# Canonical extensions of locally compact frames

Tomáš Jakl  Laboratoire J. A. Dieudonné, CNRS and Université Côte d’Azur,
06108 Nice Cedex 02, France<EMAIL_ADDRESS>
(December 1, 2018)

###### Abstract

Canonical extension of finitary ordered structures such as lattices, posets,
proximity lattices, etc., is a certain completion which entirely describes the
topological dual of the ordered structure and it does so in a purely algebraic
and choice-free way. We adapt the general algebraic technique that constructs
them to the theory of frames.

As a result, we show that every locally compact frame embeds into a completely
distributive lattice by a construction which generalises, among others, the
canonical extensions for distributive lattices and proximity lattices. This
construction also provides a new description of a construction by Marcel Erné.
Moreover, canonical extensions of frames enable us to frame-theoretically
represent monotone maps with respect to the specialisation order.

###### keywords:

canonical extensions , frames , pointfree topology , duality theory

###### MSC:

[2010] 06D22 , 06B35 , 06B23

††journal: Topology and its Applications

##  1 Introduction

Initially canonical extensions were defined by Jónsson and Tarski [21, 22] in
order to understand topological duals of Boolean algebras with additional
operations. The theory of canonical extensions was later extended to
distributive lattices, posets, proximity lattices, coherent categories and
more [15, 14, 13, 12, 16, 26, 3].

Intuitively, canonical extension of a distributive lattice algebraically
represents the lattice embedding of the compact open subsets of its Stone dual
into the complete lattice of saturated sets, that is, upsets with respect to
the specialisation order. Because the construction is purely algebraic,
canonical extension of a distributive lattice can be thought of as a point-
free and choice-free description of the frame of all saturated subsets of its
dual space.

There is nothing specific about distributive lattices and their Stone duals.
We can ask whether there is an analogous construction for frames and
topological spaces. In particular, the frame of open sets of a topological
space always embeds into the frame of saturated subsets. The problem therefore
is to give a choice-free and purely frame-theoretic or algebraic construction
of the frame of saturated subsets.

We solve this question for the class of locally compact frames. This is a wide
class of frames which includes all compact regular frames as well as coherent
frames. In fact our technique works for an even more general class of frames,
that is, for frames with enough compact fitted sublocales. We show that every
locally compact frame canonically embeds into a complete lattice, which it
meet-generates, and which is isomorphic to the frame of saturated subsets of
the dual space of the frame we started with. From this it follows that the
canonical extension of the frame is a completely distributive lattice,
although this fact is not choice-free. Moreover, we also show how to lift
perfect homomorphisms between locally compact frames to complete lattice
homomorphisms between their canonical extensions.

Our construction generalises the distributive case of Sam van Gool [26] which
solves this question for proximity lattices and, therefore, also indirectly
for stably compact frames. Moreover, it turns out that our construction is
exactly the same as the one considered by Marcel Erné in [5] in which he
constructs a certain frame extension for every preframe. We also show how one
can use canonical extensions to represent monotone maps with respect to
specialisation order, in a point-free way.

##  2 Preliminaries

For a map $f\colon X\to Y$ between sets $X$ and $Y$, it will be important
throughout the text to notationally distinguish its application to one
element, from its forward image map $f[\mkern 1.0mu\cdot\mkern
1.0mu]\colon\raisebox{2.03998pt}{\Large$\wp$}(X)\to\raisebox{2.03998pt}{\Large$\wp$}(Y)$,
and the preimage map $f^{-1}[\mkern 1.0mu\cdot\mkern
1.0mu]\colon\raisebox{2.03998pt}{\Large$\wp$}(Y)\to\raisebox{2.03998pt}{\Large$\wp$}(X)$,
that is, from the maps

$f[M]=\\{f(x)~{}|~{}x\in M\\},\enspace\text{and}\enspace f^{-1}[N]=\\{x\in
X~{}|~{}f(x)\in N\\}.$

In the following we summarise the main facts on order theory and point-free
topology which we will rely on. Proofs of most of the results summarised in
this section can be found in the standard reference book [25].

###  2.1 Topological spaces

Every topological space $(X,\tau)$ carries the so-called _specialisation
order_ on the set of its points:

$\forall x,y\in X.\quad x\leq_{\tau}y\quad\text{iff}\quad\forall U\in\tau.\
x\in U\implies y\in U.$

_Saturated_ subsets of a space are those subsets which are upward closed with
respect to the specialisation order. We denote the set of all saturated
subsets/upsets of a space $(X,\tau)$ by $\mathsf{Up}(X,\leq_{\tau})$, or
sometimes just $\mathsf{Up}(X)$. Note that a subset of a space $M\subseteq X$
is saturated if and only if $M=\bigcap\\{U\in\tau~{}|~{}M\subseteq U\\}$ or
equivalently if there is a collection $\mathscr{U}\subseteq\tau$ such that
$M=\bigcap\mathscr{U}$ (Proposition 4.2.9 in [18]). Note also that all open
subsets of a space are saturated and so $\tau\subseteq\mathsf{Up}(X)$.

###  2.2 Galois adjunctions

Order preserving function $f\colon A\to B$ between complete lattices preserves
arbitrary joins if and only if there is a monotone function $g\colon B\to A$
(in fact, meet preserving) such that

$f(a)\leq b\quad\text{iff}\quad a\leq g(b),$

or equivalently

$f(g(b))\leq b\quad\text{iff}\quad a\leq g(f(a)).$

We call $f$ the _left (Galois) adjoint_ and $g$ the _right (Galois) adjoint_.
It follows that $f=fgf$ and $g=gfg$ and so by setting $c=gf$ we obtain a
_closure operator_ $c\colon A\to A$ on $A$ which satisfies the following
conditions, for all $a,a^{\prime}\in A$:

$a\leq a^{\prime}\implies c(a)\leq c(a^{\prime}),\quad a\leq
c(a),\quad\text{and}\quad c(c(a))\leq c(a).$

The fixpoints of $c\colon A\to A$ form a complete lattice, in the order
inherited from $A$.

Note that if we start from two _antitone_ maps $f\colon A\rightleftarrows
B\colon g$ such that $a\leq g(b)$ iff $b\leq f(a)$ then we speak of an
_antitone Galois adjunction_. In this case both maps $f$ and $g$ transform
arbitrary joins into meets. In fact, antitone Galois adjunctions are
historically the more standard setting.

###  2.3 Frames

A complete lattice $L$ is a _frame_ if, for every $A\subseteq L$ and $b\in L$,

$(\bigvee A)\wedge b=\bigvee_{a\in A}(a\wedge b).$

Similarly, a _preframe_ $P$ is a poset which has finite meets, directed joins
and, for every _directed_ $D\subseteq P$ and $b\in P$,

$(\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}D)\wedge
b=\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}_{d\in
D}(d\wedge b),$

where $\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}D$ denotes
the supremum of the directed set $D$. Homomorphisms of frames (resp.
preframes) are the maps which preserve arbitrary joins (resp. directed joins)
and finite meets. We say that a poset is a _coframe_ if it is a frame in the
opposite order and, similarly, it is a _co-preframe_ if it is a preframe in
the opposite order.

Every topological space $(X,\tau)$ gives rise to a frame
$\mathcal{O}(X,\tau)=\tau$ and, vice versa, every frame $L$ gives rise to a
space $\mathsf{pt}(L)$ whose points are the frame homomorphisms $p\colon
L\to\mathbf{2}$ into the two-element frame $\mathbf{2}$. The topology of
$\mathsf{pt}(L)$ consists of the sets of the form
$\\{p\in\mathsf{pt}(L)~{}|~{}p(a)=1\\}$, for some $a\in L$. Moreover, the
mappings $X\mapsto\mathcal{O}(X)$ and $L\mapsto\mathsf{pt}(L)$ extend to a
pair of dually adjoint functors, with the actions on morphisms defined as

$\displaystyle f\colon X\to Y$
$\displaystyle\enspace\longmapsto\enspace\mathcal{O}(f)\colon\mathcal{O}(Y)\to\mathcal{O}(X),\
V\mapsto f^{-1}[V],\enspace\text{and}\enspace$ $\displaystyle h\colon L\to M$
$\displaystyle\enspace\longmapsto\enspace\mathsf{pt}(h)\colon\mathsf{pt}(M)\to\mathsf{pt}(L),\
p\mapsto p\circ h.$

In case when a frame is isomorphic to $\mathcal{O}(X)$, for some space $X$, we
say that the frame is _spatial_ or that it has _enough points_. Conversely,
spaces which are homeomorphic to $\mathsf{pt}(L)$, for some frame $L$, are
called _sober_.

###  2.4 Scott-open filters and locally compact frames

A _filter_ $F\subseteq A$ of a frame or distributive lattice111We assume that
all lattices are bounded. $A$ is an upset closed under finite meets. In
particular, $F$ is non-empty as it contains the empty meet, i.e. the top
element. Similarly, a subset $I\subseteq A$ is an _ideal_ if it is a filter in
$A^{\text{op}}$, that is, in $A$ ordered by the opposite order. The frame of
ideals ordered by set inclusion is denoted by $\mathsf{Idl}(A)$. Similarly,
the _co_ frame of filters ordered by the reverse inclusion is denoted by
$\mathsf{Filt}(A)$. Note that filtered meets, binary joins and binary meets of
filters are computed as follows:

$\bigwedge_{i}\\!{}^{\mathord{\downarrow}\mkern 1.0mu}\
F_{i}=\operatorname*{\bigcup\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}F_{i},\quad
F\vee G=F\cap G,\quad F\wedge G=\\{f\wedge g~{}|~{}f\in F,\ g\in G\\}.$

A filter $F$ of a frame $L$ is called _Scott-open_ if, whenever
$\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}D\in F$ for some
directed $D\subseteq L$, then already $d\in F$ for some $d\in D$. Note that
the poset of Scott-open filters on a frame $L$ is a sub-co-preframe of
$\mathsf{Filt}(L)$, which we will denote by $L^{\wedge}$. (In our notation, we
will always denote Scott-open filters by $\kappa$, $\lambda$, or $\chi$.)
Moreover, Scott-open filters are in a bijective correspondence with _pre_
frame homomorphisms $L\to\mathbf{2}$. Consequently, a preimage of a Scott-open
filter by a preframe homomorphism is again a Scott-open filter.

We prefer to order filters by reverse inclusion because then we can interpret
filters as a point-free representation of saturated subsets. This is justified
because such order makes the following two maps monotone (for a sober space
$X$)

$M\in\mathsf{Up}(X)\mapsto\\{U\in\mathcal{O}(X)~{}|~{}M\subseteq
U\\}\quad\text{and}\quad F\in\mathsf{Filt}(\mathcal{O}(X))\mapsto\bigcap F.$

This representation of saturated subsets as filters is not unique in general.
However, if we restrict to Scott-open filters then we obtain a unique
representation of compact saturated subsets. This is the celebrated
_Hofmann–Mislove Theorem_ [19, 24].

A frame $L$ is called _locally compact_ (or _continuous_), if
$a=\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}\\{c~{}|~{}c\ll a\\}$ for every $a\in L$, where

$c\ll a\quad\text{iff}\quad
a\leq\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}D\implies
c\leq d\text{ for some }d\in D.$

For locally compact frames this relation has also another characterisation
(e.g. see Lemma VII.6.3.2 in [25] or Proposition I-3.3 in [17]):

$\displaystyle c\ll a\quad\text{iff}\quad\exists\kappa\in L^{\wedge}\text{
s.t.\ }a\in\kappa\subseteq\mathord{\uparrow}\mkern 1.0muc$

The dual adjunction between the functors $\mathcal{O}$ and $\mathsf{pt}$
restricts to a duality of categories, with locally compact frames on one side
and _locally compact sober_ spaces on the other side. This duality is known as
the Hofmann–Lawson duality.

