# The duality between boolean valued models and topological presheaves

Moreno Pierobon, Matteo Viale

###### Abstract.

Boolean valued models for a signature $\mathcal{L}$ are generalizations of
$\mathcal{L}$-structures in which we allow the $\mathcal{L}$-relation symbols
to be interpreted by boolean truth values. For example, for elements
$a,b\in\mathcal{M}$ with $\mathcal{M}$ a $\mathsf{B}$-valued
$\mathcal{L}$-structure for some boolean algebra $\mathsf{B}$, $(a=b)$ may be
neither true nor false, but get an intermediate truth value in $\mathsf{B}$.
In this paper we expand and relate the work of Mansfield and others on the
semantics of boolean valued models, and of Monro and others on the adjunctions
between $\mathsf{B}$-valued models and $\mathsf{B}^{+}$-presheaves for a
boolean algebra $\mathsf{B}$. First of all we introduce a topological
characterization of the sheafification process for presheaves on topological
spaces induced by the dense Grothendieck topology. On the way to produce our
characterization, we also relate the notion of open continuous mapping between
topological spaces to that of complete homomorphism between complete boolean
algebras, and to that of adjoint homomorphism between boolean algebras (e.g.
an homomorphism which has a left adjoint, if seen as a functor between partial
orders/categories). Next we link these topological/category theoretic results
to the theory of boolean valued models. First of all we give a different proof
of Monro result identifying topological presheaves on Stone spaces with
boolean valued models, and sheaves (according to the dense Grothendieck
topology) with boolean valued models having the _mixing property_. Next we
give an exact topological characterization (the so called _fullness property_)
of which boolean valued models satisfy Łoś Theorem (i.e. the general form of
the Forcing Theorem which Cohen —Scott, Solovay, Vopenka— established for the
special case given by the forcing method in set theory). Then we separate the
fullness property from the mixing property, by showing that the latter is
strictly stronger. Finally we give an exact categorical characterization of
which presheaves correspond to full boolean valued models in terms of the
structure of global sections of their associated étalé space.

Both the authors acknowledge support from INDAM through GNSAGA. The second
author acknowledges support from the project: _PRIN 2017-2017NWTM8R
Mathematical Logic: models, sets, computability. MSC: _03E40,18A40, 18F20,
03C90, 03C20_._

## Summary of main results

This paper analyzes the correspondences existing between boolean valued models
and presheaves on boolean algebras. Our aim is to lay foundations for bridging
the gap between set theorists and category theorists, and to bring to light
the deep connections of the boolean valued approach to forcing with sheaf
theory. We start giving a brief summary of the main results (in some cases we
are forced to be inaccurate or cryptic as the precise formulation of many of
our theorems requires a terminology that cannot be given rightaway):

  1. (1)

We show that there exists an adjunction between the category of boolean valued
models (see Definitions 4.1, 4.2) and the category of _separated_ 111A
presheaf is separated if any compatible family of local sections admits at
most one collation. presheaves on Stone spaces with homomorphism induced by
_open_ continuous mappings (Theorem 5.5); moreover this adjunction specializes
to an equivalence of categories between the family of _extensional_ boolean
valued models (Definition 4.1) and the separated presheaves in which all local
sections are restrictions of global sections (Corollary 5.5); note that (up to
boolean isomorphism) every boolean valued model is extensional. In this
correspondence, models with the mixing property (Definition 4.13) are sheaves
according to the dense Grothendieck topology, and conversely (Proposition
5.7). Our result is the special case for boolean algebras of a more general
result by Monro [8, Theorem 5.4] which proves the existence of an equivalence
of categories between Heyting valued models and presheaves defined on
topological spaces. Our direct proof in terms of boolean valued models has the
advantage of being accessible to set theorists as well as to category
theorists.

  2. (2)

We investigate the basic features of boolean valued models and their
semantics, separating the mixing property and the fullness property
(Definition 4.10); the latter identifies those $\mathsf{B}$-valued models for
which Łoś Theorem —equivalently the Forcing Theorem— holds. Specifically we
show that every model with the mixing property is full (Proposition 4.14),
that this implication is strict (Example 4.16), and that not all boolean
valued models are full (Example 4.12). We also find a simple topological
property characterizing the fullness property (Theorem 4.11).

  3. (3)

With the aim of clarifying further the distinction between fullness and mixing
property, we characterize the fullness property elaborating more on our
correspondence between $\mathsf{B}$-valued models and presheaves, and taking
into account the notion of étalé space; in particular we use the following
characterization of topological sheaves: a fixed presheaf $\mathcal{F}$ on a
topological space $(X,\tau)$ is a topological sheaf if and only if the global
sections of the associated étalé space are exactly described by the elements
of $\mathcal{F}(X)$ [6, Section II.5, Theorem 1]. Given a $\mathsf{B}$-valued
model $\mathcal{M}$ for a first order language $\mathcal{L}$, to every
$\mathcal{L}$-formula $\varphi(\overline{x})$ we associate a presheaf
$\mathcal{G}^{\varphi}_{\mathcal{M}}$ on the clopen $N_{b_{\varphi}}$ in the
Stone space $\operatorname{St}(\mathsf{B})$ of ultrafilters of $\mathsf{B}$
(where $b_{\varphi}$ is the boolean value given to the existential formula
$\exists\overline{x}\varphi(\overline{x})$, and $N_{b}$ is the clopen subset
of $\operatorname{St}(\mathsf{B})$ given by the ultrafilters to which $b$
belongs). Roughly the local sections of $\mathcal{G}^{\varphi}_{\mathcal{M}}$
are given by the tuples $\langle\sigma_{1},\dots,\sigma_{n}\rangle$ such that
$\left\llbracket\varphi(\sigma_{1},\dots,\sigma_{n})\right\rrbracket>0_{\mathsf{B}}$.
We show that $\mathcal{M}$ is full exactly when, for every formula $\varphi$,
the étalé space associated to $\mathcal{G}^{\varphi}_{\mathcal{M}}$ has _at
least one_ global section (Theorem 6.2).

  4. (4)

To obtain the above characterizations of fullness and mixing property, we
introduce a topological construction of the sheafification process induced by
the dense Grothendieck topology (which we call the _stonean sheafification_).
This construction is performed by means of étalé spaces (as described in [6,
Section II.5] \- see Definition 3.12). On the way to produce our
characterization (given by Proposition 3.13, Corollary 3.19, Theorem 3.29) we
also relate the notion of open continuous mapping between topological spaces,
to that of complete homomorphism between complete boolean algebras, to that of
adjoint homomorphism between boolean algebras (e.g. an homomorphism of boolean
algebras which -seen as a functor between partial orders/categories- has a
left adjoint - Theorem 2.4 and Propositions 2.5 2.6 in Section 2). Furthermore
we characterize the extremally disconnected compact Hausdorff spaces as those
spaces which are the Stone-Čech compactification of any of their dense subsets
(Proposition 1.22). We also relate our construction to standard completion
processes performed in functional analysis/general topology (Examples 3.11 and
5.12) and outline in which ways our topological construction of the stonean
sheafification for topological spaces differs from the standard one described
in [6, Section II.5]. Roughly our construction is strictly grounded in
classical logic in that it is naturally correlated to a boolean valued
semantics on topological presheaves rather than a Heyting valued semantics.

## Forcing, boolean valued models, and sheaves

Boolean valued models constitute one of the classical approaches to forcing
[1, 3]. They also give a useful method for producing new first order models of
a theory; for example the standard ultrapower construction is just a special
instance of a more general procedure to construct boolean valued elementary
extensions of a given structure (see [12] or [7]). Many interesting spaces of
functions (for example $L^{\infty}(\mathbb{R}),C(X)$ etc.) can be naturally
endowed of the structure of a $\mathsf{B}$-valued model; for example set
$\left\llbracket
f=g\right\rrbracket=\left[\left\\{x\in\mathbb{R}:f(x)=g(x)\right\\}\right]_{\mathsf{MALG}}$
for $f,g\in L^{\infty}(\mathbb{R})$, where $\mathsf{MALG}$ is the complete
boolean algebra given by Lebesgue measurable sets modulo Lebesgue null sets;
$\left\llbracket f=g\right\rrbracket$ gives a natural measure of the degree of
equality subsisting between $f$ and $g$. In particular by means of boolean
valued models one can expect to employ logical methods to analyze the
properties of these function spaces. This can be fruitful, see for example
[11] or [4, 9, 10].
The aim of this paper is to systematize these results and connect them to
sheaf theory, in doing so we give a direct proof in terms of boolean valued
models of the boolean case of a result of Monro222We note that we became aware
of Monro’s results only after preparing the first version of this paper. We
thank Greta Coraglia for bringing Monro’s work to our attention. [8, Theorem
5.4, Proposition 5.6] (contribution 1 above), and we expand and develop this
result (contributions 2, 3, 4 above). Our long-stretched hope is that this is
a first step towards a fruitful transfer of techniques, ideas and results
arising in set theory in the analysis of the forcing method to other domains
where a sheaf-theoretic approach to problems is useful.
This paper (while written by scholars whose mathematical background is rooted
in set theory) is aimed primarily at readers with some expertise in category
theory and familiarity with first order logic. We emphasize that _no knowledge
or familiarity with the forcing method as presented in[1, 3, 5] is required_
to follow the proofs and statements of the main results. We will sporadically
relate our results to the forcing machinery through some examples (see in
particular Examples 4.16, 5.10, 5.12) and a few comments, but that’s all. The
reader unfamiliar with the forcing method can safely skip them without
compromising the comprehension of the main body of this article.
Let us spell out more precisely the outcomes of [4, 9, 10, 11] for what we
will do here. [4, 9, 10] show that there is a “natural” identification
between:

  * •

the unit ball of commutative real $C^{*}$-algebras of the form $C(X)$ with $X$
compact Hausdorff extremally disconnected;

  * •

the $\operatorname{\mathsf{RO}}(X)$-names for the unit interval $[0;1]$ as
computed in the forcing extension $V^{\operatorname{\mathsf{RO}}(X)}$ of $V$
given by $\operatorname{\mathsf{RO}}(X)$ (where
$\operatorname{\mathsf{RO}}(X)$ denotes the complete boolean algebra of
regular open subsets of $X$).

In [11] these results are applied to establish a weak form of Schanuel’s
conjecture on the transcendence property of the exponential function over the
complex numbers. One of the outcomes of the present paper combined with [10]
will be that the sheafification of the presheaf given by the essentially
bounded measurable functions defined on some measurable subset of $[0;1]$ is
exactly given by the sheaf associated to the boolean valued model describing
the real numbers in $[0;1]$ which exist in $V^{\mathsf{MALG}}$ (see Example
5.12). These results are just samples of the various roles boolean valued
models can play across different mathematical fields, and why we strongly
believe it can be useful to develop a flexible translation tool to relate
concepts expressed in rather different terminology in differents set-up. This
is more or less all we will say on the relation between the results of the
present paper and forcing. Let us now move to a detailed presentation of our
results.

## Structure of the paper

The paper is organized as follows:

  * •

Section 1 gives prerequisites on topology, boolean algebras, partial orders.

  * •

Section 2 explores the connections between the notion of open continuous map,
that of complete homomorphism between complete boolean algebras and that of
adjoint homomorphism between boolean algebras (the latter is an homomorphism
which seen as a functor among partial orders/categories has a right adjoint).

  * •

Section 3 presents our topological construction of the sheafification (stonean
sheafification) of a topological presheaf with respect to the dense
Grothendieck topology.

  * •

Section 4 gives a rapid presentation of the key properties of boolean valued
models.

  * •

Section 5 establishes a correspondence between boolean valued models with
morphisms and topological presheaves on Stone spaces, which specializes to a
duality between boolean valued models with the mixing property and stonean
sheaves.

  * •

Section 6 gives a categorical characterization of the fullness property of
boolean valued models in terms of properties of the bundles associated to
their presheaf structure.

###### Contents

  1. 1 Boolean algebras, partial orders, topological spaces, compactifications
    1. 1.1 Basics on boolean algebras, partial orders, topological spaces
    2. 1.2 Adjunctions of preorders
    3. 1.3 Hausdorff compactifications
    4. 1.4 Maximal filters of closed or open sets and Hausdorff compactifications
    5. 1.5 Extremally disconnected compact Hausdorff spaces as Stone-Čech compactifications
  2. 2 Dualities and adjuctions on topological spaces with open continuous maps
  3. 3 stonean sheafifications
    1. 3.1 Stonean sheaves versus topological sheaves
    2. 3.2 Stonean bundles and stonean sheaves
    3. 3.3 Stonean sheafification as an adjoint functor
  4. 4 Boolean valued models
    1. 4.1 Łoś Theorem for boolean valued models and fullness
    2. 4.2 The mixing property and fullness
  5. 5 The presheaf structure of a boolean valued model
  6. 6 A characterization of the fullness property using étalé spaces

##  1\. Boolean algebras, partial orders, topological spaces,
compactifications

In this section we gather a number of folklore results on boolean algebras,
partial orders, topological spaces we will use freely along the remainder of
the paper. We include proofs of those non-trivial results for which we are not
able to trace a reference in the literature. For non-trivial results whose
proof is omitted we give appropriate references.

###  1.1. Basics on boolean algebras, partial orders, topological spaces

We recall some basic facts and terminology about boolean algebras, Heyting
algebras, partial orders, compact spaces. The missing proofs can be found in
[3, 13].

Given a topological space $(X,\tau)$ and $A$ subset of $X$,
$\mathrm{Reg}\left(A\right)$ is the interior of the closure of $A$ in
$(X,\tau)$. $A$ is _regular open_ if
$A=\mathrm{Int}\left(\mathrm{Cl}\left(A\right)\right)=\mathrm{Reg}\left(A\right)$.
$\operatorname{\mathsf{RO}}(X,\tau)$ (in short $\operatorname{\mathsf{RO}}(X)$
if $\tau$ is clear from the context) denotes the family of regular open
subsets of $(X,\tau)$. $\operatorname{\mathsf{CLOP}}(X,\tau)$ (or just
$\operatorname{\mathsf{CLOP}}(X)$ if $\tau$ is clear from the context) denotes
the family of clopen subsets of $(X,\tau)$. Clearly
$\operatorname{\mathsf{CLOP}}(X)\subseteq\operatorname{\mathsf{RO}}(X)$.
$\mathcal{O}(X,\tau)$ (or just $\mathcal{O}(X)$ if $\tau$ is clear from the
context) is $\tau\setminus\left\\{\emptyset\right\\}$.

By the Stone Representation Theorem (see for instance [13, Theorem 2.2.32] or
[3, Theorem 7.11]), every boolean algebra $\mathsf{B}$ is isomorphic to the
boolean algebra $\operatorname{\mathsf{CLOP}}(\operatorname{St}(\mathsf{B}))$
of the clopen subsets of its Stone space $\operatorname{St}(\mathsf{B})$. The
latter is the compact Hausdorff zero-dimensional space whose points are the
ultrafilters on $\mathsf{B}$, topologized by taking as a base the family of
sets $N_{b}:=\left\\{G\in\operatorname{St}(\mathsf{B}):b\in G\right\\}$ for
some $b\in\mathsf{B}$. These sets are actually the clopen sets of
$\operatorname{St}(\mathsf{B})$.

######  Notation 1.1.

For the remainder of this paper, when dealing with Stone spaces $X$ (i.e.
$0$-dimensional compact Hausdorff spaces) it is convenient to look at points
of $X$ at times as elements of their clopen neighborhoods and at times as
ultrafilters on $\operatorname{\mathsf{CLOP}}(X)$ as the Stone duality
naturally identifies each point of $X$ to the ultrafilter on
$\operatorname{\mathsf{CLOP}}(X)$ given by its clopen neighborhoods. Thus we
will write $G\in U$ for $U\in\operatorname{\mathsf{CLOP}}(X)$ and
$G\in\operatorname{St}(\operatorname{\mathsf{CLOP}}(X))$ if we see $G$ as a
point in $X$, and $U\in G$ if we see $G$ as an ultrafilter on
$\operatorname{\mathsf{CLOP}}(X)$.

