S}}$$\scriptstyle{U^{\mathbb{T}}\tilde{\omega}}$$\textstyle{SU^{\mathbb{T}}\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\iota}$$\textstyle{SSU^{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega
U^{\mathbb{T}}}$$\textstyle{{U^{\mathbb{T}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U^{\mathbb{T}}\tilde{\nu}}$$\scriptstyle{\nu
U^{\mathbb{T}}}$$\textstyle{U^{\mathbb{T}}\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{SU^{\mathbb{T}}}$$\textstyle{{U^{\mathbb{T}}\tilde{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{{SU^{\mathbb{T}}}}$
(3.5)
$\textstyle{SSU^{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega
U^{\mathbb{T}}}$$\scriptstyle{S\pi}$$\textstyle{SU^{\mathbb{T}}\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\tilde{S}}$$\textstyle{U^{\mathbb{T}}\tilde{S}\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U^{\mathbb{T}}\tilde{\omega}}$$\textstyle{{U^{\mathbb{T}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U^{\mathbb{T}}\tilde{\nu}}$$\scriptstyle{\nu
U^{\mathbb{T}}}$$\textstyle{SU^{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{U^{\mathbb{T}}\tilde{S}}$$\textstyle{{SU^{\mathbb{T}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{U^{\mathbb{T}}\tilde{S}}\hbox
to0.0pt{ ;\hss}}$

while a _weak extension_ of $\mathbb{T}$ to ${\mathcal{C}}_{\mathbb{S}}$
comprises a functor
$\tilde{T}\colon{\mathcal{C}}_{\mathbb{S}}\rightarrow{\mathcal{C}}_{\mathbb{S}}$
and natural transformation
$\tilde{\mu}\colon\tilde{T}\tilde{T}\Rightarrow\tilde{T}$ such that
$\tilde{T}\circ F_{\mathbb{S}}=F_{\mathbb{S}}\circ T$ and $\tilde{\mu}\circ
F_{\mathbb{S}}=F_{\mathbb{S}}\circ\mu$.

Note that our “weak liftings” are exactly the simultaneous weak $\iota$\- and
$\pi$-liftings of [3]. By exactly the same constructions as in Proposition 6,
we have:

######  Proposition 11.

For monads $\mathbb{S}$, $\mathbb{T}$ on ${\mathcal{C}}$, there is a bijective
correspondence between weak distributive laws of $\mathbb{S}$ over
$\mathbb{T}$ and weak extensions of $\mathbb{T}$ to
${\mathcal{C}}_{\mathbb{S}}$.

The correspondence between weak distributive laws and weak liftings is more
interesting. It is proved by Proposition 4.4 and Theorem 4.5 of [3] in a more
general context; however, for the particular kind of weakness we are
interested in, the following more direct proof is possible.

To begin with, we define a _semialgebra_ for a monad $\mathbb{T}=(T,\eta,\mu)$
to be a pair $(X\in{\mathcal{C}},x\colon TX\rightarrow X)$ satisfying the
associativity axiom $x.Tx=x.\mu_{X}$ but not necessarily the unit axiom
$x.\eta_{X}=1_{X}$. The $\mathbb{T}$-semialgebras form a category
${\mathcal{C}}^{\mathbb{T}}_{s}$, wherein a map from $(X,x)$ to $(Y,y)$ is a
map $f\colon X\rightarrow Y$ with $y.Tf=f.x$.

######  Lemma 12.

If idempotents split in ${\mathcal{C}}$, then the full inclusion
$I\colon{\mathcal{C}}^{\mathbb{T}}\rightarrow{\mathcal{C}}^{\mathbb{T}}_{s}$
has a simultaneous left and right adjoint
$K\colon{\mathcal{C}}^{\mathbb{T}}_{s}\rightarrow{\mathcal{C}}^{\mathbb{T}}$.

###### Proof.

For any $(X,x)\in{\mathcal{C}}^{\mathbb{T}}_{s}$ we have
$x.\eta_{X}.x=x.Tx.\eta_{TX}=x.\mu_{X}.\eta_{TX}=x=x.\mu_{X}.T\eta_{X}=x.Tx.T\eta_{X}$
so that $x.\eta_{X}\colon(X,x)\rightarrow(X,x)$ is an idempotent of
$\mathbb{T}$-semialgebras. Splitting this idempotent yields a diagram

(3.6)
$\textstyle{(X,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{(Y,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{(X,x)}$

in ${\mathcal{C}}^{\mathbb{T}}_{s}$ with $pi=1_{Y}$ and $ip=x.\eta_{X}$. The
semialgebra $(Y,y)$ is in fact a $\mathbb{T}$-algebra since
$y\eta_{Y}=piy\eta_{Y}=px.Ti.\eta_{Y}=px.\eta_{X}i=pipi=1_{Y}$. Moreover, if
$(Z,z)$ is a $\mathbb{T}$-algebra and $f\colon(X,x)\rightarrow(Z,z)$, then
$f=z\eta_{Z}.f=z.Tf.\eta_{X}=f.x.\eta_{X}=fip$ so that $f$ factors through
$p$. On the other hand, if $g\colon(Z,z)\rightarrow(X,x)$, then
$g=gz\eta_{Z}=x.Tg.\eta_{Z}=x.\eta_{X}.g=ipg$ so that $g$ factors through $i$.
Thus $i$ and $p$ exhibit $(Y,y)$ as the value at $(X,x)$ of the desired left
and right adjoint $K$. ∎

######  Proposition 13.

If idempotents split in ${\mathcal{C}}$, then for any monads $\mathbb{S}$,
$\mathbb{T}$ on ${\mathcal{C}}$, there is a bijective correspondence between
weak distributive laws of $\mathbb{S}$ over $\mathbb{T}$ and weak liftings of
$\mathbb{S}$ to ${\mathcal{C}}^{\mathbb{T}}$.