###  2.5 Sublocales

Next we introduce the frame-theoretic analogue of subspaces. A subset $S$ of a
frame $L$ is a _sublocale_ if $S$ is the set of fixpoints of some _nucleus_
$\nu\colon L\to L$, where nucleus is a closure operator which preserves binary
meets.

Every sublocale $S$, in the order derived from $L$, is a frame and the nucleus
$\nu$ restricts to an _onto_ frame homomorphism $L\twoheadrightarrow S$. In
fact, every onto frame homomorphism gives rise to a nucleus, and vice versa.

Finally, the lattice of sublocales $\mathcal{S}(L)$, ordered by set inclusion,
forms a coframe with meets computed as intersections, and embeds the original
frame $L$ via the map

$a\in L\enspace\mapsto\enspace U_{a}=\\{x~{}|~{}x=x\to a\\}.$

Sublocales of the form $U_{a}$ are called _open sublocales_ and their
complements in $\mathcal{S}(L)$, which always exist, are called _closed
sublocales_. For a more in-depth discussion about sublocales, nuclei and frame
quotients we refer the reader to [25].

##  3 Abstract description

###  3.1 Canonical extensions for distributive lattices

To motivate our definition of canonical extension for frames we first recall
the construction for distributive lattices (for details see e.g. [4, 11, 12,
14]). Given a distributive lattice $A$, its canonical extension $e\colon
A\hookrightarrow A^{\delta}$ is a lattice embedding of $A$ into a complete
lattice $A^{\delta}$ uniquely described by the following two conditions

  1. (Dense)

For all $u,v\in A^{\delta}$, $u\leq v$ if, for every filter $F\subseteq A$ and
ideal $I\subseteq A$, $\bigwedge e[F]\leq u$ and $v\leq\bigvee e[I]$ implies
$\bigwedge e[F]\leq\bigvee e[I]$.

  2. (Compact)

If $\bigwedge e[F]\leq\bigvee e[I]$ for some filter $F\subseteq A$ and ideal
$I\subseteq A$, then $F\cap I\not=\emptyset$.

Density simply means that every element of the canonical extension
$A^{\delta}$ is both a join of meets and a meet of joins of elements of $A$.

Note that up until recently it was preferred to write $A^{\sigma}$ to denote
the canonical extension of $A$. However, this notation conflicts the notation
for the two different extensions, $\sigma$– and $\pi$-extension, of monotone
maps – the first defined as a join of meets and the other as a meet of joins
(and there is no reason why one should have any preference over the other).

######  Remark 3.1.

The intuition behind the two axioms is that $e\colon A\hookrightarrow
A^{\delta}$ represents the embedding of compact opens of $X$ into the complete
lattice of saturated subsets of $X$, where $X$ is the Stone dual (spectral)
space of $A$. Under this interpretation, ideals of $A$ correspond to the open
sets via the mapping $I\mapsto\bigvee e[I]$ and, similarly, filters correspond
to the compact saturated subsets via $F\mapsto\bigwedge e[F]$.

Density then simply means that every saturated subset $M$ of $X$ is uniquely
determined by compact saturated sets $K$ such that $K\subseteq M$ and also by
open sets $U$ such that $M\subseteq U$. Further, compactness means that
$K\subseteq U$ implies that there is a compact open $O$ such that $K\subseteq
O\subseteq U$.

Note that, by assuming the axiom of choice, we can identify the elements of
$A^{\delta}$ that correspond to the principal upsets $\mathord{\uparrow}\mkern
1.0mux$ in the specialisation order of $X$ as those elements which are
completely join prime. We see that $A^{\delta}$ represents the points of $X$
as well as its upsets in one structure and so it effectively contains all
information about the space $X$. [11]

Existence and uniqueness of the canonical extension $A\hookrightarrow
A^{\delta}$ follow from a general theory of polarities as described, for
example, in [10]. Let us recall some of the basics of the theory that goes
back to Birkhoff [2]. A _polarity_ is a triple $(X,Y,Z)$ where $X$ and $Y$ are
sets and $Z\subseteq X\mathclose{\times}Y$ is a relation. This data induces
the following pair of antitone maps between
$\raisebox{2.03998pt}{\Large$\wp$}(X)$ and
$\raisebox{2.03998pt}{\Large$\wp$}(Y)$, in the subset order:

$\displaystyle
p\colon\raisebox{2.03998pt}{\Large$\wp$}(X)\to\raisebox{2.03998pt}{\Large$\wp$}(Y),\quad
M\mapsto\\{y\in Y~{}|~{}\forall x\in M.\ xZy\\}$ $\displaystyle
q\colon\raisebox{2.03998pt}{\Large$\wp$}(Y)\to\raisebox{2.03998pt}{\Large$\wp$}(X),\quad
N\mapsto\\{x\in X~{}|~{}\forall y\in N.\ xZy\\}$

Since $N\subseteq p(M)$ iff $M\subseteq q(N)$, the maps $p$ and $q$ constitute
an antitone Galois adjunction and $\phi=q\circ p$ is a closure operator on
$\raisebox{2.03998pt}{\Large$\wp$}(X)$. Set
$\mathscr{G}(X,Y,Z)\subseteq\raisebox{2.03998pt}{\Large$\wp$}(X)$ to be the
set of Galois closed sets, i.e. sets $\\{M\subseteq X~{}|~{}\phi(M)=M\\}$.
There are also two maps

$f\colon X\to\mathscr{G}(X,Y,Z),\
x\mapsto\phi(\\{x\\}),\enspace\text{and}\enspace g\colon
Y\to\mathscr{G}(X,Y,Z),\ y\mapsto q(\\{y\\}).$

######  Fact 3.2 (e.g., Section 2 in [10]).

Let $(X,Y,Z)$ be a polarity. Then the complete lattice $C=\mathscr{G}(X,Y,Z)$
has the following properties.

  1. 1.

For any $u\in C$,

$u=\bigvee\\{f(x)~{}|~{}x\in X,\ f(x)\leq u\\}\enspace\text{and}\enspace
u=\bigwedge\\{g(y)~{}|~{}y\in Y,\ u\leq g(y)\\}.$

  2. 2.

For any $x\in X$ and $y\in Y$, $f(x)\leq g(y)$ iff $xZy$.

  3. 3.

For any $f^{\prime}\colon X\to C^{\prime}$ and $g^{\prime}\colon Y\to
C^{\prime}$ also satisfying the conditions (1) and (2) there is a unique
complete lattice isomorphism $\iota\colon C^{\prime}\to C$ such that
$\iota\circ f^{\prime}=f$ and $\iota\circ g^{\prime}=g$.

Consequently, for a distributive lattice $A$, setting $X=\mathsf{Filt}(A)$,
$Y=\mathsf{Idl}(A)$, and $Z=\\{(F,I)~{}|~{}F\cap I\not=\emptyset\\}$ gives us
that, for the complete lattice $C=\mathscr{G}(X,Y,Z)$, the composite

$e\colon A\xrightarrow{~{}a~{}\mapsto~{}\mathord{\downarrow}\mkern
1.0mua~{}}\mathsf{Idl}(A)\xrightarrow{\quad g\quad}C$

is the canonical extension of $A$, e.g. [12].

###  3.2 The same construction for spaces

The construction mentioned above can be rewritten purely in terms of the Stone
dual $(X,\tau)$ of the distributive lattice $A$. As we mentioned in Remark
3.1, the frame of ideals is isomorphic to $\tau$ and the coframe of filters is
isomorphic to the poset of compact saturated subsets $\mathsf{KS}(X)$ of $X$,
ordered by set inclusion. Moreover, the relation $FZI$ expresses the fact that
$K_{F}\subseteq U_{I}$ where $K_{F}$ is the compact saturated subset
corresponding to $F$ and $U_{I}$ is the open set corresponding to $I$.

This intuition allows us to obtain the frame ${\mathsf{Up}(X,\leq_{\tau})}$
entirely from this data. Furthermore, this construction works for all
topological spaces, not just those arising as spectra of distributive
lattices.

######  Proposition 3.3.

Let $(X,\tau)$ be a topological space. Then
$\mathscr{G}(\mathsf{KS}(X),\tau,Z)$ is isomorphic to
${\mathsf{Up}(X,\leq_{\tau})}$ where $KZU$ iff $K\subseteq U$.

###### Proof.

By Fact 3.2 it is enough to check that the embeddings
$f\colon\mathsf{KS}(X)\hookrightarrow\mathsf{Up}(X,\leq_{\tau})$ and
$g\colon\tau\hookrightarrow\mathsf{Up}(X,\leq_{\tau})$ satisfy the conditions
(1) and (2).

The first condition is immediate as, for a saturated subset $M$ of $X$,
$M=\bigcup\\{\mathord{\uparrow}\mkern 1.0mux~{}|~{}x\in M\\}$ and
$M=\bigcap\\{U\in\tau~{}|~{}M\subseteq U\\}$. The second condition follows
immediately from the definition of $Z$. ∎

We see that the proof of Proposition 3.3 relies on the fact that the principal
upsets in the specialisation order $\mathord{\uparrow}\mkern 1.0mux$ are
compact and saturated. In the following we will try to mimic this construction
in a purely frame-theoretic setting.

###  3.3 Canonical extension for frames

By the Hofmann–Mislove Theorem, compact saturated subsets of a sober space are
in a bijective correspondence with Scott-open filters of its lattice of opens.
Therefore, we expect that the best candidate for the canonical extension of a
frame $L$, by example of Proposition 3.3, is the lattice
$\mathscr{G}(L^{\wedge},L,Z)$ where, for $\kappa\in L^{\wedge}$ and $a\in L$,
$\kappa Za$ iff $a\in\kappa$. The expression for $Z$ represents the statement
that the compact saturated set corresponding to $\kappa$ is a subset of the
open set corresponding to $a$.

Before we understand what is the structure of the lattice we obtained this
way, let us first describe the canonical extension $L^{\delta}$ of $L$ based
on the expected mutual relationship between Scott-open filters $L^{\wedge}$
and (abstract) open sets $L$ when embedded into $L^{\delta}$. Since it is
expected that the compact saturated subset corresponding to a Scott-open
filter is obtained as the “intersection” of all the opens it contains, it is
natural to consider the following definition.

######  Definition 3.4.

Let $L$ be a frame. A monotone mapping $e\colon L\to L^{\delta}$ into a
complete lattice $L^{\delta}$ is a _canonical extension_ of $L$ if

  1. (Dense)

For all $u,v\in L^{\delta}$, $u\leq v$ if, for all $\kappa\in L^{\wedge}$ and
$a\in L$, satisfying $\bigwedge e[\kappa]\leq u$ and $v\leq e(a)$ implies
$\bigwedge e[\kappa]\leq e(a)$.

  2. (Compact)

If $\bigwedge e[\kappa]\leq e(a)$ for some $\kappa\in L^{\wedge}$ and $a\in
L$, then $a\in\kappa$.

Note that $L^{\delta}$ is not required to be distributive, nor even a frame.

######  Example 3.5.

Let us check that the definition holds for our motivating example. We check
that, for a sober space $(X,\tau)$, the embedding
$\tau\hookrightarrow\mathsf{Up}(X,\leq_{\tau})$ is a canonical extension of
$\tau$.