Given a topological space $(X,\tau)$, $\operatorname{\mathsf{RO}}(X)$ is
endowed of the structure of a complete boolean algebra by letting

  * •

$0_{\operatorname{\mathsf{RO}}(X)}:=\emptyset$,
$1_{\operatorname{\mathsf{RO}}(X)}:=X$,

  * •

for $U,V\in\operatorname{\mathsf{RO}}(X)$

$\displaystyle U\vee V:=\mathrm{Reg}\left(U\cup V\right),$ $\displaystyle
U\wedge V:=U\cap V,$ $\displaystyle\neg
U:=X\setminus\mathrm{Cl}\left(U\right).$

  * •

for any family $\left\\{U_{i}:i\in
I\right\\}\subseteq\operatorname{\mathsf{RO}}(X)$

$\displaystyle\bigvee_{i\in I}U_{i}:=\mathrm{Reg}\left(\bigcup_{i\in
I}U_{i}\right),$ $\displaystyle\bigwedge_{i\in
I}U_{i}:=\mathrm{Reg}\left(\bigcap_{i\in I}U_{i}\right).$

######  Fact 1.2.

A boolean algebra $\mathsf{B}$ is complete if and only if
$\operatorname{\mathsf{CLOP}}(\operatorname{St}(\mathsf{B}))=\operatorname{\mathsf{RO}}(\operatorname{St}(\mathsf{B}))$
(i.e. if and only if the space $\operatorname{St}(\mathsf{B})$ is extremally
disconnected).

See [13, Prop. 3.2.12] for a proof.

######  Remark 1.3.

Let $\left\\{N_{a}:a\in A\right\\}$ be a family of basic open sets in the
Stone space $\operatorname{St}(\mathsf{B})$ of a boolean algebra $\mathsf{B}$
such that $\bigvee A$ exists. Then $\bigcup_{a\in A}N_{a}$ is a dense open set
in the clopen set $N_{b}$ of $\operatorname{St}(\mathsf{B})$ if and only if
$\bigvee_{\mathsf{B}}A=b$. However, $\bigvee A=b$ holds also in case
$\bigcup_{a\in A}N_{a}\neq N_{b}$. In case $\mathsf{B}$ is complete,

$\mathrm{Reg}\left(\bigcup_{a\in A}N_{a}\right)=N_{\bigvee A}.$

Given a pre-order $(P,\leq)$, we endow it with the downward topology
$\tau_{P}$: given $X\subseteq P$,

$\downarrow X:=\\{p\in P:\textit{ there exists }x\in X\textit{ such that
}p\leq x\\}.$

$X\subseteq P$ is a _down-set_ if $X=\downarrow X$.
The family $\textrm{DOWN}(P)$ of the down-sets of $P$ is a topology for $P$,
the _downward topology_.

######  Definition 1.4.

An embedding of pre-orders $f:P\to Q$ is an order and incompatibility
preserving map, i.e. a map satisfying the following two conditions:

  * •

if $p_{1}\leq p_{2}$ in $P$ then $f(p_{1})\leq f(p_{2})$ in $Q$;

  * •

if $p_{1}\bot p_{2}$ (i.e. there is no $s\in P$ such that $s\leq p_{1}$ and
$s\leq p_{2}$) then $f(p_{1})\bot f(p_{2})$.

$f$ is _dense_ if $\operatorname{ran}(f)$ is dense in $Q$ endowed with the
downward topology.

Note that an embedding need not be injective: for example,
$U\mapsto\mathrm{Reg}\left(U\right)$ defines a non-injective (and surjective)
embedding of $\tau$ into $\operatorname{\mathsf{RO}}(X,\tau)$ provided
$\tau\supsetneq\operatorname{\mathsf{RO}}(X,\tau)$.

######  Definition 1.5.

A pre-order $P$ is:

  * •

_separative_ if, for every $p,q\in P$, $p\nleq q$ implies that there exists
$r\in P$ such that $r\leq p$ and $r\bot q$;

  * •

_upward complete_ if it admits suprema for all its non-empty subsets.

Observe the following:

  * •

Given $(X,\tau)$ a topological space, $\tau$ is an upward complete preorder.

  * •

If $\mathsf{B}$ is a boolean algebra, then
$\mathsf{B}^{+}:=\mathsf{B}\setminus\\{0_{\mathsf{B}}\\}$ with the induced
order is a separative ordered set.

  * •

A boolean algebra $\mathsf{B}$ is complete if and only if $\mathsf{B}^{+}$ is
upward complete.

  * •

If $P$ is a pre-order, we can always surject it onto a separative pre-order
via an order and incompatibility preserving map.

######  Definition 1.6.

The _boolean completion_ of a partial order $(P,\leq)$ is a complete boolean
algebra $\mathsf{B}$ such that there exists a dense embedding
$e:P\to\mathsf{B}^{+}$.
The _boolean completion_ of a boolean algebra $\mathsf{B}$ is the boolean
completion of the separative order $\mathsf{B}^{+}$.

######  Theorem 1.7.

If $P$ is a partial order, then $\operatorname{\mathsf{RO}}(P,\tau_{P})$ is
its boolean completion, as witnessed by the map

$\displaystyle e:$ $\displaystyle P\to\operatorname{\mathsf{RO}}(P).$
$\displaystyle p\mapsto\mathrm{Reg}\left(\downarrow p\right).$

Moreover the boolean completion of $P$ is unique up to isomorphism.

###### Proof.

See the proof of [3, Theorem 14.10] or [13, Theorem 4.3.1]. ∎

######  Fact 1.8.

Every boolean algebra $\mathsf{B}$ can be densely embedded in the complete
boolean algebra $\operatorname{\mathsf{RO}}(\operatorname{St}(\mathsf{B}))$
via the Stone duality map

$b\mapsto N_{b}=\left\\{G\in\operatorname{St}(\mathsf{B}):\,b\in G\right\\}$

which identifies $\mathsf{B}$ with
$\operatorname{\mathsf{CLOP}}(\operatorname{St}(\mathsf{B}))$. The image is
dense because the clopen sets form a base for the topology on
$\operatorname{St}(\mathsf{B})$.
Moreover
$\operatorname{\mathsf{RO}}(\mathsf{B}^{+},\tau_{\mathsf{B}^{+}})\cong\operatorname{\mathsf{RO}}(\operatorname{St}(\mathsf{B}))$
via the unique extension of the map defined on $\mathsf{B}^{+}$ by $b\mapsto
N_{b}$.

We also recall that a topological space is $T_{1}$ if any two distinct points
can be separated by open sets which contain exactly one of them; it is
extremally disconnected if the closure of an open set is open, it is
$0$-dimensional (or totally disconnected) if the clopen sets form a base for
the space.

######  Fact 1.9.

Let $(X,\tau)$ be a $T_{1}$ topological space. Then:

  * •

if $X$ is $0$-dimensional, it is Hausdorff;

  * •

if $X$ is Hausdorff, extremally disconnected, it is $0$-dimensional;

  * •

$X$ is extremally disconnected if and only if
$\operatorname{\mathsf{CLOP}}(X)=\operatorname{\mathsf{RO}}(X)$.

Extremally disconnected compact Hausdorff spaces are also called _stonean_
spaces.

We also need the following characterization of of the regularization operation
(see [13, Lemma 3.2.6]):

######  Lemma 1.10.

Let $(X,\tau)$ be a topological space and $A\subseteq X$. Then its
regularization $\mathrm{Reg}\left(A\right)$ is the set

$\left\\{a\in X:\,\exists\,U\text{ open neighborhood of }a\text{ such that
}A\cap U\text{ is dense in }U\right\\}.$

Using this characterization (and the fact that the image of a dense set under
a continuous dense map333For us, a map between topological spaces is _dense_
if its image is dense in the codomain. is dense) the following result is
almost immediate:

######  Fact 1.11.

Let $i:X\to Y$ be a dense topological embedding (e.g. an homeomorphism of $X$
with its image, which is dense in $Y$). Then $i$ implements an isomorphism of
$\operatorname{\mathsf{RO}}(Y)$ with $\operatorname{\mathsf{RO}}(X)$ given by
$U\mapsto i^{-1}[U]$.

###  1.2. Adjunctions of preorders

We also need to consider adjunctions of preorders. The terminology we adopt
comes from category theory (i.e. adjoint functors for categories which are
partial orders), and from the Stone duality, which establishes the
correspondence between homomorphisms of boolean algebras and continuous
functions on Stone spaces.

######  Definition 1.12.

Let $P,Q$ be partial orders and $i:P\to Q$, $\pi:Q\to P$ be order preserving
maps between them. The pair $(\pi,i)$ forms an _adjoint pair_ (or a _monotone
Galois connection_ or a _pair of residuated mappings_) if for all $p\in P$ and
$q\in Q$

$i(p)\geq q\text{ if and only if }\pi(q)\leq p.$

######  Fact 1.13.

Let $P,Q$ be partial orders and $i:P\to Q$, $\pi:Q\to P$ be order preserving
maps. The following are equivalent:

  1. (1)

$(\pi,i)$ is an adjunction;

  2. (2)

$\pi$ is defined by

$\pi(c)=\inf_{P}\left\\{b\in P:i(b)\geq c\right\\},$

and $i$ is defined by

$i(b)=\sup_{Q}\left\\{c\in Q:b\geq\pi(c)\right\\}.$

Hence any order preserving map $i:P\to Q$ ($\pi:Q\to P$) can have at most one
$\pi:Q\to P$ ($i:P\to Q$) such that $(\pi,i)$ is an adjoint pair.

The proof is an easy application of the definition and is left to the reader.

###  1.3. Hausdorff compactifications

We now recall some basic facts about Hausdorff compactifications.

######  Definition 1.14.

A triple $\langle Y,\sigma,\iota\rangle$ is a Hausdorff compactification of
some topological space $(X,\tau)$ if $(Y,\sigma)$ is a compact Hausdorff space
and $\iota:X\to Y$ is a topological embedding with a dense image.

For $(X,\tau)$ a $T_{0}$-topological space, its Stone-Čech compactification
(when it exists) is the unique Hausdorff compactification $\langle
Y,\sigma,\iota\rangle$ of $X$ that has the following universal property: any
continuous map $f:X\to K$ where $K$ is compact Hausdorff extends uniquely to a
continuous map $\beta(f):Y\to K$ such that $\beta(f)\circ\iota=f$.

The Stone-Čech compactification exists for a very large class of topological
spaces $(X,\tau)$:

######  Definition 1.15.

A topological space $(X,\tau)$ is _Tychonoff_ if for any $x\in X$ and $C$
closed subset of $X\setminus\left\\{x\right\\}$ there is a continuous
$f:X\to\mathbb{R}$ which vanishes on $x$ and gets value $1$ constantly on $C$.

######  Remark 1.16.

Every locally compact Hausdorff space is Tychonoff. Furthermore the Stone-Čech
Compactification Theorem characterizes the spaces which are topologically
homeomorphic to subsets of a compact Hausdorff space as the class of Tychonoff
spaces.

We spell out the details of one possible presentation of the Stone-Čech
compactification.

######  Notation 1.17.

Let $(X,\tau)$ be a topological space. $C\subseteq X$ is a _$0$ -set_ if there
is a continuous $f:X\to\mathbb{R}$ such that $C=f^{-1}[\left\\{0\right\\}]$.
We let $\mathcal{O}_{0}(X)$ denote the open sets which are complements of
$0$-sets.

######  Remark 1.18.

Note that, for any topological space, the family of its $0$-sets is closed
under finite unions and clopen sets are $0$-sets. Also note that elements of
$\mathcal{O}_{0}(X)$ can be described as the preimage of $[0;a)$ for some
$a<1$ and some continuous $f:X\to[0;1]$.
When $(X,\tau)$ is Tychonoff any closed set is the intersection of $0$-sets
and also the intersection of open sets which are the complement of $0$-sets.
Hence in a Tychonoff space $\mathcal{O}_{0}(X)$ forms a base of open sets
closed under finite intersections.

For any topological space $(X,\tau)$, one can define the space $\beta(X)$
whose elements are the maximal filter of $0$-sets with topology $\beta(\tau)$
generated by $\beta^{c}(U)=\left\\{F\in\beta(X):\,\exists C\in F\,C\subseteq
U\right\\}$ for $U\in\mathcal{O}_{0}(X)$. The map $\iota_{X}:X\to\beta(X)$
maps any $x\in X$ to the maximal filter given by $0$-sets containing $x$. This
map is well-defined and injective provided the (singletons of) points of $X$
are the intersection of $0$-sets, and a local homeomorphism exactly when $X$
is Tychonoff; in this latter case it can be shown that
$\langle\beta(X),\beta(\tau),\iota_{X}\rangle$ is the Stone-Čech
compactification of $(X,\tau)$ (see for instance [13, Section 7.4]).

We can sum up these informations as follows:

  * •

$\iota_{X}[X]$ is homeomorphic to $X$ if and only if $X$ is Tychonoff;

  * •

$\iota_{X}[X]$ is open in $\beta(X)$ if and only if $X$ is locally compact and
Hausdorff.

The following follows almost immediately from the above observations:

######  Fact 1.19.

Assume $(X,\tau)$ is Tychonoff. Then
$\operatorname{\mathsf{RO}}(X,\tau)\cong\operatorname{\mathsf{RO}}(\beta(X),\beta(\tau))$
via the map $U\mapsto U\cap X$ for
$U\in\operatorname{\mathsf{RO}}(\beta(X),\beta(\tau))$.

###  1.4. Maximal filters of closed or open sets and Hausdorff
compactifications

We now want to relate the points of Hausdorff compact spaces to (maximal)
filters of topological sets.

######  Definition 1.20.

Given a partial order $(P,\leq)$, a _filter_ on $P$ is an upward closed subset
$G$ such that for any $p,q\in G$ there exists $r\in G$ refining $p,q$.

The guiding example of filter on a partial order $P$ is given by the family
$G_{x}$ of open neighborhoods of a point $x\in X$ for $(X,\tau)$ a topological
space and $(P,\leq)$ the partial order $\mathcal{O}(X)$ given by non-empty
open sets of $(X,\tau)$. This filter in general is not a maximal filter
(consider the interval $[0;1]$ with the usual euclidean topology; the filter
of open neighborhoods of $1/2$ is not maximal; it can be extended to a maximal
filter containing $[0;1/2)$ and to another maximal filter containing
$(1/2;1]$). Moreover if $\mathcal{A}$ is a base of non-empty open sets for
$X$, $G_{x}\cap\mathcal{A}$ is also a filter on the partial order
$(\mathcal{A},\subseteq)$, but in general a maximal filter $G$ on
$\mathcal{A}$ may not identify a point of $X$: the intersection of the open
sets in $G$ (or even of the closure of the open sets in $G$) may be empty
(consider on $\mathbb{R}$ with the euclidean topology a maximal filter of open
sets extending $\left\\{(a,+\infty):a\in\mathbb{R}\right\\}$). This
identification of maximal filters of open sets to points of $X$ is clearly
related to the existence of compactifications of $X$. If $X$ is also compact
any maximal filter $G$ on $(\mathcal{A},\subseteq)$ determines a unique point
$x$ of $X$ with $G\supseteq G_{x}\cap\mathcal{A}$, this point $x$ is the
unique element of the closed set
$\bigcap\left\\{\mathrm{Cl}\left(U\right):U\in G\right\\}$. However different
maximal filters of open (or closed) sets could determine the same point of $X$
(as shown by the case outlined above for $1/2$ in $[0;1]$).

We now compare various possible ways to describe points of some
compactification of a topological space. Typically we can identify a point of
a compact Hausdorff space by means of:

  * •

its open neighborhoods;

  * •

its regular open neighborhoods;

  * •

its open neighborhoods which are complements of $0$-sets;

  * •

the family of closed sets to which it belongs;

  * •

the family of $0$-sets to which it belongs;

  * •

the family of clopen sets to which it belongs (in case the space is
$0$-dimensional).