###### Proof.

Given a weak distributive law $\delta\colon TS\Rightarrow ST$, we may define a
_strict_ lifting $\check{\mathbb{S}}$ of $\mathbb{S}$ to $\mathbb{T}$-_semi_
algebras by taking, as in Proposition 6, $\check{S}(X,x)=(SX,Sx.\delta_{X})$
and with the remaining data inherited from $\mathbb{S}$. We now obtain the
desired _weak_ lifting $\tilde{\mathbb{S}}$ of $\mathbb{S}$ to
${\mathcal{C}}^{\mathbb{T}}$ as the monad generated by the composite
adjunction:

$\textstyle{{({\mathcal{C}}^{\mathbb{T}}_{s})^{\check{\mathbb{S}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U^{\check{\mathbb{S}}}}$$\scriptstyle{F^{\check{\mathbb{S}}}}$$\scriptstyle{\bot}$$\textstyle{{{{\mathcal{C}}^{\mathbb{T}}_{s}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I}$$\scriptstyle{K}$$\scriptstyle{\bot}$$\textstyle{{{\mathcal{C}}^{\mathbb{T}}}\hbox
to0.0pt{ .\hss}}$

In particular, $\tilde{S}$ sends a $\mathbb{T}$-algebra $(X,x)$ to the
$\mathbb{T}$-algebra obtained as the splitting

(3.7)
$\textstyle{(SX,Sx.\delta_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X,x}}$$\textstyle{\tilde{S}(X,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{X,x}}$$\textstyle{(SX,Sx.\delta_{X})}$

of the idempotent
$Sx.\delta_{X}.\eta_{SX}\colon(SX,Sx.\delta_{X})\rightarrow(SX,Sx.\delta_{X})$
in the category of $\mathbb{T}$-semialgebras. Applying the forgetful functor
${\mathcal{C}}^{\mathbb{T}}_{s}\rightarrow{\mathcal{C}}$ to (3.7) yields the
components of the $\iota$ and $\pi$ required in (3.3), and it is clear from
the manner of definition that the lifted unit $\tilde{\nu}$ is the _unique_
map rendering the triangles in (3.4) and (3.5) commutative. As for the lifted
multiplication $\tilde{\omega}$, a short calculation shows that, for any
$\mathbb{T}$-semialgebra $(X,x)$ with $\mathbb{T}$-algebra splitting (3.6),
the maps

$\textstyle{(SX,Sx.\delta_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Sp}$$\textstyle{(SY,Sy.\delta_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Y,y}}$$\textstyle{\tilde{S}(Y,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Y,y}}$$\textstyle{(SY,Sy.\delta_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Si}$$\textstyle{(SX,Sx.\delta_{X})}$

compose to the idempotent $Sx.\delta_{X}.\eta_{SX}$, and so exhibit
$\tilde{S}(Y,y)$ as the $\mathbb{T}$-algebra splitting of
$\check{S}(X,x)=(SX,Sx.\delta_{X})$. Thus, for any $\mathbb{T}$-algebra
$(X,x)$, the maps

$\textstyle{\check{S}\check{S}(X,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{S}\pi_{X,x}}$$\textstyle{\check{S}\tilde{S}(X,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\tilde{S}(X,x)}}$$\textstyle{\tilde{S}\tilde{S}(X,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\tilde{S}(X,x)}}$$\textstyle{\check{S}\tilde{S}(X,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{S}\iota_{X,x}}$$\textstyle{\check{S}\check{S}(X,x)}$

exhibit $\tilde{S}\tilde{S}(X,x)$ as the $\mathbb{T}$-algebra splitting of
$\check{S}\check{S}(X,x)$; whence $\tilde{\omega}$ is the _unique_ map
rendering commutative the rectangles in (3.4) and (3.5), as required.

This concludes the construction of a weak lifting from a weak distributive
law. Suppose conversely we have a weak lifting of $\mathbb{S}$ to
$\mathbb{T}$-algebras. For each $\mathbb{T}$-algebra $(X,x)$ with
$\tilde{S}(X,x)=(Y,y)$, define the map $\sigma_{X,x}\colon TSX\rightarrow SX$
as the composite

$TSX\ext@arrow 01{20}0\rightarrowfill@{}{T\pi_{X,x}}TY\ext@arrow
01{20}0\rightarrowfill@{}{y}Y\ext@arrow
01{20}0\rightarrowfill@{}{\iota_{X,x}}SX$

and now define $\delta\colon TS\Rightarrow ST$ to have components (3.1).
Direct calculation shows this to be a weak distributive law. ∎

Just as before, there are various ways of describing the algebras for the
weakly lifted monad $\tilde{\mathbb{S}}$ associated to a weak distributive
law. We can consider the composite monad $\widetilde{\mathbb{ST}}$ induced by
the adjunction
$\smash{({\mathcal{C}}^{\mathbb{T}})^{\tilde{\mathbb{S}}}\leftrightarrows{\mathcal{C}}^{\mathbb{T}}\leftrightarrows{\mathcal{C}}}$,
whose underlying endofunctor $\widetilde{ST}$ is obtained by splitting the
idempotent

$ST\ext@arrow 01{20}0\rightarrowfill@{}{\eta ST}TST\ext@arrow
01{20}0\rightarrowfill@{}{\delta T}STT\ext@arrow
01{20}0\rightarrowfill@{}{S\mu}ST\hbox to0.0pt{ ,\hss}$

or we can consider the category of $\delta$-algebras defined exactly as in
Definition 7. The relation between these notions is the same as before, and we
record it as follows; for the proof, we refer the reader to [3, Proposition
3.7].

######  Lemma 14.

For any weak distributive law $\delta\colon TS\Rightarrow ST$ of $\mathbb{S}$
over $\mathbb{T}$, there are canonical isomorphisms between the category of
$\tilde{\mathbb{S}}$-algebras in ${\mathcal{C}}^{\mathbb{T}}$, the category of
$\widetilde{\mathbb{ST}}$-algebras in ${\mathcal{C}}$, and the category of
$\delta$-algebras in ${\mathcal{C}}$.