Compactness follows immediately from the Hofmann–Mislove Theorem because, for
a Scott-open filter $\kappa\subseteq\tau$ and an open $U\in\tau$,
$\bigcap\kappa\subseteq U$ iff $U\in\kappa.$ For density, let $M\not\subseteq
N$ for some upsets $M$ and $N$. Since $N$ is equal to
$\bigcap\\{U\in\tau~{}|~{}N\subseteq U\\}$, there must be a $U\in\tau$ such
that $M\not\subseteq U$ and $N\subseteq U$. Pick an $x\in M\setminus U$. Then
$\mathord{\uparrow}\mkern 1.0mux\not\subseteq U$ but $\mathord{\uparrow}\mkern
1.0mux\subseteq M$. For the Scott-open filter
$\kappa=\\{V\in\tau~{}|~{}\mathord{\uparrow}\mkern 1.0mux\subseteq V\\}$
corresponding to $\mathord{\uparrow}\mkern 1.0mux$ we see that
$\bigcap\kappa\not\subseteq U$.

We postpone the proof of existence and uniqueness of canonical extensions for
the moment and focus on the properties that follow immediately from the
definition.

Since the expression $\bigwedge e[\kappa]$ appears repeatedly in our theory,
we define a shorthand. For a map $e\colon L\to L^{\delta}$, set

$e^{\wedge}\colon L^{\wedge}\to L^{\delta},\quad\kappa\mapsto\bigwedge
e[\kappa]=\bigwedge\\{e(a)~{}|~{}a\in\kappa\\}.$

######  Proposition 3.6.

Let $L$ be a frame and $e\colon L\to L^{\delta}$ be its canonical extension.
Then

  1. 1.

$e\colon L\to L^{\delta}$ is a preframe homomorphism preserving 0.

  2. 2.

$e^{\wedge}\colon L^{\wedge}\to L^{\delta}$ is an injective co-preframe
homomorphism
(recall that $L^{\wedge}$ is taken to be ordered by reverse inclusion).

  3. 3.

For every $u\in L^{\delta}$,

$\displaystyle u$ $\displaystyle=\bigwedge\\{e(a)~{}|~{}a\in L,\ u\leq
e(a)\\}\enspace\text{and}\enspace$ $\displaystyle u$
$\displaystyle=\bigvee\\{e^{\wedge}(\kappa)~{}|~{}\kappa\in L^{\wedge},\
e^{\wedge}(\kappa)\leq u\\}.$

  4. 4.

If $e\colon L\to L^{\delta}$ is injective, then it is a frame homomorphism.

  5. 5.

If $L$ is locally compact, then $e$ is an injective frame homomorphism.

###### Proof.

(1) First we check that $e$ preserves 0 and 1. For the former, by density of
$e$, it is enough to check that whenever $\bigwedge e[\kappa]\leq e(0)$ and
$0\leq e(a)$ then also $\bigwedge e[\kappa]\leq e(a)$. From compactness
applied to the first inequality we get that $0\in\kappa$ and, from
monotonicity, $e(0)\leq e(a)$. Hence, $\bigwedge e[\kappa]\leq e(0)\leq e(a)$.
We check that $1\leq e(1)$ by a similar argument. Let $\kappa\in L^{\wedge}$
and $a\in L$ be such that $\bigwedge e[\kappa]\leq 1$ and $e(1)\leq e(a)$.
Since $1\in\kappa$ and $e$ is monotone, $\bigwedge e[\kappa]\leq e(1)\leq
e(a)$.

To check that $e$ preserves finite meets and directed joins we only need to
check one inequality because $e$ is monotone. Let $a,b\in L$. Assume that
$\bigwedge e[\kappa]\leq e(a)\wedge e(b)$ and $e(a\wedge b)\leq e(c)$ for some
$\kappa\in L^{\wedge}$ and $c\in L$. By compactness $a,b\in\kappa$. Because
$\kappa$ is a filter, $a\wedge b\in\kappa$ and so $\bigwedge e[\kappa]\leq
e(c)$.

Lastly, to check that $e(\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}D)\leq\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}e[D]$ for some directed $D\subseteq L$, assume that $\bigwedge
e[\kappa]\leq e(\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}D)$ and that $\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}e[D]\leq e(a)$ for some $\kappa\in L^{\wedge}$ and $a\in L$.
Compactness and the fact that $\kappa$ is Scott-open give that $d\in\kappa$
for some $d\in D$ and, consequently, $\bigwedge e[\kappa]\leq
e(d)\leq\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern 1.0mu}}e[D]\leq
e(a)$.

(2) For injectivity assume that $e^{\wedge}(\kappa)=e^{\wedge}(\lambda)$.
Then, for an $a\in\lambda$, $e^{\wedge}(\kappa)=e^{\wedge}(\lambda)\leq e(a)$
and so, by compactness, $a\in\kappa$. Hence, $\lambda\subseteq\kappa$. The
reverse inclusion is analogous.

$e^{\wedge}$ preserves the bottom as $e^{\wedge}(\mathord{\uparrow}\mkern
1.0mu0)\leq e(0)=0$ by (1). Next, observe that by compactness
$e^{\wedge}(\kappa)\vee e^{\wedge}(\lambda)\leq e(a)$ iff
$a\in\kappa\cap\lambda$. Consequently, if $e^{\wedge}(\chi)\leq
e^{\wedge}(\kappa\cap\lambda)$ and $e^{\wedge}(\kappa)\vee
e^{\wedge}(\lambda)\leq e(a)$, then $e^{\wedge}(\chi)\leq
e^{\wedge}(\kappa\cap\lambda)\leq e(a)$ and so $e^{\wedge}$ also preserves
binary joins.

Finally, we show that $e^{\wedge}$ preserves filtered meets. Let
$\mathscr{F}\subseteq L^{\wedge}$ be filtered, and assume that
$e^{\wedge}(\chi)\leq\bigwedge\\!{}^{\mathord{\downarrow}\mkern
1.0mu}\,e^{\wedge}[\mathscr{F}]$ and
$e^{\wedge}(\bigwedge\\!{}^{\mathord{\downarrow}\mkern
1.0mu}\,\mathscr{F})=e^{\wedge}(\operatorname*{\bigcup\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}\mathscr{F})\leq e(a)$. From the latter inequality we have that
$a\in\kappa$ for some $\kappa\in\mathscr{F}$. As a consequence
$e^{\wedge}(\chi)\leq\bigwedge\\!{}^{\mathord{\downarrow}\mkern
1.0mu}\,e^{\wedge}[\mathscr{F}]\leq e^{\wedge}(\kappa)\leq e(a)$.

(3) Let $v=\bigwedge\\{e(a)~{}|~{}a\in L,\ u\leq e(a)\\}$. We will show that
$v\leq u$. Let $\kappa\in L^{\wedge}$ and $b\in L$ be such that
$e^{\wedge}(\kappa)\leq v$ and $u\leq e(b)$. From the definition of $v$ we
immediately see that $e^{\wedge}(\kappa)\leq v\leq e(b)$.

The proof of the second part is analogous.

(4) What is left to prove is that $e$ preserves binary joins. Let
$e^{\wedge}(\kappa)\leq e(a\vee b)$ and $e(a)\vee e(b)\leq e(c)$. Since $e$ is
injective, it follows that $a\leq c$. Indeed, because $e$ preserves finite
meets, $e(a\wedge c)=e(a)\wedge e(c)=e(a)$ and so $a\wedge c=a$. Similarly,
$b\leq c$. Moreover, we know from $e^{\wedge}(\kappa)\leq e(a\vee b)$ that
$a\vee b\in\kappa$. Consequently, $c\in\kappa$ and so $e^{\wedge}(\kappa)\leq
e(c)$.

(5) It is enough to show injectivity of $e\colon L\to L^{\delta}$. Assume that
$e(a)=e(b)$ for some $a,b\in L$. Let $c\ll a$. This means that there is a
Scott-open filter ${\kappa\in L^{\wedge}}$ such that
$a\in\kappa\subseteq\mathord{\uparrow}\mkern 1.0muc$. Because
$e^{\wedge}(\kappa)\leq e(a)$, also $b\in\kappa$ by compactness and so $c\leq
b$. Consequently $a\leq b$ since $a=\bigvee\\{c~{}|~{}c\ll a\\}$. The other
direction is symmetrical. ∎

It might seem surprising how much we can prove from density and compactness
alone. This is a common feature of canonical extensions. Another common
feature is usually that the extension is injective. We showed that this is
achieved whenever the frame is locally compact. Moreover, as we discuss in
Section 5, injectivity is in fact equivalent to a certain topological
condition.

####  3.3.1 A few more observations

######  Lemma 3.7.

Let $e\colon L\to L^{\delta}$ be a canonical extension of a frame $L$, then

$\bigvee_{i\in I}e^{\wedge}(\kappa_{i})=\bigwedge e[\bigcap_{i\in
I}\kappa_{i}],$

for every collection $\\{\kappa_{i}\\}_{i\in I}\subseteq L^{\wedge}$.

###### Proof.

The “$\leq$” inequality is immediate as $e^{\wedge}(\kappa_{i})=\bigwedge
e[\kappa_{i}]\leq\bigwedge e[\bigcap_{i\in I}\kappa_{i}]$, for every $i\in I$.
We show the other inequality by density. Let $e^{\wedge}(\lambda)\leq\bigwedge
e[\bigcap_{i\in I}\kappa_{i}]$ and $\bigvee_{i\in I}e^{\wedge}(\kappa_{i})\leq
e(a)$ for some $\lambda\in L^{\wedge}$ and $a\in L$. The former assumption
implies, by compactness, that $\bigcap_{i\in I}\kappa_{i}\subseteq\lambda$ and
the latter implies that $a\in\kappa_{i}$, for all $i\in I$. Therefore,
$a\in\bigcap_{i\in I}\kappa_{i}\subseteq\lambda$ and so $\bigwedge
e[\bigcap_{i\in I}\kappa_{i}]\leq e(a)$. ∎

One can also reformulate the second axiom of canonical extensions for frames
to be more similar to the one for distributive lattices (Section 3.1).

######  Lemma 3.8.

Let $L$ be a frame and $e\colon L\to L^{\delta}$ be a monotone map satisfying
density. Then, $e$ is compact if and only if it satisfies

  1. (Compact+)

If $\bigwedge e^{\wedge}[\mathscr{F}]\leq\bigvee e[D]$ for a filtered
$\mathscr{F}\subseteq L^{\wedge}$ and directed $D\subseteq L$, then
$d\in\kappa$ for some $\kappa\in\mathscr{F}$ and $d\in D$.

###### Proof.

(Compact+) implies (Compact) since the latter is just a special case of the
former, that is, when we take $\mathscr{F}=\\{\kappa\\}$ and $D=\\{a\\}$. The
reverse direction is a consequence of Proposition 3.6. We have that

$e^{\wedge}(\bigcup\\!{}^{\mathord{\uparrow}\mkern
1.0mu}\,\mathscr{F})=\bigwedge\\!{}^{\mathord{\uparrow}\mkern
1.0mu}\,e^{\wedge}[\mathscr{F}]\leq\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}e[D]=e(\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}D),$

and so, by compactness of $e$,
$\operatorname*{\bigvee\\!{}^{\mathord{\uparrow}\mkern
1.0mu}}D\in\bigcup\mathscr{F}$. Hence, $\bigvee D\in\kappa$, for some
$\kappa\in\mathscr{F}$, and also $d\in\kappa$, for some $d\in D$, since
$\kappa$ is Scott-open. ∎

This stronger version of compactness also specialises to a well-known
consequence of the Hofmann–Mislove Theorem. For a filtered collection of
compact saturated sets $\\{K_{i}\\}_{i}$ and an open set $U$ of a sober space,
$\bigcap_{i}K_{i}\subseteq U$ if and only if $K_{i}\subseteq U$ already for
some $K_{i}$ (e.g. [23] or Theorem II-1.21 in [17]).