Note that each of the above families detects a (maximal?) filter in the
corresponding partial order under inclusion. Hence given any topological space
$(X,\tau)$ it is tempting to try to construct a Hausdorff compactification of
$X$ whose points should be the maximal filters of one of the above families
with respect to $\tau$. This is exactly what occurs in the case of the above
construction of the Stone-Čech compactification.

We are interested in comparing the above types of maximal filters and
understand when these maximal filters define the same “compactification”. We
now show that:

  * •

For any $(X,\tau)$, maximal filters on $\operatorname{\mathsf{RO}}(X)$
naturally correspond to maximal filters of open sets: assume towards a
contradiction that some maximal filter $F$ on $\operatorname{\mathsf{RO}}(X)$
admits two incompatible extensions $F_{0},F_{1}$ to maximal filters on
$\mathcal{O}(X)$. If any elements of $F_{0}$ intersects any elements of
$F_{1}$, $F_{0}\cup F_{1}$ is a prefilter strictly containing both of them.
Hence there are $U_{i}\in F_{i}$ such that $U_{0}\cap U_{1}$ is empty. This
occurs if and only if
$\mathrm{Reg}\left(U_{0}\right)\cap\mathrm{Reg}\left(U_{1}\right)$ is empty.
Now $U_{i}\cap U$ is non-empty for all $U\in F$ and this occurs if and only if
$\mathrm{Reg}\left(U_{i}\right)\cap U$ is non-empty for all $U\in F$. Since
$F$ is maximal this can be the case only if $\mathrm{Reg}\left(U_{i}\right)\in
F$ for both $i$, which contradicts $F$ being a prefilter.
We note that for $G$ ultrafilter on $\operatorname{\mathsf{RO}}(X)$ the unique
maximal filter of open sets $\bar{G}$ is given by
$\left\\{U:\mathrm{Reg}\left(U\right)\in G\right\\}$.

  * •

For any $0$-dimensional $(X,\tau)$ there is a natural identification of
maximal filters on $\operatorname{\mathsf{CLOP}}(X)$ and maximal filters of
closed sets: if $C$ is closed, $C$ is an intersection of clopen sets, hence
any maximal filter $F$ of closed sets induces a maximal filter on
$\operatorname{\mathsf{CLOP}}(X)$ given by

$\left\\{U\in\operatorname{\mathsf{CLOP}}(X):\exists C\in F\,U\supseteq
C\right\\},$

and conversely.

  * •

If $(X,\tau)$ is compact Hausdorff, maximal filters of closed sets correspond
to maximal filters of $0$-sets: Any closed set $C$ is an intersection of
$0$-sets since $(X,\tau)$ is Tychonoff. Now if $\mathcal{F}$ is a maximal
filter of closed sets, let $\mathcal{F}^{*}$ be the intersection of
$\mathcal{F}$ with the $0$-sets. It suffices to argue that $\mathcal{F}^{*}$
is a maximal filter of $0$-sets. If not there is a $0$-set $D$ such that
$D\cap E$ is non-empty for all $0$-sets in $\mathcal{F}$ and $D\cap C$ is
empty for some closed and non-empty non-$0$-set in $\mathcal{F}$. Now if
$C=\bigcap_{i\in I}C_{i}$ with each $C_{i}$ a $0$-set, we get that each
$C_{i}$ is in $\mathcal{F}$. Hence we have on the one hand that
$\emptyset=D\cap C=\bigcap_{i\in I}D\cap C_{i},$ on the other hand that
$\left\\{D\cap C_{i}:i\in I\right\\}$ is a family of closed sets with the
finite intersection property. Since $X$ is compact the latter implies that
$D\cap C$ is non-empty. We reached a contradiction.

Conversely if $\mathcal{G}$ is a maximal filter of $0$-sets,
$\bigcap\mathcal{G}$ is a singleton $\left\\{x_{\mathcal{G}}\right\\}$ (we use
here that the space is compact Hausdorff). Therefore $\mathcal{G}^{*}$ is the
intersection with the $0$-sets of the maximal filter of closed sets given by
closed sets containing $x_{\mathcal{G}}$. Hence we have a bijective
correspondence between maximal filters of closed sets and maximal filters of
$0$-sets.

  * •

If $(X,\tau)$ is extremally disconnected, maximal filters on open sets
correspond to maximal filters on $\mathcal{O}_{0}(X)$: In extremally
disconnected spaces regular open sets are closed, hence they are $0$-sets, and
the complement of $0$-sets. This gives that
$\mathcal{O}_{0}(X)\supseteq\operatorname{\mathsf{RO}}(X)$. Hence any maximal
filter on $\mathcal{O}_{0}(X)$ induces a maximal filter on
$\operatorname{\mathsf{RO}}(X)$. By the first item these overlaps with maximal
filters of open sets. We also note that whenever $\mathcal{F}$ is a maximal
filter of open sets, $\mathcal{F}\cap\mathcal{O}_{0}(X)$ is a maximal filter
on $\mathcal{O}_{0}(X)$: otherwise
$\mathcal{F}\cap\operatorname{\mathsf{RO}}(X)$ could have two incompatible
extensions to a maximal filter on $\mathcal{O}_{0}(X)$, and therefore also to
$\mathcal{O}(X)$. We already saw that the latter is impossible.

Note that if $(X,\tau)$ is extremally disconnected but not $T_{1}$, the
filters of (regular) open neighborhoods of points of $X$ might not be maximal
filters of (regular) opens. Also we have already seen that in compact
Hausdorff spaces $X$ the maximal filter of closed sets detect points of $X$,
hence they may not detect a maximal filter of open sets (in case $X$ is not
extremally disconnected).

Wrapping everything together we get the following:

######  Proposition 1.21.

Assume $(X,\tau)$ is extremally disconnected, compact, Hausdorff. Then there
are natural identifications of the families of:

  * •

maximal filters on $\operatorname{\mathsf{RO}}(X)$;

  * •

maximal filters on $\operatorname{\mathsf{CLOP}}(X)$;

  * •

maximal filters on $\mathcal{O}_{0}(X)$;

  * •

maximal filters of open sets;

  * •

maximal filters of closed sets;

  * •

maximal filters of $0$-sets.

###  1.5. Extremally disconnected compact Hausdorff spaces as Stone-Čech
compactifications

We conclude this section giving a characterization of extremally disconnected
compact Hausdorff spaces as the class of compact Hausdorff spaces which are
the Stone-Čech compactification of any of their dense subsets444We are not
able to trace the following Proposition in the literature, hence we include a
proof of it.. We will need this characterization in Section 3.

######  Proposition 1.22.

Let $(X,\tau)$ be a compact Hausdorff space. Then $X$ is extremally
disconnected if and only if it is the Stone-Čech compactification of any of
its dense subsets.

###### Proof.

Assume $(X,\tau)$ is compact Hausdorff but not extremally disconnected. Then
there is $U\in\tau$ which is regular open but not closed. This gives that
(letting $\neg U=X\setminus\overline{U}$) $U\cup\neg U$ is open dense in $X$.
Fix $x_{0}\in U$ and $x_{1}\in\neg U$. Then $f(x)=x_{0}$ if $x\in U$ and
$f(x)=x_{1}$ if $x\in\neg U$ is continuous on $U\cup\neg U$ but cannot be
extended to a continuous map on the whole of $X$: note that
$\overline{U}\cap\overline{\neg U}$ is non-empty (else $U$ is closed); if $y$
belongs to this set and $g$ extends continuously $f$ to $X$, $g(y)$ is either
$x_{0}$ or $x_{1}$, but it can be neither of them since it is an accumulation
point both for $U=f^{-1}[\left\\{x_{0}\right\\}]$ and for $\neg
U=f^{-1}[\left\\{x_{1}\right\\}]$.

The converse direction is given by the next Lemma. ∎

######  Lemma 1.23.

Let $X$ be a compact Hausdorff extremally disconnected space. Then for every
dense subset $W\subseteq X$, every compact Hausdorff space $K$ and every
continuous function $f:W\to K$, there exists an unique continuous map
$\beta(f):X\to K$ such that $\beta(f)\restriction W=f$.

###### Proof.

We can assume that $X=\operatorname{St}(\mathsf{B})$ where
$\operatorname{\mathsf{RO}}(\operatorname{St}(\mathsf{B}))=\mathsf{B}$ up to
isomorphism. In particular we identify any $G$ in $X$ with the ultrafilter on
$\operatorname{\mathsf{RO}}(X)$ given by its regular open (clopen)
neighboorhoods. Let $G\in X$. Since $W$ is dense, $W\cap V\neq\emptyset$ for
every open neighborhood $V$ of $G$. In particular, the family $\\{V\cap W:V\in
G\\}$ has the finite intersection property. Consequently the same holds for
$\\{f[V\cap W]:V\in G\\}$ and for $\\{\overline{f[V\cap W]}:V\in G\\}$. Being
$K$ compact, we have

$\bigcap_{V\in G}\overline{f[V\cap W]}\neq\emptyset.$

######  Claim 1.

Let $U$ be an open subset of $K$. Then $U\cap\bigcap_{V\in G}\overline{f[V\cap
W]}\neq\emptyset$ if and only if $\mathrm{Reg}\left(f^{-1}[U]\right)\in G$.

###### Proof.

__

$\Rightarrow$:

Since $U$ is open in $K$, $U\cap f[V\cap W]\neq\emptyset$ for every $V\in G$.
Thus (in $X$) $f^{-1}[U]\cap V\neq\emptyset$ for every $V\in G$. This entails
that (in $X$) $\mathrm{Cl}\left(f^{-1}[U]\right)\cap V\neq\emptyset$ for every
$V\in G$. Now (in $X$) $\mathrm{Reg}\left(f^{-1}[U]\right)$ is a dense subset
of $\mathrm{Cl}\left(f^{-1}[U]\right)$, and (in
$\mathrm{Cl}\left(f^{-1}[U]\right)$) $V\cap\mathrm{Cl}\left(f^{-1}[U]\right)$
is a non-empty open subset of $\mathrm{Cl}\left(f^{-1}[U]\right)$ for any
$V\in G$. Therefore $\mathrm{Reg}\left(f^{-1}[U]\right)\cap V\neq\emptyset$
for all $V\in G$, which occurs if and only if
$\mathrm{Reg}\left(f^{-1}[U]\right)\in G$.

$\Leftarrow$:

Conversely assume that $\mathrm{Reg}\left(f^{-1}[U]\right)\in G$. Then
$\mathrm{Reg}\left(f^{-1}[U]\right)\cap Z\neq\emptyset$ for all $Z\in G$.
Since $\mathrm{Reg}\left(f^{-1}[U]\right)$ is dense in
$\mathrm{Cl}\left(f^{-1}[U]\right)$, we get that
$\mathrm{Cl}\left(f^{-1}[U]\right)\cap Z\neq\emptyset$ for all $Z\in G$. Since
$f^{-1}[U]$ is dense in $\mathrm{Cl}\left(f^{-1}[U]\right)$ and
$\mathrm{Cl}\left(f^{-1}[U]\right)\cap Z\neq\emptyset$ is open in
$\mathrm{Cl}\left(f^{-1}[U]\right)$ for all $Z\in G$, we get that
$f^{-1}[U]\cap Z\neq\emptyset$ for all $Z\in G$. This gives that $U\cap
f[Z\cap W]\neq\emptyset$ for all $Z\in G$, as desired.

∎

######  Claim 2.

For any $G\in X$, there is exactly one point in $\bigcap_{V\in
G}\overline{f[V\cap W]}$.

###### Proof.

By contradiction, assume that in $\bigcap_{V\in G}\overline{f[V\cap W]}$ there
are two distinct points $y_{1},y_{2}$. Being $K$ Hausdorff, there are disjoint
open sets $U_{1},U_{2}$ such that $y_{i}\in U_{i}$, $i=1,2$. By Claim 1,
$\mathrm{Reg}\left(f^{-1}[U_{1}]\right),\mathrm{Reg}\left(f^{-1}[U_{2}]\right)\in
G$. Hence
$\mathrm{Reg}\left(f^{-1}[U_{1}]\right)\cap\mathrm{Reg}\left(f^{-1}[U_{2}]\right)$
is non-empty clopen. By density of $W$ in $X$,
$\mathrm{Reg}\left(f^{-1}[U_{1}]\right)\cap\mathrm{Reg}\left(f^{-1}[U_{2}]\right)\cap
W$ is non-empty and clopen in $W$. Since $f^{-1}[U_{1}]$ is dense open in
$\mathrm{Reg}\left(f^{-1}[U_{1}]\right)\cap W$ we get that $f^{-1}[U_{1}]$ has
non-empty intersection with $\mathrm{Reg}\left(f^{-1}[U_{2}]\right)\cap W$ and
is relatively open in it. Since $f^{-1}[U_{2}]$ is dense in
$\mathrm{Reg}\left(f^{-1}[U_{2}]\right)\cap W$, we get that $f^{-1}[U_{1}]\cap
f^{-1}[U_{2}]$ is non-empty, which is a contradiction. ∎

Thus we can define $\beta(f)(G)$ as the unique point $y_{G}$ in
$\bigcap\\{\overline{f[V\cap W]}:V\in G\\}$. We leave to the reader to check
that $\beta(f)\restriction W=f$.

Finally, we prove the continuity of $\beta(f)$.

######  Claim 3.

Assume $U$ is an open set of $K$. Then
$\beta(f)[\mathrm{Reg}\left(f^{-1}[U]\right)]\subseteq\mathrm{Cl}\left(U\right)$.

###### Proof.

Assume $G\in\mathrm{Reg}\left(f^{-1}[U]\right)$. Note that
$G\in\mathrm{Reg}\left(f^{-1}[V]\right)$ for any $V$ open neighborhood of
$\beta(f)(G)$ (by Claim 1). Hence
$\mathrm{Reg}\left(f^{-1}[V]\right)\cap\mathrm{Reg}\left(f^{-1}[U]\right)$ is
non-empty for all such $V$. As in the proof of Claim 2, we obtain that
$f^{-1}[V]\cap f^{-1}[U]$ is non-empty for all such $V$. Therefore so is
$U\cap V$ for all $V$ open neighborhood of $\beta(f)(G)$.

We conclude that $\beta(f)(G)$ is an accumulation point of $U$. ∎

Now, fix $G\in X$ and take an open neighborhood $A$ of $\beta(f)(G)$. We prove
that there exists an open neighborhood $U$ of $G$ such that
$\beta(f)[U]\subseteq A$: by normality of $K$ (being it compact Hausdorff), we
can find an open neighborhood $V$ of $\beta(f)(G)$ such that
$\mathrm{Cl}\left(V\right)\subseteq A$. By Claim 1,
$\mathrm{Reg}\left(f^{-1}[V]\right)$ is an open neighborhood of $G$. By Claim
3 its image is contained in $\mathrm{Cl}\left(V\right)\subseteq A$. ∎

##  2\. Dualities and adjuctions on topological spaces with open continuous
maps

In this section we define the type of partial orders/topological spaces over
which we consider presheaves. To state our main theorems we need a bit of
terminology.

######  Definition 2.1.

Let $\mathsf{B},\mathsf{C}$ be boolean algebras.

A map $i:\mathsf{B}\to\mathsf{C}$ is:

  * •

an _homomorphism_ if it preserves the boolean operations (equivalently if it
is order preserving);

  * •

a _complete homomorphism_ if for all $A\subseteq\mathsf{B}$ and
$a\in\mathsf{B}$ such that $\bigvee_{\mathsf{B}}A=a$, we have that
$\bigvee_{\mathsf{C}}i[A]=i(a)$;

  * •

an _adjoint homomorphism_ if it has a left adjoint
$\pi_{i}:\mathsf{C}\to\mathsf{B}$.

Define
$\pi^{*}_{i}:\operatorname{St}(\mathsf{C})\to\operatorname{St}(\mathsf{B})$ by
$\pi^{*}_{i}(G)=i^{-1}[G]$.