##  4\. Weakly lifting the power-set monad

If, in the results of the previous section, we take ${\mathcal{C}}$ to be
$\mathrm{\mathcal{S}et}$, $\mathbb{S}$ to be the power-set monad, and
$\mathbb{T}$ to be any $\mathrm{\mathcal{S}et}$-monad, then we establish a
bijection between (weak) liftings of the power-set monad to
$\mathbb{T}$-algebras and (weak) extensions of $\mathbb{T}$ to
$\mathrm{\mathcal{S}et}_{\mathbb{P}}$. Now
$\mathrm{\mathcal{S}et}_{\mathbb{P}}$ is the category $\mathrm{\mathcal{R}el}$
of _sets and relations_ , and the possibility of extending structure from
$\mathrm{\mathcal{S}et}$ to $\mathrm{\mathcal{R}el}$ was analysed by [1], as
we now recall.

###  4.1. Extending structure from sets to relations

The category $\mathrm{\mathcal{R}el}$ has sets as objects, and as morphisms
$R\colon X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$, relations
$R\subseteq X\times Y$; we write $x\mathrel{R}y$ to indicate that $(x,y)\in
R$. Identity maps are equality relations, and the composite of $R\colon
X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$ and $S\colon
Y\relbar\joinrel\mapstochar\joinrel\rightarrow Z$ is given by:

$x\mathrel{SR}z\qquad\iff\qquad(\exists y\in
Y)\,(x\mathrel{R}y)\,\wedge\,(y\mathrel{S}z)\hbox to0.0pt{ .\hss}$

Under the identification of $\mathrm{\mathcal{R}el}$ as
$\mathrm{\mathcal{S}et}_{\mathbb{P}}$, the free functor
$F_{\mathbb{P}}\colon\mathrm{\mathcal{S}et}\rightarrow\mathrm{\mathcal{S}et}_{\mathbb{P}}$
corresponds to the identity-on-objects embedding
$({\mathord{\text{--}}})_{\ast}\colon\mathrm{\mathcal{S}et}\rightarrow\mathrm{\mathcal{R}el}$
which sends a function $f\colon X\rightarrow Y$ to its graph
$f_{\ast}=\\{(x,fx):x\in X\\}\subseteq X\times Y$. We also have the reverse
relation $f^{\ast}=\\{(fx,x):x\in X\\}\subseteq Y\times X$, and in fact,
relations of these two forms generate $\mathrm{\mathcal{R}el}$ under
composition, since every $R\colon
X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$ in $\mathrm{\mathcal{R}el}$
can be written as $q_{\ast}\circ p^{\ast}$ where $p\colon X\leftarrow
R\rightarrow Y\colon q$ are the two projections.

Importantly, $\mathrm{\mathcal{R}el}$ is not just a category; each hom-set is
partially ordered by inclusion, and composition preserves the order on each
side, so making $\mathrm{\mathcal{R}el}$ a _locally partially ordered
$2$-category_. With respect to this structure, it is easy to see for any
function $f\colon X\rightarrow Y$ that $f_{\ast}$ is _left adjoint_ to
$f^{\ast}$ in $\mathrm{\mathcal{R}el}$. This observation is key to proving the
following result, essentially due to Barr [1]; for a detailed proof, see [17].

######  Proposition 15.

Any $F\colon\mathrm{\mathcal{S}et}\rightarrow\mathrm{\mathcal{S}et}$ has at
most one extension to a $2$-functor
$\tilde{F}\colon\mathrm{\mathcal{R}el}\rightarrow\mathrm{\mathcal{R}el}$. This
exists just when $F$ is weakly cartesian, and is then defined on a relation
$R\colon X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$ with projections
$p\colon X\leftarrow R\rightarrow Y\colon q$ by

(4.1) $\tilde{F}(R)=(Fq)_{\ast}(Fp)^{\ast}\colon
FX\relbar\joinrel\mapstochar\joinrel\rightarrow FY\hbox to0.0pt{ .\hss}$

Any $\alpha\colon F\Rightarrow
G\colon\mathrm{\mathcal{S}et}\rightarrow\mathrm{\mathcal{S}et}$ has at most
one extension to a $2$-natural
$\tilde{\alpha}\colon\tilde{F}\Rightarrow\tilde{G}$. This exists just when
$\alpha$ is weakly cartesian, and has components
$(\tilde{\alpha})_{X}=(\alpha_{X})_{\ast}$.

Here, a functor
$F\colon\mathrm{\mathcal{S}et}\rightarrow\mathrm{\mathcal{S}et}$ is _weakly
cartesian_ if it preserves weak pullback squares, and a natural transformation
$\alpha\colon F\Rightarrow G$ is _weakly cartesian_ if its naturality squares
are weak pullbacks; recall that a weak pullback square is one for which the
induced comparison map into the pullback is an epimorphism.

######  Corollary 16.

For any monad $\mathbb{T}=(T,\eta,\mu)$ on $\mathrm{\mathcal{S}et}$:

  1. (i)

If $T$, $\eta$ and $\mu$ are all weakly cartesian, then there is a canonical
extension of $\mathbb{T}$ to $\mathrm{\mathcal{R}el}$, and so by Proposition
6, a canonical lifting of $\mathbb{P}$ to $\mathbb{T}$-algebras;

  2. (ii)

If only $T$ and $\mu$ are weakly cartesian, then there is still a canonical
_weak_ extension of $\mathbb{T}$ to $\mathrm{\mathcal{R}el}$, and so a
canonical _weak_ lifting of $\mathbb{P}$ to $\mathbb{T}$-algebras.

The intended application of this takes $\mathbb{T}$ to be the ultrafilter
monad, but before turning to this, we consider two simpler examples.