In the next section we show, among others, that canonical extensions of
locally compact frames are completely distributive lattices. This fact relies
on the axiom of choice. One can prove a weaker property constructively for
_stably locally compact frames_ , that is, for locally compact frames which
satisfy:

$\displaystyle(\forall a,b,c)\qquad a\ll b\enspace\text{and}\enspace a\ll
c\quad\text{implies}\quad a\ll b\wedge c.$

######  Proposition 3.9.

Let $L$ be a stably locally compact frame and $e\colon L\to L^{\delta}$ its
canonical extension, then $L^{\delta}$ is a frame and coframe.

###### Proof.

The proof follows the same strategy as the proof of Theorem 3 in [11].
However, we need to repeat the same argument twice because the assumption that
$L$ is stably locally compact is used in different parts of the proof when
proving that $L^{\delta}$ is a frame and coframe, respectively.

Recall that a locally compact frame is stably locally compact if and only if
the meet of two Scott-open filters in $\mathsf{Filt}(L)$ (in the reverse
inclusion order) is also Scott-open. Observe that $e^{\wedge}\colon
L^{\wedge}\to L^{\delta}$ preserves binary meets because, by Proposition 3.6,
$e^{\wedge}(\kappa)\wedge
e^{\wedge}(\kappa^{\prime})=\bigvee\\{e^{\wedge}(\lambda)\mid\lambda\in
L^{\wedge},\ \lambda\leq\kappa,\
\lambda\leq\kappa^{\prime}\\}=e^{\wedge}(\kappa\wedge\kappa^{\prime})$.

(1) We show that $L^{\delta}$ is a frame. First we prove a restricted version
of the frame distributivity law. For $K\subseteq L^{\wedge}$ and $\kappa\in
L^{\wedge}$, we verify that

$\displaystyle e^{\wedge}(\kappa)\wedge\bigvee
e^{\wedge}[K]\leq\bigvee\\{e^{\wedge}(\kappa)\wedge
e^{\wedge}(\kappa^{\prime})\mid\kappa^{\prime}\in K\\}.$ ($\star$1)

By density it is enough to show that $e^{\wedge}(\lambda)\leq e(a)$ for every
$\lambda\in L^{\wedge}$ and $a\in L$ such that $e^{\wedge}(\lambda)\leq
e^{\wedge}(\kappa)\wedge\bigvee e^{\wedge}[K]$ and
$\bigvee\\{e^{\wedge}(\kappa)\wedge
e^{\wedge}(\kappa^{\prime})\mid\kappa^{\prime}\in K\\}\leq e(a)$. Because $L$
is stably locally compact, $e^{\wedge}(\kappa)\wedge
e^{\wedge}(\kappa^{\prime})\leq e(a)$ iff $b\wedge b^{\prime}\leq a$ for some
$b\in\kappa$ and $b^{\prime}\in\kappa^{\prime}$. Since
$e^{\wedge}(\kappa)\wedge e^{\wedge}(\kappa^{\prime})\leq a$ for every
$\kappa^{\prime}\in K$,

$\bigvee e^{\wedge}[K]\leq\bigvee\\{e(b^{\prime})\mid b^{\prime}\in L\text{
s.t. }\exists\kappa^{\prime}\in K,b\in L\colon\ b\wedge b^{\prime}\leq
a,\,b\in\kappa,\,b^{\prime}\in\kappa^{\prime}\\}.$

By Lemma 3.8, this together with $e^{\wedge}(\lambda)\leq
e^{\wedge}(\kappa)\wedge\bigvee e^{\wedge}[K]\leq\bigvee e^{\wedge}[K]$
implies that there are $b_{1},\dots,b_{n}$ and
$b^{\prime}_{1},\dots,b^{\prime}_{n}$ in $L$ such that $e^{\wedge}(\kappa)\leq
e(b_{i})$, $b_{i}\wedge b^{\prime}_{i}\leq a$ (for $i=1,\dots,n$) and
$e^{\wedge}(\lambda)\leq e(b^{\prime}_{1}\vee\ldots\vee b^{\prime}_{n})$.
Therefore, $e^{\wedge}(\lambda)\leq e^{\wedge}(\kappa)\leq e(b)$ for
$b=b_{1}\wedge\dots\wedge b_{n}$ and, because $e(\mkern 1.0mu\cdot\mkern
1.0mu)$ is a frame homomorphism (by Proposition 3.6), also

$\displaystyle e^{\wedge}(\lambda)$ $\displaystyle\leq e(b)\wedge
e(b^{\prime}_{1}\vee\ldots\vee b^{\prime}_{n})=e(b\wedge
b^{\prime}_{1})\vee\ldots\vee e(b\wedge b^{\prime}_{n})$ $\displaystyle\leq
e(b_{1}\wedge b^{\prime}_{1})\vee\ldots\vee e(b_{n}\wedge b^{\prime}_{n})\leq
e(a).$

This finishes the proof of ($\star$1). Since every element of $L^{\delta}$ is
a join of elements in $L^{\wedge}$, to show that $L^{\delta}$ is a frame, it
is enough to show that $u\wedge\bigvee e^{\wedge}[K]\leq\bigvee\\{u\wedge
e^{\wedge}(\kappa)\mid\kappa\in K\\}$ for arbitrary $u\in L^{\delta}$ and
$K\subseteq L^{\wedge}$. From the previous,

$\displaystyle u\wedge\bigvee e^{\wedge}[K]$
$\displaystyle=\bigvee\\{e^{\wedge}(\kappa)\in L^{\wedge}\mid
e^{\wedge}(\kappa)\leq u\wedge\bigvee e^{\wedge}[K]\\}$
$\displaystyle\leq\bigvee\\{e^{\wedge}(\kappa)\wedge\bigvee
e^{\wedge}[K]\mid\kappa\in L^{\wedge},\ e^{\wedge}(\kappa)\leq u\\}$
$\displaystyle\leq\bigvee_{e^{\wedge}(\kappa)\leq u}\
\bigvee_{\kappa^{\prime}\in K}e^{\wedge}(\kappa)\wedge
e^{\wedge}(\kappa^{\prime})$ $\displaystyle=\bigvee_{\kappa^{\prime}\in K}\
\bigvee_{e^{\wedge}(\kappa)\leq u}e^{\wedge}(\kappa)\wedge
e^{\wedge}(\kappa^{\prime})\leq\bigvee_{\kappa^{\prime}\in K}u\wedge
e^{\wedge}(\kappa^{\prime}).$

(2) Showing that $L^{\delta}$ is a coframe follows the same pattern. First, we
show that, for $a\in L$ and $A\subseteq L$,

$\displaystyle\bigwedge\\{e(a)\vee e(a^{\prime})\mid a^{\prime}\in A\\}\leq
e(a)\vee\bigwedge e[A].$ ($\star$2)

Let $\kappa\in L^{\wedge}$ and $c\in L$ be such that
$e^{\wedge}(\kappa)\leq\bigwedge\\{e(a)\vee e(a^{\prime})\mid a^{\prime}\in
A\\}$ and $e(a)\vee\bigwedge e[A]\leq e(c)$. Because $L$ is locally compact
and the embedding $e$ is compact, whenever $e^{\wedge}(\kappa)\leq e(a)\vee
e(a^{\prime})=e(a\vee a^{\prime})$, for an $a^{\prime}\in A$, then
$e^{\wedge}(\kappa)\leq e(b)\vee e(b^{\prime})$ for some $b,b^{\prime}\in L$
such that $b\ll a$ and $b^{\prime}\ll a^{\prime}$. Moreover, $b\ll a$ iff
there is some $\lambda\in L^{\wedge}$ such that
$a\in\lambda\subseteq\mathord{\uparrow}\mkern 1.0mub$ or, in other words,
$e(b)\leq e^{\wedge}(\lambda)\leq e(a)$ in $L^{\delta}$. Similarly, there is a
$\lambda^{\prime}\in L^{\wedge}$ such that $e(b^{\prime})\leq
e^{\wedge}(\lambda^{\prime})\leq e(a^{\prime})$ and so $e^{\wedge}(\kappa)\leq
e^{\wedge}(\lambda)\vee e^{\wedge}(\lambda^{\prime})$. Consequently, since
$e^{\wedge}(\kappa)\leq e(a)\vee e(a^{\prime})$ for every $a^{\prime}\in A$,

$\bigwedge\\{e^{\wedge}(\lambda^{\prime})\mid\lambda^{\prime}\in
L^{\wedge}\text{ s.t.}\,\,\exists a^{\prime}\in A,\lambda\in L^{\wedge}\colon
a\in\lambda,\,a^{\prime}\in\lambda^{\prime},\,\kappa\leq\lambda\vee\lambda^{\prime}\\}\leq\bigwedge
e[A]$

Since $L$ is stably locally compact, non-empty finite meets of the elements in
the set on the left-hand side form a directed family in $L^{\wedge}$.
Therefore, from $\bigwedge e[A]\leq e(a)\vee\bigwedge e[A]\leq e(c)$ it
follows, by Lemma 3.8, that there are some $\lambda_{1},\dots,\lambda_{n}$ and
$\lambda^{\prime}_{1},\dots,\lambda^{\prime}_{n}$ in $L^{\wedge}$ such that
$e^{\wedge}(\lambda_{i})\leq e(a)$,
$\kappa\leq\lambda_{i}\vee\lambda^{\prime}_{i}$ (for $i=1,\dots,n$) and
$e^{\wedge}(\lambda^{\prime})\leq e(c)$ where
$\lambda^{\prime}=\lambda^{\prime}_{1}\wedge\dots\wedge\lambda^{\prime}_{n}$.
We obtain that

$\displaystyle e^{\wedge}(\kappa)$ $\displaystyle\leq
e^{\wedge}(\lambda_{1}\vee\lambda^{\prime}_{1})\wedge\dots\wedge
e^{\wedge}(\lambda_{n}\vee\lambda^{\prime}_{n})$ $\displaystyle\leq(e(a)\vee
e^{\wedge}(\lambda^{\prime}_{1}))\wedge\dots\wedge(e(a)\vee
e^{\wedge}(\lambda^{\prime}_{n}))=e(a)\vee e^{\wedge}(\lambda^{\prime})\leq
e(c),$

where the only equality holds because $L^{\delta}$ is a frame. This finishes
the proof of ($\star$2). The rest is the same as in part (1). ∎

##  4 Existence, uniqueness and concrete description

As we explained at the beginning of Section 3.3, we expect that the canonical
extension $e\colon L\to L^{\delta}$ of a frame $L$ can be obtained by a
construction by polarities similarly to distributive lattices and spaces. That
is, we expect that

$L^{\delta}\cong\mathscr{G}(L^{\wedge},L,Z)\quad\text{where}\quad\kappa
Za\enspace\text{iff}\enspace a\in\kappa.$

(Recall the definition of $\mathscr{G}(\mkern 1.0mu\cdot\mkern 1.0mu,\mkern
1.0mu\cdot\mkern 1.0mu,\mkern 1.0mu\cdot\mkern 1.0mu)$ from Section 3.1.)