######  Notation 2.2.

Let $(X,\tau)$, $(Y,\sigma)$ be topological spaces and $f:X\to Y$ be a
continuous map. Define $\bar{\bar{k}}_{f}:\sigma\to\tau$ by
$\bar{\bar{k}}_{f}(U)=f^{-1}[U]$.

Let $\bar{k}_{f}=\bar{\bar{k}}_{f}\restriction\operatorname{\mathsf{RO}}(Y)$
and $k_{f}=\bar{k}_{f}\restriction\operatorname{\mathsf{CLOP}}(Y)$.

The continuity of $f$ grants that the range of $k_{f}$ is included in
$\operatorname{\mathsf{CLOP}}(X)$. In principle it is not clear that the range
of $\bar{k}_{f}$ is included in $\operatorname{\mathsf{RO}}(X)$; we will see
that this is the case when $f$ is an open map.

We will use the above to define contravariant functors from (variations of)
the category of topological spaces with open maps to (variations of) the
category of boolean algebras with _adjoint_ homomorphisms. First of all we
need notation to define the relevant categories and functors.

######  Notation 2.3.

__

  * •

$\mathsf{Top}_{\mathrm{open}}$ denotes the category of topological spaces with
continuous open maps.

  * •

$\mathsf{TDCH}_{\mathrm{open}}$ denotes the subcategory with the same arrows
but with objects given by the $0$-dimensional compact Hausdorff spaces.

  * •

$\mathsf{EDCH}_{\mathrm{open}}$ denotes the subcategory with the same arrows
but with objects given by the extremally disconnected compact Hausdorff
spaces.

  * •

$\mathsf{BA}_{\mathrm{adj}}$ denotes the category of boolean algebras with
arrows given by _adjoint_ homomorphisms.

  * •

$\mathsf{CBA}$ denotes the category of complete boolean algebras with arrows
given by complete homomorphisms.

$\displaystyle\mathcal{L}^{*}:$
$\displaystyle\mathsf{Top}_{\mathrm{open}}\to\mathsf{CBA}$
$\displaystyle(X,\tau)\mapsto\operatorname{\mathsf{RO}}(X)$
$\displaystyle(f:X\to
Y)\mapsto(\bar{k}_{f}:\operatorname{\mathsf{RO}}(Y)\to\operatorname{\mathsf{RO}}(X))$

$\displaystyle\mathcal{D}:$
$\displaystyle\mathsf{TDCH}_{\mathrm{open}}\to\mathsf{BA}_{\mathrm{adj}}$
$\displaystyle(X,\tau)\mapsto\operatorname{\mathsf{CLOP}}(X)$
$\displaystyle(f:X\to
Y)\mapsto(k_{f}:\operatorname{\mathsf{CLOP}}(Y)\to\operatorname{\mathsf{CLOP}}(X))$

$\displaystyle\mathcal{R}:$
$\displaystyle\mathsf{BA}_{\mathrm{adj}}\to\mathsf{TDCH}_{\mathrm{open}}$
$\displaystyle\mathsf{B}\mapsto\operatorname{St}(\mathsf{B})$
$\displaystyle(i:\mathsf{B}\to\mathsf{C})\mapsto(\pi^{*}_{i}:\operatorname{St}(\mathsf{C})\to\operatorname{St}(\mathsf{B}))$

The adjunctions/dualities we want to bring forward are summarized in the
following theorem:

######  Theorem 2.4.

The following holds:

  1. (1)

$\mathcal{D}$, $\mathcal{L}^{*}$, $\mathcal{R}$ are well defined functors.

  2. (2)

$\mathcal{D}$ implements a duality between $\mathsf{TDCH}_{\mathrm{open}}$ and
$\mathsf{BA}_{\mathrm{adj}}$ with inverse given by $\mathcal{R}$.

  3. (3)

$\mathcal{D}\restriction\mathsf{EDCH}_{\mathrm{open}}$ implements a duality
between $\mathsf{EDCH}_{\mathrm{open}}$ and $\mathsf{CBA}$ with inverse given
by $\mathcal{R}\restriction\mathsf{CBA}$.

  4. (4)

Let $\mathcal{L}=\mathcal{L}^{*}\restriction\mathsf{TDCH}_{\mathrm{open}}$.
Then $(\mathcal{L},\mathcal{R}\restriction\mathsf{CBA})$ is a contravariant
adjunction.

The proof is an easy outcome of the following propositions555We expect their
content to be folklore; however we have not been able to trace in the
literature items (iv) and (vii) of Proposition 2.5, and item (A) of
Proposition 2.6. linking the properties of (complete injective) homomorphisms
to those of (open surjective) continuous functions and of adjoint pairs:

######  Proposition 2.5.

Let $i:\mathsf{B}\rightarrow\mathsf{C}$ be an homomorphism of boolean
algebras, and $f:X\to Y$ be a continuous function between compact
$0$-dimensional Hausdorff spaces.

Then:

  1. (i)

$\pi^{*}_{i}$ is continuous and closed;

  2. (ii)

$i$ is injective if and only if $\pi^{*}_{i}$ is surjective, in which case
$\pi^{*}_{i}[N_{i(b)}]=N_{b}$;

  3. (iii)

$i=k_{\pi^{*}_{i}}$ modulo the identification of $\mathsf{D}$ with
$\operatorname{\mathsf{CLOP}}(\operatorname{St}(\mathsf{D}))$ given by
$b\mapsto N_{b}$ for $\mathsf{D}$ a boolean algebra;

  4. (iv)

$i$ has a left adjoint $\pi_{i}$ if and only if $\pi^{*}_{i}$ is an open map,
in which case $N_{\pi_{i}(c)}=\pi^{*}_{i}[N_{c}]$, and $i$ is a complete
homomorphism;

  5. (v)

$k_{f}$ is an homomorphism;

  6. (vi)

$\pi^{*}_{k_{f}}=f$ modulo the identification of $Z$ with
$\operatorname{St}(\operatorname{\mathsf{CLOP}}(Z))$ given by

$z\mapsto G_{z}=\left\\{U\in\operatorname{\mathsf{CLOP}}(Z):z\in U\right\\}$

for $Z$ compact $0$-dimensional Hausdorff space;

  7. (vii)

$k_{f}$ has a left adjoint $\pi_{f}$ if and only if $f$ is an open map, in
which case $N_{\pi_{f}(c)}=f[N_{c}]$, and $k_{f}$ is a complete homomorphism.

Furthermore the above can be meshed with the following

######  Proposition 2.6.

Let $X,Y$ be arbitrary topological spaces and $f:X\to Y$ be an open continuous
map, we have that

  1. (A)

$\mathrm{Reg}\left(f^{-1}[U]\right)=f^{-1}[\mathrm{Reg}\left(U\right)]$ for
any open set $U$ of $Y$;

  2. (B)

$\bar{k}_{f}:\operatorname{\mathsf{RO}}(Y)\to\operatorname{\mathsf{RO}}(X)$
given by $U\mapsto f^{-1}[U]$ is a complete homomorphism with right adjoint
$\bar{\pi}_{f}:\operatorname{\mathsf{RO}}(X)\to\operatorname{\mathsf{RO}}(Y)$
given by $V\mapsto f[V]$;

  3. (C)

In case $X,Y$ are $0$-dimensional compact Hausdorff
$\bar{k}_{f}\restriction\operatorname{\mathsf{CLOP}}(Y)=k_{f}$ and
$\bar{\pi}_{f}\restriction\operatorname{\mathsf{CLOP}}(X)$ is the left adjoint
of $k_{f}$;

  4. (D)

In case $X,Y$ are extremally disconnected compact Hausdorff
$\bar{k}_{f}=k_{f}$.

######  Remark 2.7.

Let $\mathsf{B}$ be the boolean algebra given by finite and cofinite subsets
of $\omega$ and $i:\mathsf{B}\to\mathcal{P}\left(\omega\right)$ be the
inclusion map. It can be checked that $i$ is a complete injective homomorphism
and $\pi^{*}_{i}$ is not an open map666This counterexample has been given by
D. Monk.. On the other hand if $i:\mathsf{B}\to\mathsf{C}$ is an homomorphism
between complete boolean algebras, $i$ has an adjoint if and only if $i$ is
complete. In particular the notion of complete homomorphisms is weaker than
that of adjoint homomorphism only when considering non-complete boolean
algebras.

First of all we prove Theorem 2.4 assuming the two propositions:

###### Proof.

The fact that $\mathcal{L}^{*}$ is a well defined functor follows from A of
Proposition 2.6. $\mathcal{D}$ is a duality by Stone duality and by (iii) and
(iv) of Proposition 2.5. $\mathcal{D}\restriction\mathsf{EDCH}$ is also a
duality by the known result that the Stone duality maps the object of
$\mathsf{EDCH}$ onto those of $\mathsf{CBA}$ bijectively up to isomorphism, by
(iii) and (iv) of Proposition 2.5, and by Proposition 2.6.
$(\mathcal{L},\mathcal{R}\restriction\mathsf{CBA})$ is an adjunction by
Proposition 2.6 and by (iii) and (iv) of Proposition 2.5. ∎

We start with the proof of Proposition 2.6.

###### Proof.

__

(A):

Assume $U$ is open and $x\in\mathrm{Reg}\left(f^{-1}[U]\right)$. Then (by
Lemma 1.10) $x$ has an open neighborhood $V$ such that $V\cap f^{-1}[U]$ is
dense in $V$. Since $f$ is open, $f[V]$ is an open neighborhood of $f(x)$. It
is enough to show that $U\cap f[V]$ is dense in $f[V]$ (again by Lemma 1.10).
Pick $Z$ open subset of $f[V]$. Then $f^{-1}[Z]$ is an open subset of $V$,
hence it has non empty intersection with $f^{-1}[U]$. If $z$ is in this latter
set, $f(z)$ is in $Z\cap U$.

Conversely assume $x\in f^{-1}[\mathrm{Reg}\left(U\right)]$. Then (by Lemma
1.10) $f(x)$ has an open neighborhood $V$ such that $U\cap V$ is dense in $V$.
It is enough to show that $f^{-1}[U]\cap f^{-1}[V]$ is dense in $f^{-1}[V]$
(again by Lemma 1.10). Pick $A$ non-empty open subset of $f^{-1}[V]$. Then
$f[A]$ is a non-empty open subset of $V$, hence it has non-empty intersection
with $U$. This easily gives that $A\cap f^{-1}[U]$ is non-empty.

(B):

By (A) we get that for $U$ regular open subset of $Y$

$\mathrm{Reg}\left(f^{-1}[U]\right)=f^{-1}[\mathrm{Reg}\left(U\right)]=f^{-1}[U],$

hence $f^{-1}[U]$ is a regular open subset of $X$ and $\bar{k}_{f}$ is well
defined.

It is immediate to check that $\bar{k}_{f}$ is order preserving.

Furthermore if $\left\\{A_{i}:i\in I\right\\}$ is a family of regular open
subsets of $Y$ and $A=\bigcup_{i\in I}A_{i}$, we have that

$\displaystyle\bigvee_{\operatorname{\mathsf{RO}}(X)}\left\\{\bar{k}_{f}(A_{i}):i\in
I\right\\}=\mathrm{Reg}\left(\bigcup_{i\in
I}f^{-1}[A_{i}]\right)=\mathrm{Reg}\left(f^{-1}[\bigcup_{i\in
I}A_{i}]\right)=\mathrm{Reg}\left(f^{-1}[A]\right)=$
$\displaystyle=f^{-1}[\mathrm{Reg}\left(A\right)]=f^{-1}[\bigvee_{\operatorname{\mathsf{RO}}(Y)}\left\\{A_{i}:i\in
I\right\\}]=\bar{k}_{f}(\bigvee_{\operatorname{\mathsf{RO}}(Y)}\left\\{A_{i}:i\in
I\right\\}),$

hence $\bar{k}_{f}$ is a complete morphism of partial orders.

It is well known that a map $k:\mathsf{B}\to\mathsf{C}$ is a complete
homomorphism of complete boolean algebras if and only if $k$ is suprema and
order preserving. Hence $\bar{k}_{f}$ is a complete homomorphism.

It is also immediate to check that for $U\in\operatorname{\mathsf{RO}}(Y)$ and
$V\in\operatorname{\mathsf{RO}}(X)$, $\bar{k}_{f}(U)=f^{-1}[U]\supseteq V$ if
and only if $U\supseteq f[V]=\bar{\pi}_{f}(V)$, hence
$(\bar{\pi}_{f},\bar{k}_{f})$ is an adjoint pair.

(C) and (D):

Trivial.

∎

We now prove Proposition 2.5:

###### Proof.

__

(i), (ii), (iii):

This is part of the standard proofs of Stone duality.

(iv):

If $\pi^{*}_{i}$ is open we have that $\pi^{*}_{i}[N_{c}]$ is clopen for all
$c\in\mathsf{C}$. Furthermore for all $b\in\mathsf{B}$ and $c\in\mathsf{C}$

$N_{i(b)}\supseteq N_{c}\text{ if and only if
}(\pi^{*}_{i})^{-1}[N_{b}]\supseteq N_{c}\text{ if and only if
}N_{b}\supseteq\pi^{*}_{i}[N_{c}].$

Hence the map $\pi_{i}:N_{c}\mapsto\pi^{*}_{i}[N_{c}]$ is the left adjoint of
$k_{\pi^{*}_{i}}$, or equivalently (letting
$\pi_{i}(c)=\bigwedge_{\mathsf{B}}\left\\{b:i(b)\geq c\right\\}$) $\pi_{i}$ is
well defined and is the right adjoint of $i$.

For the completeness of $\pi_{i}$, apply (C) of Proposition 2.6, since infima
and suprema in $\operatorname{\mathsf{CLOP}}(X)$ are computed using the same
operations of $\operatorname{\mathsf{RO}}(X)$ (when they exist).

(v):

Trivial by continuity of $f$.

(vi):

See (iii) or else:

$\displaystyle\pi^{*}_{k_{f}}(G_{x})=\left\\{V\in\operatorname{\mathsf{CLOP}}(Y):k_{f}(V)\in
G_{x}\right\\}=\left\\{V\in\operatorname{\mathsf{CLOP}}(Y):f^{-1}[V]\in
G_{x}\right\\}=$
$\displaystyle=\left\\{V\in\operatorname{\mathsf{CLOP}}(Y):x\in
f^{-1}[V]\right\\}=\left\\{V\in\operatorname{\mathsf{CLOP}}(Y):f(x)\in
V\right\\}=G_{f(x)}.$

(vii):

Use (C) of Proposition 2.6.

∎

##  3\. stonean sheafifications

In this section we define a _stonean_ sheafification process for presheaves on
topological spaces $(X,\tau)$ which will associate to them étalé spaces whose
stalks are indexed by maximal filters of open sets of $\tau$. Our stonean
sheafification construction gives an explicit topological representation of
the sheafification process induced by the dense Grothendieck topology when
applied to a presheaf on an arbitrary topological space $(X,\tau)$. We mimick
the standard construction of sheaves (as described for example in [6, II]) for
presheaves defined on topological spaces; e.g. we associate to each presheaf
$\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ an étalé bundle and
we obtain the sheafification of $\mathcal{F}$ by taking the local section of
the (one point compactification) of the bundle. In performing both the
constructions of the étalé bundle and the selection of the local section of
this bundle we are more generous in our definitions and we blow up the base
space of the bundle (which is not anymore $X$ but
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ and the type of local
sections of the bundle we put into the sheafification, which are those
continuous functions from $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$
to the one point compactification of the (domain of the) bundle which take
value almost everywhere in the (domain of the) bundle. Therefore our
sheafification process produces a variety of stalks in the corresponding étalé
space which cannot exist in the étalé space associated to the standard
sheafification process (as described for example in [6, II]): the existence of
these additional stalks is based on the use of logical principles not
available in intuitionistic logic. Looking at our stonean sheafification
process with the lenses of boolean valued semantics, it describes a canonical
process to “complete” a $\mathsf{B}$-valued models $\mathcal{M}$ to a
$\operatorname{\mathsf{RO}}(\mathsf{B}^{+})$-valued model $\mathcal{M}^{*}$
which roughly satisfies the same semantics but which also has the mixing
property. We will outline in Section 5 the correspondence between boolean
valued models and presheaves. In this correspondence, boolean valued models
with the mixing property are stonean sheaves. When dealing with a
$\mathsf{B}$-model $\mathcal{M}$ without the mixing property, it would be
convenient to find a canonical way to “mixify” $\mathcal{M}$. This is exactly
what our stonean sheafification process does (see Cor. 5.13). [6] is our
reference for unexplained terminology.