###  4.2. First example

Let $\mathbb{T}=(T,\eta,\mu)$ be the commutative monoid monad. This is an
_analytic monad_ in the sense of [15]—in fact, the terminal one—so that each
of $T$, $\eta$ and $\mu$ is weakly cartesian: thus $\mathbb{T}$ extends
_strictly_ from $\mathrm{\mathcal{S}et}$ to $\mathrm{\mathcal{R}el}$. Using
the formula (4.1), we see that the action of this extension on a relation
$R\colon X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$ is the relation
$\tilde{T}R\colon TX\relbar\joinrel\mapstochar\joinrel\rightarrow TY$ with

$x_{1}\cdots x_{n}\mathrel{\tilde{T}R}y_{1}\cdots
y_{m}\iff(\exists\,\sigma\colon\\!n\cong
m)(x_{1}\mathrel{R}y_{\sigma(1)})\wedge\dots\wedge(x_{n}\mathrel{R}y_{\sigma(n)})\hbox
to0.0pt{ .\hss}$

Plugging this in to the proof of Proposition 6, we see that the distributive
law corresponding to this extension has components $\delta_{X}\colon
TPX\rightarrow PTX$ given by

(4.2) $A_{1}\cdots A_{n}\quad\mapsto\quad\\{a_{1}\cdots a_{n}:\text{each
}a_{i}\in A_{i}\\}\hbox to0.0pt{ .\hss}$

and so that, under the identification of $\mathbb{T}$-algebras with
commutative monoids, the lifted monad $\tilde{\mathbb{P}}$ takes a commutative
monoid $(X,\cdot,1)$ to the commutative monoid with underlying set $PX$, unit
$\\{1\\}$ and multiplication $A\cdot B=\\{a\cdot b:a\in A,b\in B\\}$. The
algebras for the lifted monad $\tilde{\mathbb{P}}$ are precisely the
_commutative unital quantales_ : complete lattices $X$ endowed with a
commutative monoid structure $(X,\cdot,1)$ whose binary multiplication
preserves sups separately in each variable.

###  4.3. Second example

We now consider the finite power-set monad $\mathbb{P}_{f}$ on
$\mathrm{\mathcal{S}et}$, whose algebras are idempotent commutative monoids.
Unlike the commutative monoid monad, this does not extend strictly from
$\mathrm{\mathcal{S}et}$ to $\mathrm{\mathcal{R}el}$, due to:

######  Lemma 17.

The endofunctor and the multiplication of the finite power-set monad
$\mathbb{P}_{f}$ on $\mathrm{\mathcal{S}et}$ are weakly cartesian, but the
unit is not.

(In fact the same is true on replacing $\mathbb{P}_{f}$ by the full power-set
monad $\mathbb{P}$. In other words, $\mathbb{P}$ does not distribute over
itself; see [16].)

###### Proof.

For the endofunctor part, see, for instance, [13, Proposition 1.4]. For the
multiplication, we must show that, given a function $f\colon X\rightarrow Y$,
a finite subset $A\subseteq X$ and a finite subset
$B=\\{B_{1},\dots,B_{n}\\}\subseteq P_{f}Y$ with $f(A)=B_{1}\cup\dots\cup
B_{n}$, there exists a finite subset $\\{C_{1},\dots,C_{m}\\}\subseteq P_{f}X$
with $\\{f(C_{1}),\cdots,f(C_{m})\\}=B$ and $A=C_{1}\cup\dots\cup C_{m}$. We
have such on taking $m=n$ and $C_{i}=A\cap f^{-1}(B_{i})$. Finally, to see the
unit is not weakly cartesian, note that under the function
$\\{0,1\\}\rightarrow\\{0\\}$, the finite $\\{0,1\\}\subseteq\\{0,1\\}$ maps
to the singleton $\\{0\\}$, but is not itself a singleton. ∎

However, we still have a weak extension of $\mathbb{P}_{f}$ to
$\mathrm{\mathcal{R}el}$; this observation is apparently due to Ehrhard, and
is discussed in detail in [12]. Calculating explicitly using (4.1), we see
that the action of $\tilde{P}_{f}$ on a relation $R\colon
X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$ is given by the “Egli–Milner
relation”:

$A\mathrel{\tilde{P}_{f}R}B\iff(\forall a\in A.\,\exists b\in
B.\,a\mathrel{R}b)\wedge(\forall b\in B.\,\exists a\in A.\,a\mathrel{R}b)\hbox
to0.0pt{ .\hss}$

Thus, by Proposition 11, the weak distributive law corresponding to this weak
extension has components $\delta_{X}\colon P_{f}PX\rightarrow PP_{f}X$ given
by

${\mathcal{A}}\ \mapsto\ \\{B\subseteq X\text{
finite}:B\subseteq\textstyle\bigcup{\mathcal{A}}\text{ and }A\cap
B\neq\emptyset\text{ for all $A\in{\mathcal{A}}$}\\}\hbox to0.0pt{ .\hss}$

We now calculate the corresponding weak lifting of the power-set monad to the
category of $\mathbb{P}_{f}$-algebras. Given such an algebra $(X,x)$, we first
form the associated semialgebra $(PX,Px.\delta_{X})$, whose action map
$P_{f}PX\rightarrow PX$ is

$\displaystyle\\{A_{1},\dots,A_{n}\\}$ $\displaystyle\mapsto\\{a_{1}\cdots
a_{m}:\text{each $a_{i}$ is in some $A_{j}$, and some $a_{i}$ is in each
$A_{j}$}\\}\hbox to0.0pt{ .\hss}$

In particular, the idempotent $Px.\delta_{X}.\eta_{X}\colon PX\rightarrow PX$
takes $A\subseteq X$ to the set of non-empty finite products of elements of
$A$. Clearly $A$ is fixed by this idempotent just when it is a _subsemigroup_
—i.e., closed under binary multiplication.

It follows that the lifted monad $\tilde{\mathbb{P}}$ acts on $(X,x)$ to yield
the set $P_{\bullet}(X)$ of all subsemigroups of $X$, under the
$P_{f}$-algebra structure given as in the previous display. Reading off the
monoid structure from this, we see that the unit of $P_{\bullet}X$ is given by
$\\{1\\}$, while the binary multiplication is given by

$A\cdot B=\\{a_{1}\cdots a_{n}\cdot b_{1}\cdots b_{m}:n,m\geqslant 1\text{,
each $a_{i}\in A$ and each $b_{j}\in B$}\\}\hbox to0.0pt{ .\hss}$

In this expression, since $A$ and $B$ are already subsemigroups, we have that
$a=a_{1}\cdots a_{n}$ is itself in $A$ and $b=b_{1}\cdots b_{m}$ is itself in
$B$; so, more succinctly,

(4.3) $A\cdot B=\\{a\cdot b:a\in A,b\in B\\}\hbox to0.0pt{ ,\hss}$

i.e., the same formula that we derived in Section 4.2 for the commutative
monoid monad. It now follows from Lemma 14 that the algebras for the lifted
monad $\tilde{\mathbb{P}}$ are exactly the commutative (unital) quantales
whose multiplication is idempotent.