This follows from a standard argument, e.g. [13]. Let us denote
$\mathscr{G}(L^{\wedge},L,Z)$ by $C$. The crucial ingredient in the
construction of $C$ is the pair of antitone maps $p$ and $q$ whose
definitions, in this case, unfold as

$\displaystyle p\colon$
$\displaystyle\raisebox{2.03998pt}{\Large$\wp$}(L^{\wedge})\to\raisebox{2.03998pt}{\Large$\wp$}(L),\quad
M\mapsto\\{a\in L~{}|~{}\forall\kappa\in M.\ a\in\kappa\\}=\bigcap
M,\enspace\text{and}\enspace$ $\displaystyle q\colon$
$\displaystyle\raisebox{2.03998pt}{\Large$\wp$}(L)\to\raisebox{2.03998pt}{\Large$\wp$}(L^{\wedge}),\quad
N\mapsto\\{\kappa\in L^{\wedge}~{}|~{}\forall a\in N.\
a\in\kappa\\}=\\{\kappa~{}|~{}N\subseteq\kappa\\}.$

Notice that the image of $p$ is always a filter on $L$ and, similarly, an
image of $q$ is an ideal on $L^{\wedge}$, when $L^{\wedge}$ is taken to be
ordered by set inclusion. Therefore, it is safe to restrict the adjoint maps
to:

${\mathsf{Idl}(L^{\wedge})}$${\mathsf{Filt}(L)}$$\scriptstyle{p}$$\scriptstyle{\bot}$$\scriptstyle{q}$

Since we order $\mathsf{Filt}(L)$ by reverse inclusion, the antitone Galois
adjunction becomes monotone, with
$p\colon\mathscr{I}\mapsto\bigvee\mathscr{I}$ (evaluated in
$(\mathsf{Filt}(L),\supseteq)$) on the left and $q$ on the right. The maps
$f\colon L^{\wedge}\to C$ and $g\colon L\to C$ are defined as

$\displaystyle f$ $\displaystyle\colon\lambda\mapsto
qp(\\{\lambda\\})=q(\lambda)=\\{\kappa\in
L^{\wedge}~{}|~{}\kappa\supseteq\lambda\\}\quad\text{and}\quad$ $\displaystyle
g$ $\displaystyle\colon a\mapsto q(\\{a\\})=\\{\kappa\in
L^{\wedge}~{}|~{}a\in\kappa\\}.$

In Proposition 3.6 we showed that having a canonical extension $e\colon L\to
L^{\delta}$ of $L$ guarantees existence of a map $e^{\wedge}\colon
L^{\wedge}\to L^{\delta}$ such that, for $f=e^{\wedge}$ and $g=e$, the
conditions (1) and (2) of Fact 3.2 are satisfied (where (2) follows from
compactness). This means, by Fact 3.2.3, that if $e\colon L\to L^{\delta}$ is
a canonical extension, it must be that $L^{\delta}$ is isomorphic to $C$. The
existence of a canonical extension map $L\to L^{\delta}$ follows from the
following lemma.

######  Lemma 4.1.

The map $g\colon L\to C$ defined above is the canonical extension of $L$, that
is, it is dense and compact (in the sense of Definition 3.4).

###### Proof.

Recall that, since $C$ consists of the fixpoints of the closure operator
$q(p(\mkern 1.0mu\cdot\mkern 1.0mu))$ on the frame $\mathsf{Idl}(L^{\wedge})$,
the meets in $C$ are computed as in $\mathsf{Idl}(L^{\wedge})$. Hence,
$f(\kappa)=\bigwedge g[\kappa]$, for every $\kappa\in L^{\wedge}$, since

$\bigwedge g[\kappa]=\bigcap
g[\kappa]=\bigcap\\{g(a)~{}|~{}a\in\kappa\\}=\\{\lambda~{}|~{}\forall
a\in\kappa.\ a\in\lambda\\}=f(\kappa).$

As a consequence, $g$ is compact, by Fact 3.2.1, and density follows from Fact
3.2.2 and the fact that $f(\kappa)\leq g(a)$ iff $a\in\kappa$. ∎

To summarise, we have obtained the following.

######  Theorem 4.2.

Let $L$ be a frame. Its canonical extension $e\colon L\to L^{\delta}$ exists,
it is uniquely determined, and $L^{\delta}\cong\mathscr{G}(L^{\wedge},L,Z)$
where $\kappa Za$ iff $a\in\kappa$.

When a frame $L$ is locally compact, then we know more about the structure of
$L^{\delta}$. The embedding $L\hookrightarrow\mathsf{Up}(\mathsf{pt}(L))$ is
also a canonical extension (Example 3.5) and therefore the frames $L^{\delta}$
and $\mathsf{Up}(\mathsf{pt}(L))$ are isomorphic (by Theorem 4.2). Because the
lattice of saturated subsets is completely distributive, the lattice
$L^{\delta}$ is too.

######  Corollary 4.3.

The canonical extension of a locally compact frame is a completely
distributive lattice.

The concrete representation of $L^{\delta}$ as the fixpoints of the nucleus
$q(p(\mkern 1.0mu\cdot\mkern 1.0mu))$ on $\mathsf{Idl}(L^{\wedge})$ can be
equivalently described as the fixpoints of $p(q(\mkern 1.0mu\cdot\mkern
1.0mu))$ on $\mathsf{Filt}(L)$. Observe that a filter $F\in\mathsf{Filt}(L)$
is a fixpoint of $p(q(\mkern 1.0mu\cdot\mkern 1.0mu))$ iff
$F=\bigcap\\{\kappa~{}|~{}F\subseteq\kappa\\}$. In other words, fixpoints of
$p(q(\mkern 1.0mu\cdot\mkern 1.0mu))$ are precisely the intersections of
Scott-open filters.

######  Corollary 4.4.

$L^{\delta}$ is isomorphic to the poset of filters obtained as intersections
of Scott-open filters on $L$, ordered by reverse inclusion.

##  5 On injectivity of $e\colon L\to L^{\delta}$

Let us explore when is the canonical extension map $e\colon L\to L^{\delta}$
injective. For $a,b\in L$, by Proposition 3.6, $e(a)=e(b)$ if and only if, for
all $\kappa\in L^{\wedge}$, $e^{\wedge}(\kappa)\leq e(a)$ iff
$e^{\wedge}(\kappa)\leq e(b)$, and this expression is (by compactness of $e$)
equivalent to: $a\in\kappa$ iff $b\in\kappa$. Since Scott-open filters are in
a bijective correspondence with preframe homomorphisms to $\mathbf{2}$, $e$ is
injective if and only if the frame $L$ is spatial as a preframe. Furthermore,
injectivity of canonical extensions can be also characterised purely in terms
of sublocales.

######  Proposition 5.1.

For a frame $L$, the canonical extension $e\colon L\to L^{\delta}$ is
injective if and only if

$U\subseteq V\quad\text{iff}\quad\forall\text{compact fitted\,sublocale
}K\subseteq L,\quad K\subseteq U\text{ implies }K\subseteq V,$

for all open sublocales $U$, $V\subseteq L$.

Moreover, the Ultrafilter Principle is equivalent to the statement that every
frame with injective canonical extension is spatial.

###### Proof.

From the previous discussion we know that $e$ is injective iff distinct
elements of $L$ can be separated by Scott-open filters. By Johnstone’s
pointfree version of Hofmann–Mislove Theorem (Lemma 3.4 in [20], see also
[8]), Scott-open filters are in a bijective correspondence with compact fitted
sublocales, via the correspondence

$\kappa\mapsto
K_{\kappa}=\bigcap\\{U_{a}~{}|~{}a\in\kappa\\},\quad\text{and}\quad
K\mapsto\\{a~{}|~{}K\subseteq U_{a}\\}.$

From this it follows that the required separation property is equivalent to
$e$ being injective because, for every $a\in L$ and $\kappa\in L^{\wedge}$, we
have that $a\in\kappa$ iff $K_{\kappa}\subseteq U_{a}$.

The second part of the statement follows from Erné’s Theorem 5.2 in [5] that
the Ultrafilter Principle is equivalent to the statement that every frame
spatial as a preframe is spatial as a frame. ∎

This proposition justifies that we can call frames with injective canonical
extensions as frames which _have enough compact fitted sublocales_.444Such a
long name almost begs to be shortened to _has enough cofis_ (read as
“coffees”).

######  Remark 5.2.

This notion is closely related to Martín Escardó’s definition of compactly
generated locales [9]. He proves that a Hausdorff locale is isomorphic to a
colimit of its compact fitted sublocales if and only if its open sublocales
are determined by the compact sublocales they contain.

######  Remark 5.3.

Our construction of canonical extensions, as described in Corollary 4.4, is
exactly the same as the one used by Marcel Erné [5] to show that a preframe
$P$ is spatial precisely whenever there is a complete lattice which is, in
Erné’s terminology, its $\delta$-sober envelope. This means, in our
terminology, that there is an embedding $e\colon P\hookrightarrow C$ into a
complete lattice $C$ which is dense and compact.

In fact, from the beginning of our investigations we were hoping to find
connections between [5] and the theory of canonical extensions, as was
foreseen by Mai Gehrke (private communication).

######  Remark 5.4.

Assuming the Ultrafilter Principle, we obtain that $e\colon L\to L^{\delta}$
is injective if and only if $L$ is spatial. The left-to-right direction
follows from Proposition 5.1 and the reverse direction is a consequence of
Example 3.5 and Theorem 4.2.

##  6 Extensions of maps

In the theory of canonical extensions, when extending maps between lattices to
maps between their canonical extensions, two constructions are usually
considered. The so-called $\sigma$-extension and $\pi$-extension are also used
to lift additional operations on the lattice to its canonical extension and,
depending on which extension is taken, certain types of equations are
preserved. Moreover, we will be mostly concerned with extending so-called
_perfect_ maps and homomorphisms.

Before we define the two extensions let us introduce a new convention. Writing
the applications of the maps $e\colon L\to L^{\delta}$ and $e^{\wedge}\colon
L^{\wedge}\hookrightarrow L^{\delta}$ everywhere gets tiring at times. For
this reason, by abuse of notation, we will write elements of $L^{\delta}$ in
the image of $e$ without explicitly writing the application of $e(\mkern
1.0mu\cdot\mkern 1.0mu)$ and, similarly, for the elements of $L^{\wedge}$ and
applications of $e^{\wedge}(\mkern 1.0mu\cdot\mkern 1.0mu)$. For example, for
a monotone map $f\colon L\to M$, $a\in L$, and $\kappa\in L^{\wedge}$, the
expressions

$\kappa\leq a\quad\text{and}\quad\bigwedge f[\kappa]\leq f(a)$

are interpreted as

$e^{\wedge}(\kappa)\leq e(a)\quad\text{and}\quad\bigwedge e[f[\kappa]]\leq
e(f(a)),\quad\text{respectively.}\quad$

Furthermore, if $f$ is a preframe homomorphism, then taking the preimage of a
Scott-open filter gives again a Scott-open filter. In such case, we denote the
corresponding monotone map by

$f^{\dagger}\colon M^{\wedge}\to L^{\wedge},\quad\lambda\mapsto
f^{-1}[\lambda].$

Then, with the convention from above, $f^{\dagger}(\lambda)\leq u$ means
$e^{\wedge}(f^{\dagger}(\lambda))\leq u$.

With this in mind, let us define the two extensions $f^{\sigma},f^{\pi}\colon
L^{\delta}\to M^{\delta}$ for a monotone map $f\colon L\to M$ between two
frames $L$ and $M$ as

$\displaystyle f^{\sigma}\colon u$ $\displaystyle\mapsto\bigvee\\{\bigwedge
f[\kappa]~{}|~{}\kappa\in L^{\wedge},\ \kappa\leq u\\},$ $\displaystyle
f^{\pi}\colon u$ $\displaystyle\mapsto\bigwedge\\{f(a)~{}|~{}a\in L,\ u\leq
a\\}.$

These definitions are adapted from the usual definitions of extensions of
monotone maps for canonical extensions of distributive lattices, see e.g.
[11].

In the following we show that calling those maps “extensions” is justified by
the fact that, under mild assumptions, they both indeed extend the original
map.