###  3.1. Stonean sheaves versus topological sheaves

In this section we introduce and analyze the type of sheaves one is naturally
led to consider when looking at boolean valued models. We stick whenever
possible to the notation of [6]. Let us start giving an explicit definition of
the type of (pre)sheaves we will consider:

######  Definition 3.1.

Let $(P,\leq)$ be a partial order.

  * •

A _presheaf_ $\mathcal{F}:P^{\mathrm{op}}\to\mathsf{Set}$ is a contravariant
functor from $P$ (seen as a category whose arrows are determined by the order
relation) to the category $\mathsf{Set}$ (with arrow induced by functions).

  * •

A _topological presheaf_ $\mathcal{F}$ on a topological space $(X,\tau)$ is a
presheaf $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ (where
$\mathcal{O}(X)=\tau\setminus\left\\{\emptyset\right\\}$ is ordered by
inclusion).

######  Notation 3.2.

Let $(P,\leq)$ be a partial order.

  * •

$A\subseteq P$ is a _dense covering_ of $p\in P$ if it is a predense subset of
$P\restriction p$ e.g. for every $q\in P$ refining $p$ there is $r\in A$ and a
common refinement $s$ of $r$ and $q$.

  * •

Let $(P,\leq)$ be an upward complete partial order. $A\subseteq p$ is a
_complete covering_ of $p$ if $\bigvee_{P}A=p$.

Let $\mathcal{F}:P^{\mathrm{op}}\to\mathsf{Set}$ be a presheaf.

  * •

A _compatible family_ $\left\\{f_{i}:i\in I\right\\}$ for $\mathcal{F}$ is a
family of elements $f_{i}\in\mathcal{F}(p_{i})$ such that for all $i,j$ and
$r$ refining $p_{i},p_{j}$ $f_{i}\restriction r=f_{j}\restriction r$ (letting
$\mathcal{F}(s\leq t)(f)=f\restriction s$ for $f\in\mathcal{F}(t)$).

  * •

A _collation_ of a compatible family $\left\\{f_{i}:i\in I\right\\}$ on some
$p$ is some $f\in\mathcal{F}(p)$ such that $f\restriction r=f_{i}\restriction
r$ for all $i\in I$ and $r$ refining $p,p_{i}$.

######  Definition 3.3.

Let $(P,\leq)$ be a partial order, and
$\mathcal{F}:P^{\mathrm{op}}\to\mathsf{Set}$ be a presheaf.

  * •

$\mathcal{F}$ is a777We adopt the terminology of [6, III Section 2 item (e)
pp. 115]. _sheaf for the dense Grothendieck topology on $P$_ when for every
$p\in P$ and every compatible family $\left\\{f_{i}:i\in I\right\\}$ for
$\mathcal{F}$ with $f_{i}\in\mathcal{F}(p_{i})$, if $A=\left\\{p_{i}:i\in
I\right\\}$ is a dense covering of $p$, then there is a unique collation
$f\in\mathcal{F}(p)$ on $p$ of the family.

  * •

Assume further that $P$ is upward complete. $\mathcal{F}$ is a888We adopt the
terminology of [6, III Section 2 item (d) pp. 114-115]. _sheaf for the sup
Grothendieck topology on $P$_ when for every $p\in P$ and every compatible
family $\left\\{f_{i}:i\in I\right\\}$ for $\mathcal{F}$ with
$f_{i}\in\mathcal{F}(p_{i})$, if $A=\left\\{p_{i}:i\in I\right\\}$ is a
complete covering of $p$, there is a unique collation $f\in\mathcal{F}(p)$ of
the family.

Let $(X,\tau)$ be a topological space and
$\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ be a presheaf.

  * •

$\mathcal{F}$ is a _topological sheaf_ on $(X,\tau)$ if $\mathcal{F}$ is a
sheaf for the sup Grothendieck topology on $\mathcal{O}(X)$.

  * •

$\mathcal{F}$ is a _stonean sheaf_ on $(X,\tau)$ if $\mathcal{F}$ is a sheaf
for the dense Grothendieck topology on $\mathcal{O}(X)$.

The standard notion of sheaf on a topological space as defined in [6, II
Sections 5, 6] corresponds to our notion of topological sheaf. However a
topological sheaf need not be a stonean sheaf (see Example 3.11 below). On the
other hand the converse is the case:

######  Fact 3.4.

Let $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ be a stonean
sheaf. Then $\mathcal{F}$ is also a topological sheaf.

###### Proof.

Assume $\left\\{f_{i}:i\in I\right\\}$ is compatible for $\mathcal{F}$ with
$f_{i}\in\mathcal{F}(U_{i})$. Let $A=\bigcup\left\\{U_{i}:i\in I\right\\}$.
Then $\left\\{U_{i}:i\in I\right\\}$ is a dense covering of $A$. Since
$\mathcal{F}$ is a stonean sheaf we have that there is a unique
$f\in\mathcal{F}(A)$ which collates the family $\left\\{U_{i}:i\in
I\right\\}$. ∎

The following is the key property relating the two notion of sheaf:

######  Lemma 3.5.

Let $(X,\tau)$ be a topological space whose regular open sets form a base for
the topology. Let $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ be
a presheaf. $\mathcal{F}$ is a stonean sheaf on $(X,\tau)$ if and only if
$\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X)^{+}$ is a sheaf for the
sup Grothendieck topology on $\operatorname{\mathsf{RO}}(X)^{+}$.

###### Proof.

The key point is that (since the regular open sets form a base for the
topology), $A$ is predense below $U$ in $\mathcal{O}(X)$ if and only if

$D_{A}=\left\\{V\in\operatorname{\mathsf{RO}}(X):V\subseteq W\cap U\text{ for
some }W\in A\right\\}$

is such that $\bigcup D_{A}$ is a dense open subset of
$\mathrm{Reg}\left(U\right)$. Hence in $\operatorname{\mathsf{RO}}(X)^{+}$ we
have that

$\bigvee_{\operatorname{\mathsf{RO}}(X)}D_{A}=\mathrm{Reg}\left(U\right).$

Now let $\left\\{f_{i}:i\in I\right\\}$ be compatible for $\mathcal{F}$ with
$U=\bigcup\left\\{U_{i}:i\in I\right\\}$ and each
$f_{i}\in\mathcal{F}(U_{i})$. Without loss of generality (since the regular
open sets form a base for the topology) it suffice to consider just the case
in which each $U_{i}$ is regular open. Then
$f\in\mathcal{F}(\mathrm{Reg}\left(U\right))$ is a collation of
$\left\\{f_{i}:i\in I\right\\}$ with respect to the dense topology on
$\mathcal{O}(X)$ if and only if it is such with respect to the complete
topology on $\operatorname{\mathsf{RO}}(X)^{+}$.

∎

We outline a few key properties of _stonean sheaves_ :

######  Proposition 3.6.

Assume $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ is a stonean
sheaf with $\mathcal{F}(X)\neq\emptyset$. Then
$\mathcal{F}(U)=\left\\{f\restriction U:f\in\mathcal{F}(X)\right\\}$ for all
$U\in\mathcal{O}(X)$.

###### Proof.

For any $U\in\mathcal{O}(X)$,
$\left\\{U,X\setminus\mathrm{Cl}\left(U\right)\right\\}$ is a dense covering
for $X$. Let $g\in\mathcal{F}(X)$ and $f\in\mathcal{F}(U)$. Then
$\left\\{g\restriction X\setminus\mathrm{Cl}\left(U\right),f\right\\}$ is a
compatible family indexed by a dense covering of $X$. If $h\in\mathcal{F}(X)$
is a collation of this family, $h\restriction U=f$. ∎

We can also see that a stonean sheaf necessarily is a topological sheaf on an
extremally disconnected compact Hausdorff space, at least under very mild
assumptions on its local sections:

######  Fact 3.7.

Assume $\mathcal{F}$ is a stonean sheaf on some Hausdoff compact space
$(X,\tau)$ such that for some Hausdorff space $(Y,\sigma)$ with more than one
point and for every regular open set $U$ of $X$, $\mathcal{F}(U)$ consists of
continuous maps from $U$ to $Y$ and includes at least two distinct constant
maps. Then $(X,\tau)$ is also extremally disconnected.

###### Proof.

Assume $(X,\tau)$ is compact Hausdorff but not extremally disconnected. Then
there is $U\in\tau$ which is regular open but not closed. This gives that
(letting $\neg U=X\setminus\overline{U}$) $\left\\{U,\neg U\right\\}$ is a
dense cover in $X$. Fix $x_{0}\neq x_{1}\in Y$ such that $f_{0}:x\mapsto
x_{0}$ is in $\mathcal{F}(U)$ and $f_{1}:x\mapsto x_{1}$ is in
$\mathcal{F}(\neg U)$. Then $\left\\{f_{0},f_{1}\right\\}$ is a compatible
family indexed by a dense cover which cannot have a collation on $X$: note
that $\overline{U}\cap\overline{\neg U}$ is non-empty (else $U$ is closed); if
$y$ belongs to this set and $g$ extends continuously $f_{0}\cup f_{1}$ to $X$,
$g(y)$ is either $x_{0}$ or $x_{1}$, but it can be neither of them since $y$
is an accumulation point both for $U=f_{0}^{-1}[\left\\{x_{0}\right\\}]$ and
for $\neg U=f_{1}^{-1}[\left\\{x_{1}\right\\}]$. ∎

###  3.2. Stonean bundles and stonean sheaves

We stick to the notation of [6, II Sections 5, 6]. The unique notational
difference is that we will write
$(\Lambda^{0}_{\mathcal{F}},p^{0}_{\mathcal{F}})$ (instead of
$(\Lambda_{\mathcal{F}},p_{\mathcal{F}})$) for the étalé space associated to
the presheaf $\mathcal{F}$ and $\Gamma^{0}_{p}$ (instead of $\Gamma_{p}$) for
the sheaf associated to the bundle $p$. This choice is motivated by our choice
to use a similar notation ($(\Lambda^{1}_{\mathcal{F}},p^{1}_{\mathcal{F}})$
and $\Gamma^{1}_{p},\Gamma^{1/2}_{p}$, respectively) for our constructions.

######  Definition 3.8.

Let $X$ be a topological space. A _bundle_ over $X$ is a continuous (possibly
non surjective) map $p:E\to X$.
A _(global) section_ of the bundle999Global sections can exist only if the
bundle is surjective. $p:E\to X$ is a continuous map $s:X\to E$ such that
$p\circ s$ is the identity on $X$. If $U$ is an open set of $X$, a _(local)
section_ of $p$ over $U$ is a section of the bundle $p\restriction
p^{-1}[U]:p^{-1}[U]\to U$. If $p:E\to X$ is a bundle and $x\in X$, the _fiber_
at $x$ is $p^{-1}[\left\\{x\right\\}]$.
A bundle $p:E\to X$ is an _étalé space_ if it is a local homeomorphism in the
following sense: for every $e\in E$ there exists an open neighborhood $V$ of
$e$ such that $p[V]$ is open in $X$ and $p\restriction V:V\to p[V]$ is an
homeomorphism.

######  Definition 3.9.

[6, Section II.5] Let $(X,\tau)$ be a topological space and
$\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$ be a presheaf.

$\Lambda^{0}_{\mathcal{F}}:=\coprod_{x\in
X}\mathcal{F}_{x}=\left\\{[f]_{\sim_{x}}:x\in X,f\in\mathcal{F}(U),x\in
U\in\mathcal{O}(X)\right\\}$

(where $[f]_{\sim x}$ is the equivalence class induced by $f\equiv_{\sim x}g$
if and only if $f\restriction U=g\restriction U$ for some $x\in
U\in\mathcal{O}(X)$),

$\begin{aligned} p^{0}_{\mathcal{F}}:&\Lambda_{\mathcal{F}}\to X\\\
&[f]_{\sim_{x}}\mapsto x\end{aligned}.$

Each $f\in\mathcal{F}(U)$ determines

$\begin{aligned} \dot{f}:&U\to\Lambda^{0}_{\mathcal{F}}\\\
&x\mapsto[f]_{\sim_{x}}\end{aligned},$

which is a section of $p^{0}_{\mathcal{F}}$ over $U$ ($\dot{f}$ is injective:
if $x\neq y$, $[f]_{\sim_{x}}\neq[f]_{\sim_{y}}$, since
$\mathcal{F}_{x},\mathcal{F}_{y}$ are disjoint).
$\Lambda^{0}_{\mathcal{F}}$ is topologized by taking as a (semi)base101010The
topology generated by this semibase shows that this semibase is a base. of
open sets the family of all the image sets $\dot{f}[U]$ for
$U\in\mathcal{O}(X)$ and $f\in\mathcal{F}(U)$.

This topology on $\Lambda^{0}_{\mathcal{F}}$ in general is not Hausdorff, but
renders the map $p^{0}_{\mathcal{F}}$ continuous and the sections $\dot{f}$
continuous and open. In particular, it is easy to see that
$p^{0}_{\mathcal{F}}:\Lambda^{0}_{\mathcal{F}}\to X$ is an étalé space over
$X$.
Conversely, any bundle $p:E\to X$ over a topological space $X$ gives rise to a
sheaf, the _sheaf of sections_ of $E$. It is defined as follows in [6]:

######  Definition 3.10.

[6, Section II.6] Let $p:E\to X$ be a continuous map between the topological
spaces $(X,\tau)$, $(E,\sigma)$. For $U$ a non-empty open subset of $X$, let

$\Gamma^{0}_{p}(U):=\left\\{s:U\to E:s\emph{ is a section of $p$ over
$U$}\right\\}.$

Let us spend a few words about this correspondence between a presheaf
$\mathcal{F}$ on $(X,\tau)$ and the sheaf $\Gamma^{0}_{p^{0}_{\mathcal{F}}}$.
Let $\mathcal{A}$ be a base for $(X,\tau)$. Assume
$h:U\to\Lambda^{0}_{\mathcal{F}}$ is a section in
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}(U)$. Then, for every $x\in U$, there is an
open neighborhood $x\in U_{x}\subseteq U$ with $U_{x}\in\mathcal{A}$ such that
$h(x)$ is the germ at $x$ of some element $f_{x}\in\mathcal{F}(U_{x})$, i.e.
$h(x)=[f_{x}]_{\sim_{x}}=\dot{f}_{x}(x)$. Now, $h$ is continuous and
$\dot{f_{x}}[U_{x}]$ is a basic open subset of $\Lambda^{0}_{\mathcal{F}}$. By
continuity, there exists an open subset $V_{x}\in\mathcal{A}$ with $x\in
V_{x}\subseteq U_{x}$ such that $h[V_{x}]\subseteq\dot{f_{x}}[U_{x}]$. This
means that $h$ coincides with $\dot{f_{x}}$ on $V_{x}$. Therefore every
section of the bundle $\Gamma^{0}_{p^{0}_{\mathcal{F}}}$ associated to
$\mathcal{F}$ is locally a section induced by an element of $\mathcal{F}$.

For any presheaf $\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$,
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$
is always a topological sheaf but in general not a stonean sheaf. This is
exactly what occurs in the following Example which separates the notion of
stonean sheaf from that of topological sheaf.