This example can be extended in various directions. Firstly, recall that a
_normal band_ is an idempotent semigroup satisfying the axiom $xyzw=xzyw$. The
free normal band on a set $X$ is the set $P_{f}^{\ast\ast}X$ of _bipointed_
finite subsets of $X$ under the multiplication $(A,a,b)\cdot(B,c,d)=(A\cup
B,a,d)$. The induced monad $\mathbb{P}_{f}^{\ast\ast}$ does not have weakly
cartesian unit, but has weakly cartesian endofunctor and multiplication; so we
have a weak lifting of $\mathbb{P}$ to the category of normal bands. Like
before, $\smash{\tilde{\mathbb{P}}}$ takes a normal band $X$ to the normal
band $P_{\bullet}X$ of sub-semigroups under the binary operation (4.3). This
construction is also given in [30], but without the monadic context.

A second direction of extension is to consider the monad $\mathbb{T}_{S}$ on
$\mathrm{\mathcal{S}et}$ whose algebras are semimodules over a given
commutative semiring $S$. In [4, Theorem 8.10] conditions are given on $S$
which characterise _precisely_ when the associated $\mathbb{T}_{S}$ has weakly
cartesian endofunctor and multiplication; under these conditions, then, we
obtain a weak lifting of the power-set monad to the category of $S$-modules.
Our two preceding examples fit into this framework on taking
$S=(\mathbb{N},\times,+)$ and $S=(\\{0,1\\},\wedge,\vee)$; as was shown in [4,
Example 9.5], other legitimate choices of $S$ include
$(\mathbb{Q}_{+},\times,+)$ and $(\mathbb{R}_{+},\times,+)$.

##  5\. The Vietoris monad and weak distributive laws

###  5.1. Recovering the Vietoris monad

We now prove our main theorem, recovering the Vietoris monad as the weak
lifting of the power-set monad associated to the canonical weak distributive
law of $\mathbb{P}$ over $\bbbeta$. We begin with the following well known
result; see, for example, [10, Examples III.1.12.3 and Proposition
III.1.12.4].

######  Lemma 18.

The endofunctor and multiplication of the monad $\bbbeta$ are weakly
cartesian, but the unit is not.

As such, we have a canonical weak extension of $\bbbeta$ to
$\mathrm{\mathcal{R}el}$. The action of
$\tilde{\beta}\colon\mathrm{\mathcal{R}el}\rightarrow\mathrm{\mathcal{R}el}$
on a relation $R\colon X\relbar\joinrel\mapstochar\joinrel\rightarrow Y$ is
given by

(5.1)
${\mathcal{F}}\mathrel{\tilde{\beta}R}{\mathcal{G}}\iff(A\in{\mathcal{F}}\implies
R(A)\in{\mathcal{G}})$

where we write $R(A)$ for $\\{y\in Y:(\exists a\in A)(a\mathrel{R}y)\\}$; see,
for example, [10, Examples III.1.10.3(3)]. Corresponding to this weak
extension, we have a weak distributive law of $\mathbb{P}$ over $\bbbeta$;
calculating from the above expression, we see that its components
$\delta_{X}\colon\beta PX\rightarrow P\beta X$ are given by

(5.2) $\delta_{X}(\mathbf{F})=\\{\,{\mathcal{F}}\in\beta
X:\textstyle\bigcup{\mathcal{A}}\in{\mathcal{F}}\text{ for all
}{\mathcal{A}}\in\mathbf{F}\,\\}\hbox to0.0pt{ .\hss}$

We now wish to calculate the associated weak lifting of $\mathbb{P}$ to
$\bbbeta$-algebras, i.e., to compact Hausdorff spaces. We begin with:

######  Lemma 19.

Let $(X,\xi\colon\beta X\rightarrow X)$ be a $\bbbeta$-algebra. The action map
$\beta PX\rightarrow PX$ of the semialgebra $(PX,P\xi.\delta_{X})$ is given by

(5.3) $\mathbf{F}\ \mapsto\
\textstyle\bigcap_{{\mathcal{A}}\in\mathbf{F}}\overline{\textstyle\bigcup{\mathcal{A}}}$

where $(\overline{\phantom{x}})$ denotes closure in the topology on $X$. The
idempotent function
$P\xi.\delta_{X}.\eta_{PX}\colon(PX,P\xi.\delta_{X})\rightarrow(PX,P\xi.\delta_{X})$
sends each $B\in PX$ to its closure.

###### Proof.

Given $x\in X$, we have
$x\in\bigcap_{{\mathcal{A}}\in\mathbf{F}}\overline{\bigcup{\mathcal{A}}}$ if
and only if each open neighbourhood of $x$ meets each $\bigcup{\mathcal{A}}$,
if and only if there exists an ultrafilter containing each
$\bigcup{\mathcal{A}}$ and converging to $x$. But by (5.2), this happens just
when $x\in P\xi(\delta_{X}(\mathbf{F}))$. Finally, since $\eta_{PX}\colon
PX\rightarrow\beta PX$ sends $B$ to the ultrafilter $\\{{\mathcal{A}}\subseteq
PX:B\in{\mathcal{A}}\\}$, the idempotent $P\xi.\delta_{X}.\eta_{PX}$ sends
each $B\in PX$ to
$\bigcap_{B\in{\mathcal{A}}}\overline{\bigcup{\mathcal{A}}}=\overline{B}$. ∎

######  Theorem 20.

The Vietoris monad on the category of compact Hausdorff spaces is the weak
lifting of the power-set monad associated to the canonical weak distributive
law of the power-set monad over the ultrafilter monad.