######  Lemma 6.1.

Let $h\colon L\to M$ be a preframe homomorphism and $f\colon L\to M$ a
monotone map between two frames. The diagrams

${L}$${M}$${L^{\delta}}$${M^{\delta}}$$\scriptstyle{h}$$\scriptstyle{e_{L}}$$\scriptstyle{e_{M}}$$\scriptstyle{h^{\sigma}}$
and
${L}$${M}$${L^{\delta}}$${M^{\delta}}$$\scriptstyle{f}$$\scriptstyle{e_{L}}$$\scriptstyle{e_{M}}$$\scriptstyle{f^{\pi}}$

commute.

###### Proof.

Let $a\in L$. The inequality $\kappa\leq a$ implies that $a\in\kappa$ and
$\bigwedge h[\kappa]\leq h(a)$. Hence, $h^{\sigma}(a)\leq h(a)$. We prove the
reverse direction by density of $e_{M}$, let $\lambda\leq h(a)$ and
$h^{\sigma}(a)\leq b$ for some $\lambda\in M^{\wedge}$ and $b\in M$. From the
first inequality, by compactness, $h(a)\in\lambda$ and so $a\in
h^{-1}[\lambda]=h^{\dagger}(\lambda)$. Since $h^{\dagger}(\lambda)\leq a$, we
have that $\bigwedge h[h^{\dagger}(\lambda)]\leq h^{\sigma}(a)\leq b$.
Finally, observe that $\lambda\leq\bigwedge h[h^{\dagger}(\lambda)]$ in
$M^{\delta}$, because $h[h^{\dagger}(\lambda)]\subseteq\lambda$, and so
$\lambda\leq b$.

For the second square,
$f^{\pi}(a)=\bigwedge\\{f(a^{\prime})~{}|~{}a^{\prime}\in L,\ a\leq
a^{\prime}\\}=f(a)$. ∎

In the following we assume that the order between maps is always the pointwise
order. It is immediate to see that the operators $(\mkern 1.0mu\cdot\mkern
1.0mu)^{\sigma}$ and $(\mkern 1.0mu\cdot\mkern 1.0mu)^{\pi}$ are monotone with
respect to this order.

######  Lemma 6.2.

For monotone maps $f\colon L\to M$ and $g\colon M\to N$ between frames,

  1. 1.

$f^{\sigma}\leq f^{\pi}$,

  2. 2.

$f^{\sigma}=f^{\pi}$ whenever $f$ is a preframe homomorphism,

  3. 3.

$g^{\pi}f^{\pi}\leq(gf)^{\pi}$, and

  4. 4.

$(\mathsf{id}_{L})^{\sigma}=(\mathsf{id}_{L})^{\pi}=\mathsf{id}_{L^{\delta}}$.

###### Proof.

(1) Let $u\in L^{\delta}$ and let $\kappa\leq u\leq a$ for some $\kappa\in
L^{\wedge}$ and $a\in L$. Then $a\in\kappa$ and so $\bigwedge f[\kappa]\leq
f(a)$.

(2) For the reverse direction, let $\lambda\leq f^{\pi}(u)$ and
$f^{\sigma}(u)\leq b$ for some $\lambda\in M^{\wedge}$ and $b\in M$. Let us
denote the filter $\\{a\in L~{}|~{}u\leq a\\}$ by $F_{u}$. By density of
$e_{M}$, $\lambda\leq f^{\pi}(u)=\bigwedge f[F_{u}]$ is equivalent to
$f[F_{u}]\subseteq\lambda$ and hence also to $F_{u}\subseteq f^{-1}[\lambda]$.
By our assumption, $f$ is a preframe homomorphism and so
$f^{\dagger}(\lambda)=f^{-1}[\lambda]$ is a Scott-open filter. Consequently,
$f^{\dagger}(\lambda)\leq\bigwedge F_{u}=u$ and, therefore,
$\lambda\leq\bigwedge
f[f^{\dagger}(\lambda)]=f^{\sigma}(f^{\dagger}(\lambda))\leq f^{\sigma}(u)\leq
b$.

(3) Let $\chi\leq g^{\pi}(f^{\pi}(u))$ and $(gf)^{\pi}(u)\leq c$ for some
$\chi\in N^{\wedge}$ and $c\in N$. Because $M^{\delta}$ is join-generated by
elements of $M^{\wedge}$, the statement $f^{\pi}(u)=\bigwedge f[F_{u}]\leq b$
is equivalent to

$(\forall\lambda\in M^{\wedge})\quad
f[F_{u}]\subseteq\lambda\quad\text{implies}\quad b\in\lambda$

(this is because, by compactness, $\lambda\leq\bigwedge f[F_{u}]$ iff
$f[F_{u}]\subseteq\lambda$). In other words, $f^{\pi}(u)\leq b$ iff $b\in B$
where

$B=\bigcap\\{\lambda\in M^{\wedge}~{}|~{}f[F_{u}]\subseteq\lambda\\}.$

Therefore, $\chi\leq g^{\pi}(f^{\pi}(u))=\bigwedge\\{g(b)~{}|~{}b\in M\text{
s.t.\ }f^{\pi}(u)\leq b\\}=\bigwedge g[B]$. On the other hand
$(gf)^{\pi}(u)=\bigwedge g[f[F_{u}]]$. Therefore, from $f[F_{u}]\subseteq B$
we obtain the desired $\chi\leq\bigwedge g[B]\leq\bigwedge g[f[F_{u}]]\leq b$.

(4) By (2) it is enough to show the identity for $\mathsf{id}^{\pi}$, which
follows from the definition of $\pi$-extensions as
$\mathsf{id}^{\pi}(u)=\bigwedge F_{u}=u$. ∎

In the rest of the text, in cases when $f^{\sigma}$ is equal to $f^{\pi}$, we
denote the extension map by $f^{\delta}$.

###  6.1 Perfect maps

In order to proceed with our theory, we have to assume further properties
about our maps. We say that a monotone map $f\colon L\to M$ between frames is
_perfect_ if, for every $\kappa\in L^{\wedge}$, the smallest filter containing
$f[\kappa]$ is Scott-open. In other words, $f$ is perfect if the map

$f^{\wedge}\colon\kappa\mapsto\mathord{\uparrow}\mkern 1.0mu\\{\bigwedge
M~{}|~{}M\subseteq f[\kappa]\text{ and $M$ is finite}\\}$

is a well defined map $L^{\wedge}\to M^{\wedge}$.

Note that, in the locally compact case, this definition agrees with the other
definitions for frame homomorphism known under the same name.

######  Lemma 6.3.

Let $h\colon L\to M$ be a frame homomorphism with $L$ locally compact. The
following three statements are equivalent:

  1. 1.

$c\ll a$ implies $h(c)\ll h(a)$, for every $a,c\in L$.

  2. 2.

$h$ is perfect.

  3. 3.

The right adjoint of $h$ preserves directed joins.

###### Proof.

The strategy of the proof is the same as in the proof of Theorem IV-1.4 in
[17]. It needs to be slightly adapted for Scott-open _filters_. ∎

Observe that if a monotone map $f\colon L\to M$ is perfect then its
$\sigma$-extension can be written in a more compact way:

$\displaystyle f^{\sigma}\colon u$
$\displaystyle\mapsto\bigvee\\{f^{\wedge}(\kappa)~{}|~{}\kappa\in L^{\wedge},\
\kappa\leq u\\}.$

######  Lemma 6.4.

Let $f\colon L\to M$ and $g\colon M\to N$ be monotone maps between frames. If
$f$ is perfect and $g$ preserves finite meets then

  1. 1.

the diagrams

${L^{\wedge}}$${M^{\wedge}}$${L^{\delta}}$${M^{\delta}}$$\scriptstyle{f^{\wedge}}$$\scriptstyle{e^{\wedge}_{L}}$$\scriptstyle{e^{\wedge}_{M}}$$\scriptstyle{f^{\sigma}}$
and
${L^{\wedge}}$${M^{\wedge}}$${L^{\delta}}$${M^{\delta}}$$\scriptstyle{f^{\wedge}}$$\scriptstyle{e^{\wedge}_{L}}$$\scriptstyle{e^{\wedge}_{M}}$$\scriptstyle{f^{\pi}}$

commute, and

  2. 2.

$(gf)^{\sigma}\leq g^{\sigma}f^{\sigma}$.

###### Proof.

(1) For a $\kappa\in L^{\wedge}$, clearly
$f^{\sigma}(\kappa)=\bigvee\\{f^{\wedge}(\kappa^{\prime})~{}|~{}\kappa^{\prime}\in
L^{\wedge},\ \kappa^{\prime}\leq\kappa\\}=f^{\wedge}(\kappa)$ and
$f^{\pi}(\kappa)=\bigwedge\\{f(a)~{}|~{}a\in L,\ \kappa\leq a\\}=\bigwedge
f[\kappa]$ which is equal to $f^{\wedge}(\kappa)$ when interpreted by the map
$e^{\wedge}\colon L^{\wedge}\hookrightarrow L^{\delta}$.

(2) Let $u\in L^{\delta}$ and let $\kappa\in L^{\wedge}$ be such that
$\kappa\leq u$. By monotonicity of $f^{\sigma}$ and by (1) we have that
$f^{\wedge}(\kappa)=f^{\sigma}(\kappa)\leq f^{\sigma}(u)$. Therefore,

$\bigwedge g[f^{\wedge}(\kappa)]\leq\bigvee\\{\bigwedge
g[\lambda]~{}|~{}\lambda\leq f^{\sigma}(u)\\}=g^{\sigma}(f^{\sigma}(u)).$

Moreover, because $g$ preserves finite meets, the leftmost expression
simplifies as follows

$\displaystyle\bigwedge g[f^{\wedge}(\kappa)]$
$\displaystyle=\bigwedge\\{g(\bigwedge M)~{}|~{}M\subseteq f[\kappa]\text{ and
$M$ is finite}\\}$ $\displaystyle=\bigwedge\\{\bigwedge g[M]~{}|~{}M\subseteq
f[\kappa]\text{ and $M$ is finite}\\}=\bigwedge gf[\kappa].$

Since $(gf)^{\sigma}(u)=\bigvee\\{\bigwedge gf[\kappa]~{}|~{}\kappa\leq u\\}$
we get that $(gf)^{\sigma}(u)\leq g^{\sigma}(f^{\sigma}(u))$. ∎

######  Proposition 6.5.

Let $f\colon L\rightleftarrows M\colon g$ be monotone maps such that $f$ is
the left adjoint of $g$. If $f$ is perfect then $f^{\sigma}$ is the left
adjoint of $g^{\pi}$.

###### Proof.

Recall that $f$ and $g$ being adjoint means that $fg\leq\mathsf{id}$ and
$\mathsf{id}\leq gf$. Therefore, by Lemma 6.2, $f^{\sigma}g^{\pi}\leq
f^{\pi}g^{\pi}\leq(fg)^{\pi}\leq\mathsf{id}^{\pi}=\mathsf{id}$. Because right
adjoins preserve arbitrary meets, we can use Lemma 6.4 to show the other
inequality: $\mathsf{id}=\mathsf{id}^{\sigma}\leq(gf)^{\sigma}\leq
g^{\sigma}f^{\sigma}\leq g^{\pi}f^{\sigma}$. ∎

######  Theorem 6.6.

If $h\colon L\to M$ is a perfect frame homomorphism between two frames, then
$h^{\delta}\colon L^{\delta}\to M^{\delta}$ is a complete lattice
homomorphism, that is, it preserves arbitrary joins and meets.

###### Proof.

From Proposition 6.5 we know that $h^{\delta}$ is a left adjoint. Therefore,
$h^{\delta}$ preserves arbitrary joins. What is left to show is that
$h^{\delta}$ also preserves arbitrary meets.