######  Example 3.11.

Consider the presheaf $\mathcal{G}$ of continuous real valued functions
defined on some _regular open_ subset of $[0;1]$ with range a bounded subset
of $\mathbb{R}$. This is not a sheaf as the collation of a compatible family
of bounded continuous functions could results in an unbounded continuous
function. Nonetheless the étalé space $\Lambda^{0}_{\mathcal{F}}$ associated
to this sheaf can be identified with the germs of continuous functions
$f:U\to\mathbb{R}$ at points $x\in[0;1]$ with $U$ an open neighborhood of $x$,
and the (topological - according to our notation) sheaf
$\Gamma^{0}\Lambda^{0}_{\mathcal{F}}$ of local sections on
$\Lambda^{0}_{\mathcal{F}}$ corresponds to the topological sheaf of real
valued continuous functions defined on some open subset of $[0;1]$.

Consider now the family of partial constant functions $f_{n}$ for
$n\in\mathbb{N}$ with $f_{n}$ constant with value $n$ on the interval
$(\frac{1}{n},\frac{1}{n+1})$. This gives a specific family of elements of
$\mathcal{F}$ which can be collated in $\Gamma^{0}\Lambda^{0}_{\mathcal{F}}$.
Their collation $f=\bigcup_{n\in\mathbb{N}}f_{n}$ however is defined only on
the dense open subset $(0;1]\setminus\\{\frac{1}{n}:n\in\mathbb{N}\\}$ and
does not have a germ at $0$, because it does not admit a continuous extension
to any open neighborhood of $0$. In particular it does not identify a global
section of $\Lambda^{0}_{\mathcal{F}}$.

We conclude that $\Gamma^{0}\Lambda^{0}_{\mathcal{F}}$ is a topological sheaf
which is not stonean exactly because the compatible family
$\left\\{f_{n}:n\in\mathbb{N}\right\\}$ is indexed by a covering of
$\mathbb{R}$ for the dense topology which does not have a collation defined on
$\mathbb{R}$.

$f$ is a function regular enough to be of interest in our analysis, hence we
want to be able to include it as part of the global sections of the étalé
space we wish to associate to $\mathcal{F}$. To reach this objective we
associate to $\mathcal{F}$ the étalé bundle $\Lambda^{1}_{\mathcal{F}}$ (see
Definition 3.12) given by the germs of functions in $\mathcal{F}$ induced by
maximal filters of open intervals, or equivalently by ultrafilters on the
complete boolean algebra $\operatorname{\mathsf{RO}}([0;1])$ given by the
regular open subsets of $[0;1]$. In this case it is possible to show that the
global sections of $\Lambda^{1}_{\mathcal{F}}$ induce continuous functions
from $\operatorname{St}(\operatorname{\mathsf{RO}}([0;1]))$ to the one point
compactification of $\mathbb{R}$. In this set-up, $f$ defines a unique global
section of $\Lambda^{1}_{\mathcal{F}}$ which induces the following continuous
function
$\beta(f):\operatorname{St}(\operatorname{\mathsf{RO}}([0;1]))\to\mathbb{R}\cup\left\\{\infty\right\\}$:

$G\mapsto\begin{cases*}n\text{ if }(\frac{1}{n},\frac{1}{n+1})\in G\\\
\infty\text{ if for no $n$ }(\frac{1}{n},\frac{1}{n+1})\in G.\end{cases*}$

We will then use this étalé space/bundle to define a stonean sheaf
$\Gamma^{1}\Lambda^{1}_{\mathcal{F}}$ which will comprise $\beta(f)$ among its
global sections.

We will detail more on this as we unfold our concepts (the reader is referred
also to [13, Section 5.2]).

Our focus from now on will be on the definition of the construction which will
bring from $\mathcal{F}$ to $\Gamma^{1}_{\Lambda^{1}_{\mathcal{F}}}$, rather
than the usual one bringing from $\mathcal{F}$ to
$\Gamma^{0}_{\Lambda^{0}_{\mathcal{F}}}$. Loosely speaking the variant of the
sheafification process we are interested in will be able to comprise as global
sections of the étalé space of $\mathcal{F}$ also the pointwise limits of
continuous functions; for an arbitrary topological sheaf $\mathcal{F}$ the
sheaf of sections of the étalé space $\Lambda^{1}_{\mathcal{F}}$ may not be at
all isomorphic to $\mathcal{F}$, as the latter is always a stonean sheaf even
if the former is not.

We want to devise a sheafification process which given a presheaf
$\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$, yields first a
bundle
$p^{1}_{\mathcal{F}}:\Lambda^{1}_{\mathcal{F}}\to\operatorname{St}(\operatorname{\mathsf{RO}}(X))$
and then a stonean sheaf $\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ given by local
sections of $p^{1}_{\mathcal{F}}$ which can be seen as the natural “stonean”
sheafification of $\mathcal{F}$. The success of our approach is outlined by
Theorem 3.29 which states that $\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ is the left
adjoint of the inclusion functor of the full subcategory of stonean sheaves
into the category of topological presheaves on some topological
space111111Since the base space of the presheaves vary, the notion of morphism
in this category is delicate, see Section 3.3, Def. 3.25..

It is convenient for the rest of this section to look at a boolean algebra at
times as an algebraic structure and at other times as the clopens of its Stone
space. We will freely do so in the sequel. Hence we write $b\in G$ when we
think of $\mathsf{B}$ as a boolean algebra, $b$ as an element of $\mathsf{B}$,
$G$ as an ultrafilter on $\mathsf{B}$, or $G\in b$ when we think of $G$ as an
element of $\operatorname{St}(\mathsf{B})$ and $b$ as a clopen subset of
$\operatorname{St}(\mathsf{B})$. We also consider $\mathsf{B}$ a dense
subalgebra of $\operatorname{\mathsf{RO}}(\mathsf{B}^{+})$ and we note that
for any ultrafilter $G$ on $\operatorname{\mathsf{RO}}(\mathsf{B}^{+})$
$G\cap\mathsf{B}$ is a ultrafilter on $\mathsf{B}$.

We start by introducing a new notion of stonean étalé space associated to a
given topological presheaf.

######  Definition 3.12.

Let $(X,\tau)$ be a topological space and
$\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$ be a presheaf. Given
$G$ ultrafilter in $\operatorname{St}(\operatorname{\mathsf{RO}}(X,\tau))$,
let $\bar{G}$ be the unique maximal filter of $\mathcal{O}(X)$ induced by $G$
(i.e. $U\in\bar{G}$ if and only if $\mathrm{Reg}\left(U\right)\in G$).

We let for $G\in\operatorname{St}(\operatorname{\mathsf{RO}}(X,\tau))$:

  * •

for $f\in\mathcal{F}(U_{f}),g\in\mathcal{F}(U_{g})$ and
$U_{f},U_{g}\in\bar{G}$, $f\equiv_{G}g$ if and only if for some $U\in\bar{G}$
refining $U_{f},U_{g}$

$D_{f,g}=\left\\{V\in\mathcal{O}(X):\,f\restriction V=g\restriction
V\right\\}$

is dense below $U$ (where $\mathcal{F}(U\leq V)(f)=f\restriction U$);

  * •

$[f]_{G}$ be the equivalence class of $f$ induced by $\equiv_{G}$;

  * •

$\mathcal{F}_{G}:=\left\\{[f]_{G}:f\in\mathcal{F}(U_{f}),U_{f}\in\bar{G}\right\\}$.

We also let:

  * •

$\Lambda^{1}_{\mathcal{F}}:=\coprod_{G\in\operatorname{St}(\operatorname{\mathsf{RO}}(X))}\mathcal{F}_{G}$;

  * •

$p^{1}_{\mathcal{F}}$ be defined by

$\begin{aligned}
p^{1}_{\mathcal{F}}:&\Lambda^{1}_{\mathcal{F}}\to\operatorname{St}(\operatorname{\mathsf{RO}}(X))\\\
&[f]_{G}\mapsto G\end{aligned}.$

Each $f\in\mathcal{F}(U)$ determines

$\begin{aligned}
\dot{f}:&\mathrm{Reg}\left(U\right)\to\Lambda^{1}_{\mathcal{F}}\\\
&G\mapsto[f]_{G}\end{aligned}.$

We topologize $\Lambda^{1}_{\mathcal{F}}$ with topology
$\tau^{1}_{\mathcal{F}}$ by taking as a semibase of open sets the family

$B^{1}_{\mathcal{F}}=\\{\dot{f}[q]:q\in\operatorname{\mathsf{RO}}(X),\,q\subseteq\mathrm{Reg}\left(U\right),\,f\in\mathcal{F}(U),\,U\in\mathcal{O}(X)\\}.$

We start to unfold the properties of $\Lambda^{1}_{\mathcal{F}}$:

######  Proposition 3.13.

Let $(X,\tau)$ be a topological space and
$\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$ be a presheaf. Then

  * •

$B^{1}_{\mathcal{F}}$ is a base of compact open sets for
$(\Lambda^{1}_{\mathcal{F}},\tau^{1}_{\mathcal{F}})$; hence
$(\Lambda^{1}_{\mathcal{F}},\tau^{1}_{\mathcal{F}})$ is locally compact.

  * •

For any $G\in\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ the stalk
$\mathcal{F}_{G}$ over $G$ in $\Lambda^{1}_{\mathcal{F}}$ is a discrete and
closed subspace.

  * •

$p^{1}_{\mathcal{F}}:\,\Lambda^{1}_{\mathcal{F}}\to\operatorname{St}(\operatorname{\mathsf{RO}}(X))$
is the bundle associated to
$(\Lambda^{1}_{\mathcal{F}},\tau^{1}_{\mathcal{F}})$, and each $\dot{f}$ is a
local section over $\mathrm{Reg}\left(U\right)$ whenever $f\in\mathcal{F}(U)$;
hence $\Lambda^{1}_{\mathcal{F}}$ is an étalé space over
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$.

  * •

$\Lambda^{1}_{\mathcal{F}}$ is Hausdorff and $0$-dimensional, in particular it
is Tychonoff.

  * •

If $\Lambda^{1}_{\mathcal{F}}$ has an infinite stalk, it is non-compact.

###### Proof.

__

$B^{1}_{\mathcal{F}}$ is a base for $\tau^{1}_{\mathcal{F}}$:

It suffices to show that finite intersections of elements of
$B^{1}_{\mathcal{F}}$ are unions of elements of $B^{1}_{\mathcal{F}}$. We
sketch the argument for binary intersections and leave to the reader the
general case. Note that $[f]_{G}=[h]_{K}$ if and only if $G=K$ and $D_{f,g}$
is dense below $U$ for some $U\in G$. This gives that for
$q\subseteq\mathrm{Reg}\left(U_{f}\right)$,
$r\subseteq\mathrm{Reg}\left(U_{g}\right)$

$\displaystyle\dot{f}[q]\cap\dot{g}[r]=$
$\displaystyle\left\\{[f]_{G}:\,[f]_{G}=[g]_{G},\,q\wedge r\in G\right\\}=$
$\displaystyle=$ $\displaystyle\left\\{[f]_{G}:\,\exists\,W\subseteq q\cap
r,\,D_{f,g}\text{ is dense below }W\text{ and }W\in\bar{G}\right\\}$
$\displaystyle=$
$\displaystyle\bigcup\left\\{\dot{f}[\mathrm{Reg}\left(W\right)]:\,W\subseteq
q\cap r,\,D_{f,g}\text{ is dense below }W,\,W\in\mathcal{O}_{X}\right\\}.$

$\dot{f}$ is a local section over $\mathrm{Reg}\left(U\right)$ whenever
$f\in\mathcal{F}(U)$:

It is clear that $\dot{f}$ is injective and
$(p^{1}_{\mathcal{F}}\circ\dot{f})(G)=G$ for any
$G\in\mathrm{Reg}\left(U\right)$; furthermore for any
$V\subseteq\mathrm{Reg}\left(U\right)$ open set of
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$

$\dot{f}[V]=\bigcup\left\\{\dot{f}[q]:\,q\subseteq
V,q\in\operatorname{\mathsf{RO}}(X)=\operatorname{\mathsf{CLOP}}(\operatorname{St}(\operatorname{\mathsf{RO}}(X)))\right\\}.$

This easily gives that $\dot{f}$ is open and continuous on its domain.

$(\Lambda^{1}_{\mathcal{F}},\tau^{1}_{\mathcal{F}})$ is locally compact:

This is an immediate consequence of the already established fact that each
local section $\dot{f}$ is an homeomorphism of the compact subspace
$\mathrm{Reg}\left(U_{f}\right)$ of $\operatorname{St}(\mathsf{B})$ with the
open subspace $\dot{f}[\mathrm{Reg}\left(U_{f}\right)]$ of
$\Lambda^{1}_{\mathcal{F}}$.

$p^{1}_{\mathcal{F}}$ is continuous:

For any $U$ open in $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$,

$(p^{1}_{\mathcal{F}})^{-1}[U]=\bigcup\left\\{\dot{f}[\mathrm{Reg}\left(V\right)]:\mathrm{Reg}\left(V\right)\subseteq
U,\,f\in\mathcal{F}(V)\right\\},$

hence it is a union of basic open sets.

The stalks of $\Lambda^{1}_{\mathcal{F}}$ are discrete and closed:

The complement of the stalk on some
$G\in\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ is given by the union
of the images of sections in $\Gamma^{0}_{p^{1}_{\mathcal{F}}}$ on
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))\setminus\left\\{G\right\\}$,
hence the stalk on $G$ is closed; any stalk is also discrete (the intersection
of a local section with a stalk is a point).

$\Lambda^{1}_{\mathcal{F}}$ is Hausdorff and $0$-dimensional:

The delicate part is to establish the Hausdorff separation property. In this
task the only interesting case is how to separate points $[f]_{G}\neq[g]_{G}$
on the same stalk over $G\in\operatorname{St}(\mathsf{B})$ with
$f\in\mathcal{F}(U_{f})$, $g\in\mathcal{F}(U_{g})$,
$\mathrm{Reg}\left(U_{f}\right),\mathrm{Reg}\left(U_{g}\right)\in G$.
Let (in $\operatorname{\mathsf{RO}}(X)$)
$b=\bigvee_{\operatorname{\mathsf{RO}}(X)}D_{f,g}$. Then either $b\in G$ or
$\neg b\in G$. Clearly $b\notin G$ else
$p=b\wedge\mathrm{Reg}\left(U_{f}\right)\wedge\mathrm{Reg}\left(U_{g}\right)$
is in $G$ and $D_{f,g}$ is dense below it, yielding $[f]_{G}=[g]_{G}$. Hence
$\neg b\in G$. Now if $V\subseteq p=\neg
b\wedge\mathrm{Reg}\left(U_{f}\right)\wedge\mathrm{Reg}\left(U_{g}\right)$ is
open in $X$, we have that $V\not\in D_{f,g}$. Hence $f\restriction V\neq
g\restriction V$ for all $V\subseteq p$ in $\mathcal{O}(X)$, therefore
$[f]_{H}\neq[g]_{H}$ for all $H\in p$. This gives that $\dot{f}[p]$ and
$\dot{g}[p]$ are disjoint, and each is a compact open neighborhood
respectively of $[f]_{G}$, $[g]_{G}$.

Since each $\dot{f}[\mathrm{Reg}\left(U_{f}\right)]$ is compact, it is also
closed in $\Lambda^{1}_{\mathcal{F}}$ (because $\Lambda^{1}_{\mathcal{F}}$ is
Hausdorff), hence the family of local sections
$\dot{f}[\mathrm{Reg}\left(U\right)]$ for $f\in\mathcal{F}(U)$ provides a base
of clopen sets for $\Lambda^{1}_{\mathcal{F}}$.

$\Lambda^{1}_{\mathcal{F}}$ is non-compact if some stalk of
$\Lambda^{1}_{\mathcal{F}}$ is infinite:

An infinite stalk of $\Lambda^{1}_{\mathcal{F}}$ is a closed and discrete
(hence non-compact) subset; hence $\Lambda^{1}_{\mathcal{F}}$ is not compact,
being Hausdorff and containing a closed non-compact subset.