###### Proof.

Let $X$ be a compact Hausdorff space seen as a $\bbbeta$-algebra
$(X,\xi\colon\beta X\rightarrow X)$. The $\bbbeta$-algebra $\tilde{P}(X,\xi)$
is obtained by splitting the idempotent $P\xi.\delta_{X}.\eta_{PX}$ on
$(PX,P\xi.\delta_{X})$; so by the previous lemma, its underlying set is the
set $VX$ of _closed_ subsets of $X$, and its $\bbbeta$-algebra structure is
given by the same formula as in (5.3). By naturality in (3.3), the action of
$\tilde{P}$ on maps is given by direct image, while by the formulae in (3.4),
the unit and multiplication of $\tilde{\mathbb{P}}$ are once again given by
inclusion of singletons and set-theoretic union.

As such, it remains only to show that the $\bbbeta$-algebra structure on
$\tilde{P}(X,\xi)$ describes the Vietoris topology; in light of Lemma 19, we
must thus show that any ${\mathcal{F}}\in\beta VX$ converges in the Vietoris
topology to the unique point
$L=\textstyle\bigcap_{A\in{\mathcal{F}}}\overline{\textstyle\bigcup A}$. This
follows from Lemma 22 below, since the Vietoris topology on $VX$ is the Lawson
topology on the continuous lattice $(VX,\supseteq)$.∎

###  5.2. Vietoris algebras

The composite monad associated to the weak distributive law of $\mathbb{P}$
over $\mathbb{V}$ is easily seen to be the well known _filter monad_
$\mathbb{F}$; as such, Lemma 14 asserts a canonical isomorphism between the
categories of $\mathbb{V}$-algebras in $\mathrm{\mathcal{K}{\mathcal{H}}aus}$
and of $\mathbb{F}$-algebras in $\mathrm{\mathcal{S}et}$. This was originally
proven as [28, Theorem 6.3] and is, in fact, how Wyler identified the
$\mathbb{V}$-algebras as the continuous lattices—by first identifying the
$\mathbb{F}$-algebras as such (a result originally proved by Day [6]).

Now Lemma 14 also identifies $\mathbb{V}$-algebras with $\delta$-algebras for
the weak distributive law $\delta\colon\beta P\Rightarrow P\beta$, i.e., as
sets $X$ endowed with $\bbbeta$-algebra structure $\xi\colon\beta X\rightarrow
X$ and $\mathbb{P}$-algebra structure $i\colon PX\rightarrow X$ subject to
commutativity in

(5.4) $\textstyle{\beta
PX\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{X}}$$\scriptstyle{\beta
i}$$\textstyle{P\beta
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P\xi}$$\textstyle{PX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\beta
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{X\hbox
to0.0pt{ .\hss}}$

In [28], Wyler does note that a $\mathbb{V}$-algebra is a $\bbbeta$-algebra
and a $\mathbb{P}$-algebra subject to some compatibility—see, for example,
Proposition 6.4 of _ibid._ —but does not express this in terms of the square
(5.4). In fact, by using (5.4) it is easy to give a _direct_ proof that
Vietoris algebras are continuous lattices, as we will now do.

In what follows, given a filter ${\mathcal{F}}$ on a topological space, we
write $\operatorname{adh}{\mathcal{F}}$ for
$\bigcap_{A\in{\mathcal{F}}}\overline{A}$; recall that, for an ultrafilter
${\mathcal{F}}$, the points in $\operatorname{adh}{\mathcal{F}}$ are precisely
those to which ${\mathcal{F}}$ converges. On the other hand, for a filter
${\mathcal{F}}$ on a complete lattice, we write $\liminf{\mathcal{F}}$ for
$\sup\\{\inf A:A\in{\mathcal{F}}\\}$.

######  Proposition 21.

Let $X$ be a complete lattice and a compact Hausdorff space, seen as a
$\bbbeta$-algebra $\xi\colon\beta X\rightarrow X$ via ultrafilter convergence
and as a $\mathbb{P}$-algebra $i\colon PX\rightarrow X$ by taking infima. The
following are equivalent:

  1. (i)

$(X,\xi,i)$ is a $\delta$-algebra, i.e., renders (5.4) commutative.

  2. (ii)

$\liminf{\mathcal{F}}=\inf\operatorname{adh}{\mathcal{F}}$ for any filter
${\mathcal{F}}$ on $X$.

###### Proof.

_(ii) $\Rightarrow$ (i)_. Let $\mathbf{F}\in\beta{\mathcal{P}}X$. By Lemma 19,
the upper path around (5.4) takes $\mathbf{F}$ to
$\inf\operatorname{adh}\bigcup\mathbf{F}$ where $\bigcup\mathbf{F}$ is the
filter generated by $\bigcup{\mathcal{A}}$ for each
${\mathcal{A}}\in\mathbf{F}$. The lower path takes $\mathbf{F}$ to the limit
point $\inf\operatorname{adh}{\mathcal{F}}$ of the ultrafilter ${\mathcal{F}}$
generated by the sets ${\mathcal{A}}^{i}=\\{\inf A:A\in{\mathcal{A}}\\}$ for
all ${\mathcal{A}}\in\mathbf{F}$. So given (ii), it suffices to show that
$\liminf\bigcup\mathbf{F}=\liminf{\mathcal{F}}$, which follows since
$\inf{\mathcal{A}}^{i}=\inf(\bigcup{\mathcal{A}})$ for all
${\mathcal{A}}\in\mathbf{F}$.

_(i) $\Rightarrow$ (ii)_ We first show that for all $a\in X$, the principal
upset $\mathop{\uparrow}a$ and downset $\mathop{\downarrow}a$ are closed.
Consider for any ultrafilter ${\mathcal{F}}$ on $X$ the ultrafilter
$\mathbf{F}$ on ${\mathcal{P}}X$ generated by the sets
$\\{a,B\\}=\\{\\{a,b\\}:b\in B\\}$ for all $B\in{\mathcal{F}}$. Note that:

  * •

If $\mathop{\downarrow}a\in{\mathcal{F}}$, then for each $B\in{\mathcal{F}}$
also $B_{a}=B\cap{\mathop{\downarrow}a}\in{\mathcal{F}}$. Thus $B\supseteq
B_{a}=i_{!}(\\{a,B_{a}\\})\in Pi(\mathbf{F})$, so that
$Pi(\mathbf{F})={\mathcal{F}}$ and $\xi(Pi(\mathbf{F}))=\xi({\mathcal{F}})$.