Let $\\{u_{i}\\}_{i\in I}\subseteq L^{\delta}$, and let
$\lambda\leq\bigwedge_{i}h^{\delta}(u_{i})$ and
$h^{\delta}(\bigwedge_{i}u_{i})\leq b$ for some $\lambda\in M^{\wedge}$ and
$b\in M$. Denote the filter $\\{a\in L~{}|~{}u_{i}\leq a\\}$ by $F_{i}$. The
inequality $\lambda\leq\bigwedge_{i}h^{\delta}(u_{i})\leq
h^{\delta}(u_{i})=\bigwedge h[F_{i}]$ implies that, by compactness of $e_{M}$,
$h[F_{i}]\subseteq\lambda$ and so $F_{i}\subseteq
h^{-1}[\lambda]=h^{\dagger}(\lambda)$. Because, for all $i\in I$,
$h^{\dagger}(\lambda)\leq\bigwedge F_{i}=u_{i}$, we obtain that
$h^{\dagger}(\lambda)\leq\bigwedge_{i}u_{i}$. By compactness of $e_{L}$,
$G\subseteq h^{\dagger}(\lambda)$, where $G=\\{a\in
L~{}|~{}\bigwedge_{i}u_{i}\leq a\\}$, and so $h[G]\subseteq\lambda$.
Consequently, $\lambda\leq\bigwedge h[G]=h^{\delta}(\bigwedge_{i}u_{i})$. ∎

From Lemmas 6.2 and 6.4, we obtain that $(\mkern 1.0mu\cdot\mkern
1.0mu)^{\delta}$ preserves the identity morphisms and that $(hg)^{\delta}\leq
h^{\delta}g^{\delta}\leq(hg)^{\delta}$, whenever $h$ and $g$ are perfect frame
homomorphisms. Combining all this with Corollary 4.3 and Theorem 6.6 gives the
following.

######  Corollary 6.7.

The mapping $L\mapsto L^{\delta}$ and $h\mapsto h^{\delta}$ constitutes a
functor from the category of locally compact frames and perfect homomorphisms
to the category of completely distributive lattices and complete lattice
homomorphisms.

######  Remark 6.8.

In the theory of canonical extensions, $\sigma$– and $\pi$-extensions are
often used in order to lift operations on a given lattice to its canonical
extension. It seems that most of the arguments would go through for extensions
of perfect operations on a frame. However, we leave checking whether this is
indeed the case for future investigations.

###  6.2 Perfect sublocales

The map extensions can be used to lift certain sublocales of $L$ to sublocales
of $L^{\delta}$. We say that a sublocale $S\subseteq L$ is _perfect_ if the
associated quotient homomorphism $L\twoheadrightarrow S$ is perfect.

######  Lemma 6.9.

For a perfect sublocale $S\subseteq L$ of a locally compact frame $L$,
$S^{\delta}$ is isomorphic to a sublocale of $L^{\delta}$.

###### Proof.

We show that the extension $h^{\delta}$ of the perfect onto frame homomorphism
$h\colon L\twoheadrightarrow S$ is also onto. Since $S^{\wedge}$ join-
generates $S^{\delta}$ and since $h^{\delta}$ preserves joins, it is enough to
show that $h^{\delta}$ maps onto $S^{\wedge}$. Let $\lambda\in S^{\wedge}$,
then
$h^{\delta}(h^{\dagger}(\lambda))=\bigwedge\\{h(a)~{}|~{}h^{\dagger}(\lambda)\leq
a\\}=\bigwedge h[h^{\dagger}(\lambda)]$ which is equal to $\lambda$ because
$h$ is onto and so $h[h^{\dagger}(\lambda)]=\lambda$. ∎

Note that perfect sublocales have been previously studied, e.g., in [6, 7] it
is shown that the pointfree analogue of the patch topology for a stably
compact space is precisely the frame of all perfect nuclei.

##  7 Monotone maps and spectra

One of the reasons for studying canonical extensions of frames is that it
allows us to represent monotone functions with respect to the specialisation
order directly in the category of frames.

Let $(X,\tau)$ and $(Y,\rho)$ be locally compact sober spaces. It is known
that monotone maps $(X,\leq_{\tau})\to(Y,\leq_{\rho})$ are in a bijective
correspondence with continuous maps $(X,\mathsf{Up}(X))\to(Y,\rho)$. Moreover,
since $\tau^{\delta}\cong\mathsf{Up}(X)$ (by Example 3.5 and Theorem 4.2), we
also have that

$\textbf{Top}((X,\mathsf{Up}(X)),\,(Y,\rho))\cong\textbf{Frm}(\rho,\,\mathsf{Up}(X))\cong\textbf{Frm}(\rho,\tau^{\delta}),$

where $\textbf{Top}(X,Y)$ is the set of continuous maps $X\to Y$ and
$\textbf{Frm}(L,M)$ is the set of frame homomorphisms $L\to M$. In other
words, we obtain:

######  Proposition 7.1.

For locally compact frames $L$ and $M$, monotone maps
$\mathsf{pt}(L)\to\mathsf{pt}(M)$, with respect to the specialisation orders,
are in a bijective correspondence with the frame homomorphisms $M\to
L^{\delta}$.

Next, for a locally compact frame $L$ and its topological dual $(X,\tau)$, we
know that the embeddings $e\colon L\hookrightarrow L^{\delta}$ and
$L\hookrightarrow\mathsf{Up}(\mathsf{pt}(L))\cong\mathsf{Up}(X)$ are
isomorphic but this does not mean that the spaces $(X,\mathsf{Up}(X))$ and
$\mathsf{pt}(L^{\delta})$ are homeomorphic. The embedding
$L\hookrightarrow\mathsf{Up}(X)$ arises from the continuous map
$(X,\mathsf{Up}(X))\to(X,\tau)$ but, as we will see, the onto map
$\mathsf{pt}(e)\colon\mathsf{pt}(L^{\delta})\twoheadrightarrow\mathsf{pt}(L)$
can in general be a different map.

In the following we examine the relationship between the spaces
$(X,\mathsf{Up}(X))$ and $\mathsf{pt}(L^{\delta})$. First, let us look at the
interplay between the spectra of the original frame and its canonical
extension.

######  Observation 7.2.

Let $L$ be any frame and let $\mathbf{2}$ be the two element frame.

  1. 1.

$\mathbf{2}$ is isomorphic to its canonical extension $\mathbf{2}^{\delta}$.

  2. 2.

All frame homomorphisms $L\to\mathbf{2}$ are perfect.

###### Proof.

(1) Since $\mathbf{2}^{\wedge}\cong\mathbf{2}$, it is easy to check that the
identity mapping $\mathbf{2}\to\mathbf{2}$ is both dense and compact. (2) The
image of a Scott-open filter under a homomorphism $L\to\mathbf{2}$ is
automatically a Scott-open filter on $\mathbf{2}$. ∎

Let $L$ be a locally compact frame. The frame embedding $e\colon
L\hookrightarrow L^{\delta}$ gives rise to a continuous map

$\displaystyle\mathsf{pt}(e)\colon$
$\displaystyle\mathsf{pt}(L^{\delta})\to\mathsf{pt}(L),\quad r\colon
L^{\delta}\to\mathbf{2}\enspace\longmapsto\enspace r\circ e\colon
L\to\mathbf{2}.$ An immediate consequence of Observation 7.2 is that there is
also a map in the opposite direction:
$\displaystyle\mathsf{pt}(L)\to\mathsf{pt}(L^{\delta}),\quad p\colon
L\to\mathbf{2}\enspace\longmapsto\enspace p^{\delta}\colon
L^{\delta}\to\mathbf{2}^{\delta}\cong\mathbf{2}.$

It is also immediate to check that $p^{\delta}(u)$, for a $u\in L^{\delta}$,
is computed as $\bigwedge p[F_{u}]$ where $F_{u}=\\{a\in L~{}|~{}u\leq a\\}$
(we follow the convention from Section 6).

As a result, we obtain that $\mathsf{pt}(e)$ is onto. Indeed, let $p\colon
L\to\mathbf{2}$ be a frame homomorphism. For an $a\in L$,

$(\mathsf{pt}(e)(p^{\delta}))(a)=p^{\delta}(e(a))=\bigwedge p[F_{a}]=\bigwedge
p[\mathord{\uparrow}\mkern 1.0mua]=p(a).$

This is no surprise as $e$ is injective. However, because $p^{\delta}$
preserves all meets (Corollary 6.7), we see that $\mathsf{pt}(e)$ stays onto
even if we restrict it to the subspace $\mathsf{pt}_{\bigwedge}(L^{\delta})$
of frame homomorphism $L^{\delta}\to\mathbf{2}$ which preserve all meets. This
makes sense since $L^{\delta}$ lives in the category of completely
distributive lattices and so its spectrum should be computed there.

To show that $\mathsf{pt}(e)$ is injective, when restricted to
$\mathsf{pt}_{\bigwedge}(L^{\delta})$, let $r\colon L^{\delta}\to\mathbf{2}$
be a complete lattice homomorphism and let $u\in L^{\delta}$. We have that

$(r\circ e)^{\delta}(u)=\bigwedge_{a\in F_{u}}(re)^{\delta}(a)=\bigwedge_{a\in
F_{u}}r(a)=r(u)$

where the first equality holds because $(r\circ e)^{\delta}$ preserves meets
and $u$ is equal to the meet $\bigwedge F_{u}$ in $L^{\delta}$, the second
equality follows from Lemma 6.1 and the last equality holds because $r$
preserves meets.

We have showed that there is a bijection between the points of
$\mathsf{pt}(L)$ and $\mathsf{pt}_{\bigwedge}(L^{\delta})$. This bijection on
points extends to a homeomorphism, once we equip $\mathsf{pt}(L)$ with the
topology of upsets.

######  Proposition 7.3.

Let $L$ be a locally compact frame and $X$ be its dual space. Then,
$\mathsf{pt}_{\bigwedge}(L^{\delta})$ (seen as a subspace of
$\mathsf{pt}(L^{\delta})$) is homeomorphic to $(X,\mathsf{Up}(X))$.

###### Proof.

We know that $L^{\delta}\cong\mathsf{Up}(X)$. What is left to show is that the
points of $\mathsf{pt}_{\bigwedge}(L^{\delta})$ separate opens. Let $u\not\leq
v$ in $L^{\delta}$. By density of canonical extensions, there are $\kappa\in
L^{\wedge}$ and $a\in L$ such that $\kappa\leq u$, $v\leq a$ and
$a\notin\kappa$. By a variant of prime filter extension theorem (see, e.g.
Proposition VII.6.2 in [25]), there exists a frame homomorphism $p\colon
L\to\mathbf{2}$ such that $p[\kappa]=\\{1\\}$ and $p(a)=0$. Consequently,
$p^{\delta}(u)\geq p^{\delta}(\kappa)=\bigwedge p[\kappa]=1$ and
$p^{\delta}(u)\leq p^{\delta}(a)=0$. ∎

##  8 Comparison with earlier approaches

###  8.1 Canonical extensions of distributive lattices revisited

In this section we show how our construction generalises some of the known
constructions of canonical extensions. First, let $A$ be a distributive
lattice. Since $\mathsf{Idl}(A)$ is coherent and therefore also locally
compact, we have the embedding

$i\colon A\xhookrightarrow{\enspace a\ \mapsto\ \mathord{\downarrow}\mkern
1.0mua\enspace}\mathsf{Idl}(A)\xhookrightarrow{\quad
e\quad}\mathsf{Idl}(A)^{\delta}.$

Moreover, $\mathsf{Filt}(A)$ is known to be isomorphic to
$\mathsf{Idl}(A)^{\wedge}$ via the map that sends $F$ to the Scott-open filter
$\kappa_{F}=\\{I~{}|~{}I\cap F\not=\emptyset\\}$ (e.g., as a special case of
Lemma 15 in [23]). Therefore, for a filter $F$,

$\bigwedge i[F]=\bigwedge\\{e(\mathord{\downarrow}\mkern 1.0mua)~{}|~{}a\in
F\\}=\bigwedge\\{e(I)~{}|~{}I\cap F\not=\emptyset\\}=\bigwedge e[\kappa_{F}]$

where the middle equality follows from the fact that, whenever $I\cap
F\not=\emptyset$, then $e(\mathord{\downarrow}\mkern 1.0mua)\leq e(I)$ for any
$a\in I\cap F$. As a consequence, from density and compactness of $e$ we
obtain density and compactness of the distributive lattice embedding $i\colon
A\hookrightarrow\mathsf{Idl}(A)^{\delta}$ (as defined in Section 3.1). In
other words, the embedding $i\colon A\hookrightarrow\mathsf{Idl}(A)^{\delta}$
is a canonical extensions of the distributive lattice $A$ and those are
uniquely determined.