∎

Given a locally compact Hausdorff space $(X,\tau)$ let $\beta_{0}(X)$ denote
its one point compactification $X\cup\left\\{\infty\right\\}$ with
neighborhood of $\infty$ given by the complements of compact subsets of $X$.

We can now introduce the type of sheaf we want to associate to bundles.
Compactifications will play a crucial role in our ability to describe global
sections; thus we limit the scope of our definition to bundles $p:E\to X$ with
$E$ locally compact Hausdorff121212Actually we could be slightly more liberal
in Definition 3.14 and allow $E$ to be Tychonoff space (in which case in the
rest of Definition 3.14 we should replace $\beta_{0}(E)$ with $\beta(E)$). In
order for Definition 3.14 to work properly, we only need to be able to define
an Hausdorff compactification of $E$, which is the case if the latter is
Tychonoff. However to define the stonean sheafification of a presheaf
$\mathcal{F}$, it suffices to consider $E$ locally compact Hausdorff, since so
is $\Lambda^{1}_{\mathcal{F}}$..

######  Definition 3.14.

A continuous map $p:E\to X$ is an _extremally disconnected bundle_ if:

  * •

$E$ is locally compact and $X$ is Hausdorff, extremally disconnected, and
compact;

  * •

$p[E]$ is a dense open subset of $X$.

Let $p:E\to X$ be an extremally disconnected bundle;
$\Gamma^{1}_{p}:\mathcal{O}(X)^{\text{op}}\to\mathsf{Set}$ is defined by:

  * •

$\Gamma^{1}_{p}(U)$ is the set of continuous $f:U\to\beta_{0}(E)$ such that:

    * –

$\left\\{x\in U:\,f(x)\in E\right\\}$ contains an open dense subset $D_{f}$ of
$U$;

    * –

$f\restriction D_{f}$ is a local section of $p$ (e.g. $f\restriction
D_{f}\in\Gamma^{0}_{p}(D_{f})$);

  * •

$\Gamma^{1}_{p}(U\subseteq V)(f):=f\restriction U$ for every
$f\in\Gamma^{1}_{p}(V)$.

$\Gamma^{1/2}_{p}$ is given by
$\Gamma^{1}_{p}\restriction\operatorname{\mathsf{RO}}(X,\tau)^{+}$.

######  Proposition 3.15.

Let $p:E\to X$ be an extremally disconnected bundle. Let also
$U\in\mathcal{O}(X)$ and $f\in\prod_{\left\\{G\in
U\right\\}}(p^{-1}[\left\\{G\right\\}]\cup\left\\{\infty\right\\})$. The
following are equivalent:

  * •

$f$ is a local section of $p$ over $U$ for $\Gamma^{1}_{p}$;

  * •

$f:U\to\beta_{0}(E)$ is open and continuous and
$f^{-1}[\left\\{\infty\right\\}]$ is nowhere dense in $U$.

###### Proof.

One implication is trivial. Now assume $f:U\to\beta_{0}(E)$ is open and
continuous and $f^{-1}[\left\\{\infty\right\\}]$ is nowhere dense in $U$. Then

$D_{f}=\left\\{G\in U:f(G)\in E\right\\}$

is open and dense in $U$. Since $f$ is open and continuous for each $G\in
D_{f}$ there is some $g_{G}:U_{G}\to E$ local homeomorphism with $G\in
V_{G}\subseteq U_{G}$, $f(G)=g_{G}(G)$, $V_{G}$ clopen subset of $D_{f}$,
$U_{G}$ clopen subset of $X$, $g_{G}[U_{G}]$ clopen subset of $E$,
$f[V_{G}]\subseteq g_{G}[U_{G}]$. Hence $f\restriction D_{f}=\bigcup_{G\in
D_{f}}g_{G}\restriction V_{G}$ is a local section. Since $D_{f}$ is a dense
subset of $U$, we conclude that $f\in\Gamma^{1}_{p}(U)$. ∎

######  Proposition 3.16.

For any extremally disconnected bundle $p:E\to X$, $\Gamma^{1}_{p}$ is a
stonean sheaf and $\Gamma^{1/2}_{p}$ is a sheaf both for the sup-topology on
$\operatorname{\mathsf{RO}}(X)$ and for the dense topology on
$\operatorname{\mathsf{RO}}(X)^{+}$.

###### Proof.

We prove the thesis in details for $\Gamma^{1/2}_{p}$ with respect to the sup-
topology on $\operatorname{\mathsf{RO}}(X)$, we leave to the reader the
details for the other assertions.

Since $X$ is Hausdorff extremally disconnected compact,
$\mathrm{Reg}\left(U\right)=\mathrm{Cl}\left(U\right)$ for every open
$U\subseteq X$; therefore (by Proposition 1.22) for any open $U\subseteq X$
its Stone-Čech compactification $\beta(U)$ is (homeomorphic to)
$\mathrm{Reg}\left(U\right)$ and any continuous function $f$ defined on some
open subset $U\subseteq X$ and range in a compact Hausdorff space admits a
unique continuous extension $\beta(f)$ to
$\mathrm{Cl}\left(U\right)=\mathrm{Reg}\left(U\right)=\beta(U)$.

Assume now $\left\\{f_{i}:U_{i}\to\beta_{0}(E):i\in I\right\\}$ is a matching
family for $\Gamma^{1/2}_{p}$. Then, letting $f=\bigcup_{i\in I}f_{i}$, $f$
has domain $U=\bigcup_{i\in I}U_{i}$ and $\beta(f)$ is continuous with domain
the clopen set

$\bigvee_{\operatorname{\mathsf{RO}}(X)}\left\\{U_{i}:i\in
I\right\\}=\mathrm{Cl}\left(U\right)$

and collates the matching family in $\Gamma^{1/2}_{p}$; furthemore
$D_{f}=\bigcup_{i\in I}D_{f_{i}}\subseteq U$ is a dense open subset of
$\mathrm{Cl}\left(U\right)$ on which $f$ takes values in $E$, and
$f\restriction D_{f}$ is a local section of $\Lambda^{0}_{p}$. Hence
$\beta(f)$ is the unique continuous extension of the local section
$f\restriction D_{f}\in\Gamma^{0}_{p}(D_{f})$ to a function with range in
$\beta_{0}(E)$, and $\beta(f)$ can take as value the unique point
$\infty\in\beta_{0}(E)\setminus E$ only on the nowhere dense closed set
$\mathrm{Cl}\left(U\right)\setminus D_{f}$. ∎

######  Proposition 3.17.

Let $X$ be an extremally disconnected compact Hausdorff space. $\mathcal{F}$
is a stonean sheaf on $X$ if and only if (up to a natural isomorphism)
$\mathcal{F}\cong\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ and each local section of
$\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ takes values only in
$\Lambda^{1}_{\mathcal{F}}$.

###### Proof.

Since $X$ is extremally disconnected compact Hausdorff space we have that up
to homeomorphism $\mathcal{F}$ and $\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ are
stonean sheaves on $X$.

stonean sheaves are uniquely determined by the global sections, hence we just
need to find a natural identification of $\mathcal{F}(X)$ with
$\Gamma^{1}_{p^{1}_{\mathcal{F}}}(\operatorname{St}(\operatorname{\mathsf{RO}}(X))$.

Now an element of
$\Gamma^{1}_{p^{1}_{\mathcal{F}}}(\operatorname{St}(\operatorname{\mathsf{RO}}(X))$
is some $f:\operatorname{St}(\operatorname{\mathsf{RO}}(X))\to\beta_{0}(E)$
such that on some $D_{f}$ dense subset of
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$, $f\restriction D_{f}$
takes vaules in $E$, $f=\beta(f\restriction D_{f})$ and furthermore
$D_{f}=\bigcup_{i\in I}V_{i}$ so that:

  * •

$f\restriction V_{i}=\dot{f}_{i}$ for some $f_{i}\in\mathcal{F}(V_{i})$,

  * •

$V_{i}$ is regular open in $X$ for all $i\in I$.

This gives that the family $\left\\{f_{i}:i\in I\right\\}$ is a compatible
family in $\mathcal{F}$ and $\left\\{V_{i}:i\in I\right\\}$ is a dense
covering of $X$. Since $\mathcal{F}$ is a stonean sheaf, there is some
$h\in\mathcal{F}(X)$ such that $h\restriction V_{i}=f_{i}$ for all $i\in I$.
Then $\dot{h}:\operatorname{St}(\operatorname{\mathsf{RO}}(X))\to\beta_{0}(E)$
takes values only in $E$ and

$\dot{h}(G)=[h]_{G}=[f_{i}]_{G}=\dot{f}_{i}(G)=f(G)$

for all $G\in V_{i}$ and $i\in I$. Hence $f$ is $h$ since they are both
continuous and overlap on the dense set $D_{f}$ (modulo the identification of
$X$ with $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$). ∎

######  Corollary 3.18.

Let $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ be a stonean
sheaf with $X$ extremally disconnected compact Hausdorff. Then (up to natural
isomorphism) there is an extremally disconnected and surjective bundle
$p_{\mathcal{F}}:E_{\mathcal{F}}\to X$ such that for each $U$ open in $X$
$\mathcal{F}(U)$ is the family of sections $f:U\to E_{\mathcal{F}}$ and the
pointwise image of these sections form a base of clopen sets for
$E_{\mathcal{F}}$.

######  Corollary 3.19.

Let $X$ be extremally disconnected compact Hausdorff. Then the pair
$\Gamma^{1}\circ\Lambda^{1}$ is left adjoint of the inclusion functor between
the category $\textrm{Presh}(X,\mathsf{Set})$ and the category
$\textrm{Sh}^{*}(X,\mathsf{Set})$ of stonean sheaves on $X$.

The above can be proved using the general methods on sheafification of
presheaves with respect to Grothendieck topologies as given in [6, III].

###  3.3. Stonean sheafification as an adjoint functor

We aim for a more general adjunction result than the one given in Corollary
3.19 as the latter does not describe the relation between
$\textrm{Presh}^{*}(X,\mathsf{Set})$ and
$\textrm{Sh}^{*}(\operatorname{St}(\operatorname{\mathsf{RO}}(X)),\mathsf{Set})$
given by the functor $\Gamma^{1}\circ\Lambda^{1}$ when $X$ is not extremally
disconnected, compact Hausdorff.

Toward this aim we need to describe what are the natural morphisms between
presheaves defined on different topological spaces.

We have to give an ad-hoc definition (or at least one not present in [6, I]),
since such morphisms are comparing functors defined on different categories.

Since our focus is on the sheafification process induced by the dense
topology, we should consider only morphisms from $\mathcal{F}_{0}$ to
$\mathcal{F}_{1}$ for $\mathcal{F}_{i}:X_{i}\to\mathsf{Set}$ which map
compatible families of functions in $\mathcal{F}_{0}$ indexed by a dense
covering of some open set of $X_{0}$ to compatible functions of functions in
$\mathcal{F}_{1}$ indexed by a dense covering of the corresponding (via the
morphism) open set of $X_{1}$. Furthermore these morphisms should map a
collation of such a compatible families existing in $\mathcal{F}_{0}$ to a
collation of the pointwise image under the morphism of such a compatible
family existing in $\mathcal{F}_{1}$. If these requests are not met by a
potential morphism, this potential morphism may not commute with the
sheafification process applied on each of the $\mathcal{F}_{i}$. We will see
that these are the unique requests we must impose.

To define these morphisms we first need to view presheaves as presheaves of
local sections (e.g. presheaves $\mathcal{F}$ such that $\mathcal{F}(U)$ is a
set of maps with domain $U$). This can be done using the functor
$\Gamma^{0}\circ\Lambda^{0}$ (see [6, II]).

######  Fact 3.20.

Let $\mathcal{F}:\mathcal{A}\to\mathsf{Set}$ be a presheaf defined on some
$\mathcal{A}\subseteq\mathcal{O}(X)$. Then $\mathcal{F}$ is naturally
isomorphic in the category $\textrm{Presh}(\mathcal{A},\mathsf{Set})$ to the
functor $\mathrm{ls}(\mathcal{F})$ defined on objects by
$U\mapsto\left\\{(\Gamma^{0}\circ\Lambda^{0})_{U}(f):f\in\mathcal{F}(U)\right\\}$
where

$(\Gamma^{0}\circ\Lambda^{0})_{U}(f)=\left\\{\langle
x,[f]_{\sim_{x}}\rangle:\,x\in U\right\\}$

for any $U\in\mathcal{A}$ and $f\in\mathcal{F}(U)$ and on arrows by

$(U\subseteq V)\mapsto(f\in\mathrm{ls}(\mathcal{F})(V)\mapsto f\restriction
U).$

Furthermore it is useful to note that the stonean sheafification of some
presheaf $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ depends
only on $\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X)$ (at least when
the regular opens are a base for the topology).

######  Fact 3.21.

Let $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ be a presheaf on
$(X,\tau)$, and
$\mathcal{G}:\operatorname{\mathsf{RO}}(X)^{\mathrm{op}}\to\mathsf{Set}$ be
also a presheaf (of local sections). Define
$\mathrm{ext}(\mathcal{G}):\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ on
objects by

$\mathrm{ext}(\mathcal{G})(U)=\mathcal{G}(\mathrm{Reg}\left(U\right))\restriction
U,$

and on arrows

$\mathrm{ext}(\mathcal{G})(U\subseteq
V):f\in\mathrm{ext}(\mathcal{G})(V)\mapsto f\restriction U.$

Then with respect to the categories of presheaves on $\mathcal{O}(X)$ and of
presheaves on $\operatorname{\mathsf{RO}}(X)$:

  * •

$\mathrm{ext}$ is left adjoint to
$\mathcal{F}\mapsto\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X)$,
$\mathrm{ext}(\mathcal{F})\restriction\operatorname{\mathsf{RO}}(X)=\mathcal{F}$,
and $\mathrm{ext}(\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X))$ is a
subobject of $\mathcal{F}$ in $\textrm{Presh}(X,\mathsf{Set})$.

  * •

If $\operatorname{\mathsf{RO}}(X)$ is a dense subset of $\mathcal{O}(X)$, then
for any $\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$

$\Gamma^{1}\circ\Lambda^{1}\circ\mathrm{ext}(\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X))=\Gamma^{1}\circ\Lambda^{1}(\mathcal{F}).$

These facts show that morphisms on presheaves in order to commute with stonean
sheafifications should not depend on their behaviour outside of the regular
open sets.

We should also expect that a morphism
$\iota:\mathcal{F}_{0}\to\mathcal{F}_{1}$ naturally induces an order morphism
$k_{\iota}:\mathcal{O}(X_{0})\to\mathcal{O}(X_{1})$. If $\iota$ maps
compatible families indexed by dense coverings to compatible families indexed
by dense coverings, we must have that $k_{\iota}$ maps dense coverings to
dense coverings. In view of Proposition 2.6 we may consider only $k_{\iota}$
of the form $\bar{\bar{k}}_{f}$ for some continuous open $f:X_{1}\to X_{0}$ or
(more generally) such that
$k_{\iota}\restriction\operatorname{\mathsf{RO}}(X)$ is a complete
homomorphism131313Note that there can be complete homomorphisms
$k:\operatorname{\mathsf{RO}}(X)\to\operatorname{\mathsf{RO}}(Y)$ which are
not of the form $k_{f}$ for any continuous open $f:Y\to X$. For example let
$k:\mathcal{P}\left(\left\\{0,1\right\\}\right)\to\operatorname{\mathsf{RO}}([0;1])$
be given by $k(\left\\{0\right\\})=[0;1/2)$, $k(\left\\{1\right\\})=(1/2;1]$.
Then no continuous function $f:[0;1]\to\left\\{0,1\right\\}$ can have $k$ as
$k_{f}$..

Also we can naturally lift using continuous maps (or complete homomorphisms on
regular open algebras) a presheaf (of local sections) to a presheaf (of local
sections) as follows:

######  Notation 3.22.

Let $f:Y\to X$ be a continuous open function,
$\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$ be a presheaf,
$\Theta:\mathcal{F}_{0}\to\mathcal{F}_{1}$ be a natural transformation of
presheaves on $X$.