  * •

If $\mathop{\uparrow}a\in{\mathcal{F}}$, then $\\{a\\}\in Pi(\mathbf{F})$
whence $\xi(Pi(\mathbf{F}))=a$.

In either case,
$P\xi(\delta_{X}(\mathbf{F}))=\bigcap_{B\in{\mathcal{F}}}\overline{\\{a\\}\cup
B}=\\{a\\}\cup\bigcap_{B\in{\mathcal{F}}}\overline{B}=\\{a,\xi({\mathcal{F}})\\}$
by Lemma 19, and so
$i(P\xi(\delta_{X}(\mathbf{F})))=a\wedge\xi({\mathcal{F}})$. So by the
assumption, if $\mathop{\downarrow}a\in{\mathcal{F}}$ then
$\xi({\mathcal{F}})=a\wedge\xi({\mathcal{F}})$ so that
$\xi({\mathcal{F}})\in\mathop{\downarrow}a$; while if
$\mathop{\uparrow}a\in{\mathcal{F}}$ then $a=a\wedge\xi({\mathcal{F}})$ so
that $\xi({\mathcal{F}})\in\mathop{\uparrow}a$. This proves that both
$\mathop{\downarrow}a$ and $\mathop{\uparrow}a$ are closed.

We now prove (ii). Given a filter ${\mathcal{F}}$ on $X$, the family of
subsets of ${\mathcal{P}}X$ given by ${\mathcal{F}}$ together with
$\mathop{\downarrow}A$ for each $A\in{\mathcal{F}}$ has the finite
intersection property; let $\mathbf{F}$ be any ultrafilter on ${\mathcal{P}}X$
which extends it. Now, for each $A\in{\mathcal{F}}$ we have
$\mathop{\downarrow}A\in\mathbf{F}$ and so
$A=\bigcup\mathop{\downarrow}A\in\bigcup\mathbf{F}$. On the other hand, each
${\mathcal{A}}\in\mathbf{F}$ meets ${\mathcal{F}}$, say in $A$ and so
$\bigcup{\mathcal{A}}\supseteq A$ is in ${\mathcal{F}}$. So
$\bigcup\mathbf{F}={\mathcal{F}}$, and so by Lemma 19 the upper path around
(5.4) takes $\mathbf{F}$ to $\inf\operatorname{adh}{\mathcal{F}}$. As for the
lower path, $\beta i(\mathbf{F})$ contains $\\{\inf A:A\in{\mathcal{F}}\\}$
and $\mathop{\uparrow}(\inf A)$ for each $A\in{\mathcal{F}}$; so $\xi(\beta
i(\mathbf{F}))$ is contained in the intersection of closed sets
$\bigcap\mathop{\uparrow}(\inf A)=\mathop{\uparrow}(\liminf{\mathcal{F}})$,
but also in the closed set
$\mathop{\downarrow}(\liminf{\mathcal{F}})\supseteq\\{\inf
A:A\in{\mathcal{F}}\\}$ and so must equal $\liminf{\mathcal{F}}$. Thus
$\inf\operatorname{adh}{\mathcal{F}}=\lim\inf{\mathcal{F}}$ as desired. ∎

The remainder of the argument is standard continuous lattice theory, contained
in, say, [8]; we include it here for the sake of a self-contained
presentation.

######  Lemma 22.

An ultrafilter ${\mathcal{F}}$ on a continuous lattice converges in the Lawson
topology to the unique point $\ell=\liminf{\mathcal{F}}$.

###### Proof.

We first show ${\mathcal{F}}$ contains every subbasic open neighbourhood of
$\ell$. First, if $\ell\in s^{+}$, i.e., $s\ll\ell$, then $s\ll\inf A$ for
some $A\in{\mathcal{F}}$, and so $s\ll a$ for all $a\in A$; whence $A\subseteq
s^{+}$ and so $s^{+}\in{\mathcal{F}}$. Next, if $\ell\in s^{-}$, i.e.,
$s\nleqslant\ell$ then $s\nleqslant\inf A$ for all $A\in{\mathcal{F}}$. So for
each $A\in{\mathcal{F}}$, we have $s\nleqslant a$ for some $a\in A$, i.e.,
each $A\in{\mathcal{F}}$ meets $s^{-}$, and so, since ${\mathcal{F}}$ is an
ultrafilter, we have $s^{-}\in{\mathcal{F}}$. Thus ${\mathcal{F}}$ converges
to $\ell$; suppose it also converges to $y$. Then for each $s\ll y$ we have
$s^{+}\in{\mathcal{F}}$ and so $s\leqslant\inf s^{+}\leqslant\ell$. Since
$y=\bigvee\\{s:s\ll y\\}$ we must have $y\leqslant\ell$. We claim
$\ell\leqslant y$, i.e., $\inf A\leqslant y$ for each $A\in{\mathcal{F}}$. If
not, then $\inf A\nleqslant y$ for some $A\in{\mathcal{F}}$, so that $(\inf
A)^{-}$ is in ${\mathcal{F}}$. So $(\inf A)^{-}$ and $A$ are disjoint sets in
${\mathcal{F}}$, a contradiction. ∎

######  Proposition 23.

Let $X$ be a complete lattice and a compact Hausdorff space. The following are
equivalent:

  1. (i)

$\liminf{\mathcal{F}}=\inf\operatorname{adh}{\mathcal{F}}$ for any filter
${\mathcal{F}}$ on $X$;

  2. (ii)

$X$ is a continuous lattice and its topology is the Lawson topology.

###### Proof.

_(i) $\Rightarrow$ (ii)_. By the proof of Proposition 21, each principal upset
$\mathop{\uparrow}x$ is closed in $X$. We now show that, if $U$ is up-closed
and open and $x\in U$, then $\inf U\ll x$. Indeed, suppose $x\leqslant\sup D$
for some directed $D\subseteq X$. Then $\mathop{\uparrow}(\sup D)\subseteq U$
since $U$ is up-closed, and so $\emptyset=(X\setminus
U)\cap\mathop{\uparrow}(\sup D)=(X\setminus
U)\cap\bigcap\\{\mathop{\uparrow}d:d\in D\\}$. By compactness of $X$ and
downward-directedness of $\\{\mathop{\uparrow}d:d\in D\\}$, it follows that
$\emptyset=(X\setminus U)\cap\mathop{\uparrow}d$ for some $d\in D$; thus $d\in
U$ and so $\inf U\leqslant d$ as required.