######  Proposition 8.1.

Let $A$ be a distributive lattice. Its canonical extension $e\colon
A\hookrightarrow A^{\delta}$ is isomorphic to $\mathsf{Idl}(A)^{\delta}$.

Note also that homomorphisms between distributive lattices lift to perfect
frame homomorphisms between their frames of ideals and so Theorem 6.6 also
applies for them.

###  8.2 Canonical extensions of join-strong proximity lattices

Sam J. van Gool defined in [26] canonical extensions for _join-strong
proximity lattices_ , that is, for lattices $A$ equipped with a relation
$R\subseteq A\mathclose{\times}A$ such that, for every
$a,a^{\prime},b,b^{\prime}\in A$,555Sam J. van Gool shows that the categories
of distributive join-strong proximity lattices and stably compact spaces are
dually equivalent, which strengthens the result of [23] where also the dual of
the axiom (5) is required. The condition $R\subseteq(\leq)$ can be dropped if
we slightly modify the definition of round ideals and filters (see [23]).

  1. 1.

$R\circ R=R$ and $R\subseteq(\leq)$,

  2. 2.

$0Ra$ and $aR1$,

  3. 3.

$aRb$ and $a^{\prime}Rb$ iff $(a\vee a^{\prime})Rb$,

  4. 4.

$aRb$ and $aRb^{\prime}$ iff $aR(b\wedge b^{\prime})$,

  5. 5.

$aR(b\vee c)$ implies that $\exists b^{\prime},c^{\prime}$ such that
$c^{\prime}Rc$, $b^{\prime}Rb$ and $aR(b^{\prime}\vee c^{\prime})$.

In case when the lattice part of a join-strong proximity lattice $(A,R)$ is
distributive, we can use a similar argument as above, to show that the lattice
embedding666We know that $i\colon A\hookrightarrow\mathsf{RIdl}(A)^{\delta}$
is a lattice homomorphism because $a\mapsto I_{a}$ is a lattice homomorphism
by Proposition 17 in [23], and also that $e$ is an injective frame
homomorphism because $\mathsf{RIdl}(A)$ is locally compact by Theorem 11 in
[23].

$i\colon A\xhookrightarrow{\enspace a\ \mapsto\
I_{a}\enspace}\mathsf{RIdl}(A)\xhookrightarrow{\quad
e\quad}\mathsf{RIdl}(A)^{\delta}$

satisfies Sam van Gool’s conditions of canonical extensions for join-strong
proximity lattices. Here $\mathsf{RIdl}(A)$ denotes the frame of all _round_
ideals, that is, ideals $I$ such that $a\in I$ implies $a^{\prime}\in I$ for
some $aRa^{\prime}$. The ideal $I_{a}$ is taken to be $\\{b~{}|~{}bRa\\}$. The
argument also relies on the fact that the preframe of Scott-open filters of
$\mathsf{RIdl}(A)$ is isomorphic to the frame of round filters (Lemma 15 in
[23]).

######  Proposition 8.2.

Let $(A,R)$ be a distributive join-strong proximity lattice. Its canonical
extension $e\colon A\hookrightarrow(A,R)^{\delta}$ is isomorphic to
$\mathsf{RIdl}(A)^{\delta}$.

Sam van Gool also shows that his canonical extension lifts the natural
morphisms of join-strong proximity lattices (the so-called _j-morphisms_) to
maps between the corresponding canonical extensions. Such lifted maps preserve
all meets and, when assuming the axiom of choice, it can be shown that they
also preserve all joins [26]. Since j-morphisms between proximity lattices
correspond exactly to frame homomorphisms between the corresponding stably
compact frames, our Theorem 6.6 improves van Gool’s result in sense that we do
not need to use any choice principle for perfect morphisms.

###  8.3 The Boolean case

It turns out that canonical extensions of Boolean algebras ${B\hookrightarrow
B^{\delta}}$, which we showed how to obtain by frame-theoretic means in
Section 8.1, also have a different frame-theoretic description. First, recall
that a frame is _subfit_ if it satisfies the following condition

$(\forall a,b)\qquad a\not\leq b\implies\exists c\ \ a\vee
c=1\enspace\text{and}\enspace b\vee c\not=1$

Let us refer to a recent result by Richard Ball and Aleš Pultr [1] for subfit
frames.

######  Fact 8.3.

Let $L$ be a subfit frame and $M$ a complete Boolean algebra. If $L$ embeds
into $M$ in such a way that, whenever $x<y$ in $M$, then there are $a<b$ in
$L$ such that

$x\wedge b\leq a\quad\text{and}\quad y\vee a\geq b,$

then $M$ is isomorphic to $\mathcal{S}_{c}(L)$, which is the frame of
sublocales of $L$ which are joins of closed sublocales.

Moreover, Ball and Pultr [1] also show that
$\mathcal{S}_{c}(\mathcal{O}(X))\cong\raisebox{2.03998pt}{\Large$\wp$}(X)$ for
every T1-space $X$. On the other hand, we know that $B^{\delta}$ is a complete
Boolean algebra which is, assuming the Axiom of Choice, isomorphic to
$\raisebox{2.03998pt}{\Large$\wp$}(\mathsf{spec}(B))$, where
$\mathsf{spec}(B)$ is the Stone dual to $B$. This means that the canonical
extension of $B$ (seen as a lattice), the canonical extension of the frame
$\mathsf{Idl}(B)$, and $\mathcal{S}_{c}(\mathsf{Idl}(B))$ are all isomorphic.
In fact, we can prove the same statement without the detour to spaces and
without the axiom of choice.

######  Theorem 8.4.

Let $B$ be a Boolean algebra, then $B^{\delta}$ (resp.
$\mathsf{Idl}(B)^{\delta}$) is isomorphic to
$\mathcal{S}_{c}(\mathsf{Idl}(B))$.

###### Proof.

We show that the embedding
$e\colon\mathsf{Idl}(B)\hookrightarrow\mathsf{Idl}(B)^{\delta}$ satisfies the
conditions of Fact 8.3. Observe that $\mathsf{Idl}(B)^{\delta}$ is a complete
Boolean algebra since $B^{\delta}$ is (Theorem 6.3 in [12]) and since
$B^{\delta}\cong\mathsf{Idl}(B)^{\delta}$ (Proposition 8.1).

Next, let $x<y$ in $\mathsf{Idl}(B)^{\delta}$. By density of $e$, there is an
ideal $I\in\mathsf{Idl}(B)$ such that

$\displaystyle x\leq e(I)\quad\text{and}\quad y\not\leq e(I).$ (1)

Next, by density of the lattice canonical extension $i\colon
B\hookrightarrow\mathsf{Idl}(B)\hookrightarrow\mathsf{Idl}(B)^{\delta}$, there
is a filter $F\in\mathsf{Idl}(B)$ such that

$\displaystyle\bigwedge i[F]\leq y\quad\text{and}\quad\bigwedge i[F]\not\leq
e(I)=\bigvee i[I].$ (2)

Hence, compactness of $i$ gives that $F\cap I=\emptyset$ or, in other words,
that $f\not\leq i$ for every $f\in F$ and $i\in I$. Since $f\not\leq i$ iff
$i\vee\neg f\not=1$, we see that the join of ideals $I\vee\neg F$, where $\neg
F=\\{\neg f~{}|~{}f\in F\\}$, is not equal to the ideal
$\mathord{\downarrow}\mkern 1.0mu1$. Set $a=e(I\vee\neg F)$ and $b=1$. Finally
we check that the pair $a<b$ satisfies the two required inequalities. By (1)
we have that

$x\wedge b=x\leq e(I)\leq e(I\vee\neg F)=a$

and by (2) we have that

$y\vee a\geq y\vee e(\neg F)\geq\bigwedge i[F]\vee e(\neg F)=1=b.$

To see why the penultimate equality holds, let $\bigwedge i[F]\vee e(\neg
F)=\bigwedge i[F]\vee\bigvee i[\neg F]\leq i(z)$ for some $z\in B$. We see
that $z\in F$ by compactness of $i$. Moreover, because $i(\neg z)\leq\bigvee
i[\neg F]\leq i(z)$ and so $\neg z\leq z$, it must be that $z$ is equal to 1
(in $B$). ∎

Note that the present theorem cannot be generalised to distributive lattices
because canonical extensions of those are not complete Boolean algebras in
general, as required by Fact 8.3.

##  9 Final remarks

In this paper we have outlined the basics of the theory of canonical
extensions for frames. By the example of how powerful the theory of canonical
extensions has shown to be for posets and lattices, it is natural to ask
whether this can happen for frames too. For example, already with the present
work we can lift operations on frames to their canonical extensions. As in the
classical theory of canonical extensions, one should expect that such lifted
operations will preserve certain types of equations and also that the original
operations correspond to certain relations on the spectrum of the frame (in
the spirit of [15, 11]).

However, the current results are only a starting point. It is not clear yet
how to generalise the theory to arbitrary frames. Our construction works well
for locally compact frames or, more generally, for frames with enough compact
fitted sublocales (Proposition 5.1).

Another missing piece of the puzzle concerns $\sigma$– and $\pi$-extensions of
maps. All continuous maps between sober spaces lift to continuous maps between
their canonical extensions (the preimage of an upset, by a continuous map, is
always an upset). For this reason we would expect that homomorphisms between
frames lift to complete lattice homomorphisms between the corresponding
canonical extensions. However, we were not able to prove this without assuming
perfectness. Maybe a more general construction of canonical extensions needs
to be considered first.

Finally, it is well known that a complete Boolean algebra is completely
distributive if and only if it is atomic, which is a fact that relies on a
non-constructive choice principle. For this reason, one cannot expect to prove
that the canonical extension of an arbitrary frame is completely distributive.
This is because in the special case when we take the frame $\mathsf{Idl}(B)$,
for a Boolean algebra $B$, the canonical extension $\mathsf{Idl}(B)^{\delta}$
is known to be a complete Boolean algebra (Theorem 6.3 in [12]) and so proving
complete distributivity for $\mathsf{Idl}(B)^{\delta}$ requires non-
constructive principle. It remains open, however, whether canonical extensions
of frames are always frames and coframes. Note that this is always the case
for canonical extensions of distributive lattices (Theorem 3 in [11]).

## Acknowledgement

I would like to thank the participants of the Workshop on Algebra, Logic and
Topology in Coimbra 2018 for many stimulating discussions. In particular, I am
grateful to Mai Gehrke for suggesting this research topic to me and giving me
a lot of invaluable feedback in the process. I also appreciate comments and
suggestions of the anonymous referee which greatly improved the presentation
of the paper.

Last but not least, I would like to thank Aleš Pultr for introducing me to the
ever-fascinating world of pointfree topology.

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