We now show that $X$ is continuous. Let $x\in X$ and let ${\mathcal{F}}$ be
the neighbourhood filter of $x$. Since $X$ is Hausdorff, we have
$\operatorname{adh}{\mathcal{F}}=\\{x\\}$ and so by (i) that
$\liminf{\mathcal{F}}=x$. Clearly $\liminf{\mathcal{F}}$ is the supremum of
$\\{\inf U:U\text{ up-closed in }{\mathcal{F}}\\}$, and by above $\inf U\ll x$
for each such $U$. It follows that $x=\sup\\{s:s\ll x\\}$ so that $X$ is
continuous. Finally, the condition
$\liminf{\mathcal{F}}=\inf\operatorname{adh}{\mathcal{F}}$ applied to an
ultrafilter implies by Lemma 22 that the topology on $X$ is the Lawson
topology.

_(ii) $\Rightarrow$ (i)_. We first show $\inf A=\inf\overline{A}$ for any
$A\subseteq X$. Clearly $\inf\overline{A}\leqslant\inf A$; while if
$x\in\overline{A}$, then $x$ is the convergence point of some ultrafilter
${\mathcal{F}}$ containing $A$, whence $x=\liminf{\mathcal{F}}\geqslant\inf
A$, so that $\inf A\leqslant\inf\overline{A}$. We now prove (i). Given a
filter ${\mathcal{F}}$, we have for each $A\in{\mathcal{F}}$ that $\inf
A=\inf\overline{A}\leqslant\inf\operatorname{adh}{\mathcal{F}}$ and so
$\liminf{\mathcal{F}}\leqslant\inf\operatorname{adh}{\mathcal{F}}$. It remains
to show $\inf\operatorname{adh}{\mathcal{F}}\leqslant\liminf{\mathcal{F}}$. By
continuity, we can write $\inf\operatorname{adh}{\mathcal{F}}$ as
$\bigvee\\{s:s\ll\inf\operatorname{adh}{\mathcal{F}}\\}$, so it suffices to
show $s\ll\inf\operatorname{adh}{\mathcal{F}}$ implies
$s\leqslant\liminf{\mathcal{F}}$. We prove the contrapositive: if
$s\nleqslant\liminf{\mathcal{F}}$ then
$s\not\ll\inf\operatorname{adh}{\mathcal{F}}$. Now if
$s\nleqslant\liminf{\mathcal{F}}$, then $s\nleqslant\inf A=\inf\overline{A}$
for each $A\in{\mathcal{F}}$. Thus, for each $A\in{\mathcal{F}}$ there is some
$a\in\overline{A}$ with $s\nleqslant a$ and hence $s\not\ll a$. This says that
the closed set $X\setminus s^{+}$ meets $\bar{A}$ for each
$A\in{\mathcal{F}}$; whence
$\\{\overline{A}:A\in{\mathcal{F}}\\}\cup\\{X\setminus s^{+}\\}$ has the
finite intersection property, so that by compactness, $X\setminus s^{+}$ meets
$\bigcap_{A\in{\mathcal{F}}}\overline{A}=\operatorname{adh}{\mathcal{F}}$.
This means $s\not\ll a$ for some $a\in\operatorname{adh}{\mathcal{F}}$, and so
$s\not\ll\inf\operatorname{adh}{\mathcal{F}}$ as desired. ∎

So $\delta$-algebras are continuous lattices, and it is easy to identify the
corresponding $\delta$-algebra maps as the inf- and directed-sup preserving
functions. We thus recover:

######  Theorem 24.

[28] The category of $\mathbb{V}$-algebras is isomorphic to the category of
continuous lattices with inf- and directed sup-preserving maps.

###  5.3. Variations

It is natural to consider variations on Theorem 20, involving different weak
distributive laws on possibly different categories. Treating these in detail
must await further work, but it is worth sketching a couple of possibilities.

On the one hand, we may replace $\mathbb{P}$ by the _non-empty_ power-set
monad $\mathbb{P}_{+}$ on $\mathrm{\mathcal{S}et}$, while keeping $\bbbeta$
the same. In this case, we expect to obtain a weak distributive law of
$\mathbb{P}_{+}$ over $\bbbeta$ whose corresponding weak lifting to the
category $\mathrm{\mathcal{K}{\mathcal{H}}aus}$ of $\bbbeta$-algebras is the
_proper_ Vietoris monad $\mathbb{V}_{+}$. This monad, considered in [29],
sends a compact Hausdorff space $X$ to its set of non-empty closed subsets
under the Vietoris topology, and has as its algebras the continuous
_semilattices_.

On the other hand, we can replace $\mathbb{P}$ by the _upper-set monad_
$\mathbb{P}^{\uparrow}$ on the category of posets, and $\bbbeta$ by the _prime
filter monad_ $\mathbb{Pf}$. As in [7], this latter monad has as algebras the
_compact pospaces_ —compact spaces $X$ with a partial order $\leqslant$ which
is closed in $X\times X$. In this case, via the partially ordered version of
Barr’s relation lifting [17, Section 3.3], we expect to find a weak
distributive law of $\mathbb{P}^{\uparrow}$ over $\mathbb{Pf}$, with
corresponding weak lifting the “ordered Vietoris monad”
$\mathbb{V}^{\uparrow}$ on compact pospaces. This takes a compact pospace $X$
to its space $V^{\uparrow}X$ of closed upper-sets ordered by reverse
inclusion, with a modified version of the Vietoris topology; see [8, Example
VI-3.10]. As explained in [9], the $\mathbb{V}^{\uparrow}$-algebras are, once
again, the continous lattices. In fact, [9] describes other Vietoris-like
monads, and it is natural to hope that these may arise in a similar manner;
but again, we leave this to future work.

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