We let:

  * •

$f^{*}(\mathcal{F}):\mathcal{O}(Y)^{\mathrm{op}}\to\mathsf{Set}$ be the
presheaf defined by

$U\mapsto\left\\{g\circ f:g\in\mathcal{F}(f[U])\right\\}$

for $U\in\mathcal{O}(Y)$, and by

$(U\subseteq V)\mapsto\left\\{g\circ f\restriction
U:g\in\mathcal{F}(f[V])\right\\}$

for $U\subseteq V$ and both in $\mathcal{O}(Y)$.

We let for $g\in\mathcal{F}(f[U])$ $f^{*}(g)$ be the section $g\circ f$ of
$f^{*}(\mathcal{F})(U)$.

  * •

$f^{*}(\Theta)$ be defined by

$\displaystyle f^{*}(\Theta)_{U}:$ $\displaystyle f^{*}(\mathcal{F}_{0})(U)\to
f^{*}(\mathcal{F}_{1})(U)$ $\displaystyle g\circ f\mapsto\Theta_{f[U]}(g)\circ
f$

for $U$ in $\mathcal{O}(Y)$ and $g\in\mathcal{F}_{0}(f(U))$.

Conversely let
$i:\operatorname{\mathsf{RO}}(X)\to\operatorname{\mathsf{RO}}(Y)$ be a
complete homomorphism. Then it admits
$\pi_{i}:\operatorname{\mathsf{RO}}(Y)\to\operatorname{\mathsf{RO}}(X)$ left
adjoint map.

We let:

  * •

$i_{*}(\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X)):\operatorname{\mathsf{RO}}(Y)^{\mathrm{op}}\to\mathsf{Set}$
be defined by

$U\mapsto\mathcal{F}(\pi_{i}(U))$

for $U\in\operatorname{\mathsf{RO}}(Y)$, and by

$(U\subseteq V)\mapsto\mathcal{F}(\pi_{i}(U)\subseteq\pi_{i}(V))$

for $U\subseteq V$ and both in $\operatorname{\mathsf{RO}}(Y)$.

  * •

$i_{*}(\Theta\restriction\operatorname{\mathsf{RO}}(X))$ be defined by

$\displaystyle i_{*}(\Theta)_{U}:$ $\displaystyle i_{*}(\mathcal{F}_{0})(U)\to
i_{*}(\mathcal{F}_{1})(U)$ $\displaystyle h\mapsto\Theta_{\pi_{i}(U)}(h)$

for $U\in\operatorname{\mathsf{RO}}(Y)$ and $h\in\mathcal{F}_{0}(\pi_{i}(U))$.

$i_{*}(\mathcal{F})$ is defined to be
$\mathrm{ext}(i_{*}(\mathcal{F}\restriction\operatorname{\mathsf{RO}}(X)))$
and $i_{*}(\Theta)$ is defined as
$\mathrm{ext}(i_{*}(\Theta\restriction\operatorname{\mathsf{RO}}(X)))$.

It is clear that (up to natural isomorphism) $f^{*}=(k_{f})_{*}$.

We see below that both of the above lifting operators are well defined and
commute well with the stonean sheafification operation.

######  Notation 3.23.

Given a topological space $(X,\tau)$ we let
$i_{X}:\operatorname{\mathsf{RO}}(X)\to\operatorname{\mathsf{CLOP}}(\operatorname{St}(\operatorname{\mathsf{RO}}(X)))$
be the Stone duality map

$V\mapsto
N_{V}=\left\\{G\in\operatorname{St}(\operatorname{\mathsf{RO}}(X)):V\in
G\right\\}.$

In case $(X,\tau)$ is compact Hausdorff we also define the natural projection
$\pi_{X}:\operatorname{St}(\operatorname{\mathsf{RO}}(X))\to X$ which maps $G$
to the unique point in $\bigcap\left\\{\mathrm{Cl}\left(U\right):U\in
G\right\\}$.

For $\mathcal{F}$ a presheaf on $(X,\tau)$ and $\Theta$ a natural
transformation on presheaves on $X$, we have that $(i_{X})_{*}(\mathcal{F})$
is a presheaf on $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ and
$(i_{X})_{*}(\Theta)$ is a natural transformations of presheaves on
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$. Furthermore in case $X$ is
compact Hausdorff $(\pi_{X})^{*}(\mathcal{F})$ is up to isomorphism equal to
$(i_{X})_{*}(\mathcal{F})$ and the same holds for $(\pi_{X})^{*}(\Theta)$ and
$(i_{X})_{*}(\Theta)$ (now in the category of natural transformations on
presheaves on $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$).

The key property of these lifting operators is that they commute well with
stonean sheafification (Recall Definition 2.1 for the definition of
$\pi^{*}_{i}:\operatorname{St}(\operatorname{\mathsf{RO}}(Y)\to\operatorname{St}(\operatorname{\mathsf{RO}}(X))$)

######  Lemma 3.24.

Let $\Theta:\mathcal{F}_{0}\to\mathcal{F}_{1}$ be a natural transformation for
presheaves $\mathcal{F}_{j}:\mathcal{O}(X_{j})\to\mathsf{Set}$ and
$i:\operatorname{\mathsf{RO}}(X_{0})\to\operatorname{\mathsf{RO}}(X_{1})$ be a
complete homomorphism.

Then:

  * •

$(\pi^{*}_{i})^{*}(\Gamma^{1}\circ\Lambda^{1}(\mathcal{F}_{0}))\cong\Gamma^{1}\circ\Lambda^{1}(i_{*}(\mathcal{F}_{0}))$,

  * •

$(\pi^{*}_{i})^{*}(\Gamma^{1}\circ\Lambda^{1}((i_{X_{0}})_{*}(\Theta)))\cong\Gamma^{1}\circ\Lambda^{1}((i_{X_{1}})_{*}\circ
i_{*}(\Theta))$.

###### Proof.

A detailed proof of all items requires a careful diagram chasing, which we do
not want to pursue in its entirety. We give a few key hints of why everything
works well for presheaves
$\mathcal{F}:\mathcal{O}(X)^{\mathrm{op}}\to\mathsf{Set}$. The case of natural
transformations is notationally much more involved but does not need any
further idea, hence we dare to leave it to the reader.

Let $g:A\to\beta(\Lambda^{1}_{i_{*}(\mathcal{F})})$ be a local section of
$\Gamma^{1}\circ\Lambda^{1}(i^{*}(\mathcal{F}))$ for some $A$ open in
$\operatorname{St}(\operatorname{\mathsf{RO}}(Y))$. Then $g$ is the collation
in $\Gamma^{1}\circ\Lambda^{1}(i^{*}(\mathcal{F}))$ of local sections
$\left\\{g_{i}:i\in I\right\\}$ such that:

  * •

each

$g_{i}=\dot{h}_{i}=\left\\{\langle H,[h_{i}]_{H}\rangle:\,V_{i}\in
H,\,h_{i}\in\mathcal{F}(i(V_{i}))\right\\}$

for some $h_{i}\in\mathcal{F}(i(V_{i}))$ and
$V_{i}\in\operatorname{\mathsf{RO}}(Y)$;

  * •

$\bigcup\left\\{N_{V_{i}}:i\in I\right\\}$ is a dense subset of $A$ in
$\operatorname{St}(\operatorname{\mathsf{RO}}(Y))$.

We note that this imposes that $\left\\{h_{i}:i\in I\right\\}$ is a compatible
family of local sections in $\mathcal{F}$ each one defined on the open set
$i(V_{i})$ with $\bigcup\left\\{N_{i(V_{i})}:i\in I\right\\}$ a dense subset
of the open set $B=\mathrm{Reg}\left(\bigcup_{i\in I}N_{i(V_{i})}\right)$ in
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ and also with
$(\pi^{*}_{i})^{-1}[B]$ containing $A$ has a dense subset (since
$\pi^{*}_{i}[N_{V}]=N_{\pi_{i}}(V)$ for any
$V\in\operatorname{\mathsf{RO}}(Y)$). Since
$\Gamma^{1}\circ\Lambda^{1}(\mathcal{F})$ is a stonean sheaf, there is a
unique collation $\bar{g}$ of $\left\\{\dot{h}_{i}:i\in I\right\\}$ defined on
$\mathrm{Reg}\left(B\right)$. Then
$(\pi^{*}_{i})^{*}(\bar{g})=\bar{g}\circ\pi^{*}_{i}$ and $g$ coincide on the
dense in $A$ open set $\bigcup\left\\{N_{V_{i}}:i\in I\right\\}$, hence they
must be equal on $A$.

A similar reasoning shows the converse inclusion.

The case of natural transformations is notationally more involved and is also
left to the reader. ∎

We condense these observations in the following definition:

######  Definition 3.25.

Let $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ be presheaves
$\mathcal{F}_{i}:\mathcal{O}(X_{i})^{\mathrm{op}}\to\mathsf{Set}$ for $i=0,1$.

$\iota=(i_{\iota},\Theta_{\iota})$ defines a morphism of $\mathcal{F}_{0}$ to
$\mathcal{F}_{1}$ if:

  * •

$i_{\iota}:\operatorname{\mathsf{RO}}(X_{0})\to\operatorname{\mathsf{RO}}(X_{1})$
is a complete homomorphism;

  * •

$\Theta_{\iota}:(i_{\iota})_{*}(\mathcal{F}_{0})\to\mathcal{F}_{1}$ is a
natural transformation in the category $\textrm{Presh}(X_{1},\mathsf{Set})$.

######  Definition 3.26.

$\textrm{Presh}^{*}(\mathsf{Set})$ is the category of presheaves on the open
sets of some topological space with morphisms given by Definition 3.25.

$\textrm{Sh}^{*}(\mathsf{Set})$ is the subcategory of
$\textrm{Presh}^{*}(\mathsf{Set})$ given by stonean sheaves
$\mathcal{F}:\mathcal{O}(X)^{\text{op}}\to\mathsf{Set}$ for some compact
Hausdorff extremally disconnected $(X,\tau)$.

We are now in the position to state our adjunction result for the
sheafification process induced by $\Gamma^{1}\circ\Lambda^{1}$.

To formulate it we just need to define precisely how
$\Gamma^{1}\circ\Lambda^{1}$ acts on the morphisms of
$\textrm{Presh}^{*}(\mathsf{Set})$.

######  Definition 3.27.

Let $\iota=(i_{\iota},\Theta_{\iota})$ be a morphism of $\mathcal{F}_{0}$ to
$\mathcal{F}_{1}$ induced by a complete homomorphism
$i_{\iota}:\operatorname{\mathsf{RO}}(X_{0})\to\operatorname{\mathsf{RO}}(X_{1})$
and a natural transformation $\Theta_{\iota}$ of
$\textrm{Presh}(X_{1},\mathsf{Set})$ from $(i_{\iota})_{*}(\mathcal{F}_{0})$
to $\mathcal{F}_{1}$.

Then

$\Gamma^{1}\circ\Lambda^{1}(\iota)=\iota^{1}=(i^{1}_{\iota},\Theta^{1}_{\iota})$

is defined as follows:

  * •

$i^{1}_{\iota}=k_{\pi^{*}_{i_{\iota}}}:\operatorname{\mathsf{RO}}(\operatorname{St}(\operatorname{\mathsf{RO}}(X_{0})))\to\operatorname{\mathsf{RO}}(\operatorname{St}(\operatorname{\mathsf{RO}}(X_{1})))$
which maps $N_{U}$ to $N_{i_{\iota}(U)}$ (recall Notation 2.2).

  * •

$\Theta^{1}_{\iota}$ is
$\Gamma^{1}\circ\Lambda^{1}((i_{X_{1}})_{*}(\Theta_{\iota}))$.

######  Lemma 3.28.

For each presheaf morphism $\iota=\langle i_{\iota},\Theta_{\iota}\rangle$ of
$\mathcal{F}_{0}$ to $\mathcal{F}_{1}$, $\Theta^{1}_{\iota}$ as defined in
3.27 is a well-defined natural transformation.

###### Proof.

This is a tiresome check we leave to the reader. The key point is that the
required compositions and liftings commute with collations of local sections
in view of Lemma 3.24. ∎

######  Theorem 3.29.

$\Gamma^{1}\circ\Lambda^{1}$ is a well defined functor.

Furthermore the inclusion functor of $\textrm{Sh}^{*}(\mathsf{Set})$ into
$\textrm{Presh}^{*}(\mathsf{Set})$ is right adjoint to
$\Gamma^{1}\circ\Lambda^{1}$.

The proof of this theorem is a lengthy and tiresome check that (any of the
standard set of) the clauses defining an adjunction are satisfied by
$\Gamma^{1}\circ\Lambda^{1}$.

We briefly outline some delicate points of the proof, but we omit a detailed
argument since it would only amount to a few pages of uninformative and
tiresome calculations.

###### Proof.

The first delicate point is to check that $\Gamma^{1}\circ\Lambda^{1}$ is well
defined on morphisms $\iota$. This is the content of Lemma 3.24.

The functoriality of $\Gamma^{1}\circ\Lambda^{1}$ is a tiresome check we leave
to the reader.

The key ideas in the proof that $\Gamma^{1}\circ\Lambda^{1}$ is a left adjoint
follow again from Lemma 3.24. ∎

There are some key observations to make:

######  Remark 3.30.

__Assume $\mathsf{B}=\operatorname{\mathsf{RO}}(X,\tau)$ for some Tychonoff
topological space $(X,\tau)$ and
$\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$ is a separated
presheaf. In particular
$\operatorname{\mathsf{RO}}(\operatorname{St}(\mathsf{B}))=\mathsf{B}$ and
thus $\Lambda^{1}_{\mathcal{F}}$ is an étalé space on
$\operatorname{St}(\mathsf{B})$ and $p^{1}_{\mathcal{F}}$ is an extremally
disconnected bundle. Then $\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ and
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}$ can have really little to share:

  * •

If $X$ is connected there cannot be a continuous map of $X$ into
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ since the latter is
$0$-dimensional.

  * •

Since $X$ admits a base of regular open sets, we have a natural projection of
a dense subset of $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$ to $X$
mapping $G$ to $x$ if $G$ extends the filter of regular open neighborhoods of
$X$ (however this filter need not be a maximal filter of open sets: consider
the case of the filter of regular open neighborhoods of $1/2$ for $[0;1]$ with
euclidean topology).

In particular certain maximal filters on $\operatorname{\mathsf{RO}}(X)$ do
not correspond to points of $X$ (if $X$ is non-compact), or there could be
points in $X$ whose filter of open neighborhoods admits different extensions
to maximal filters on $\operatorname{\mathsf{RO}}(X)$.
Assume now $(X,\tau)$ is compact Hausdorff and extremally disconnected. Then
$\operatorname{St}(\operatorname{\mathsf{RO}}(X))=\operatorname{St}(\operatorname{\mathsf{CLOP}}(X))$
is homeomorphic to $X$ via the map

$x\mapsto G_{x}=\left\\{U\in\operatorname{\mathsf{RO}}(X,\tau):\,x\in
U\right\\}.$

Fix any presheaf $\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$
with base set $X$. We can use Proposition 1.22 to argue that $[f]_{G}=[g]_{G}$
holds in $\Lambda^{1}_{\mathcal{F}}$ if and only if $[f]_{G}=[g]_{G}$ holds in
$\Lambda^{0}_{\mathcal{F}}$. This yields that $\Lambda^{0}_{\mathcal{F}}$ and
$\Lambda^{1}_{\mathcal{F}}$ (as well as $p^{0}_{\mathcal{F}}$ and
$p^{1}_{\mathcal{F}}$) are the same.

Now $\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ and $\Gamma^{0}_{p^{0}_{\mathcal{F}}}$
are both sheaves $\mathcal{O}(X)^{\textrm{op}}\to\mathsf{Set}$. However they
may be different (see the following example).

######  Example 3.31.

Let $(X,\tau)$ be a compact Hausdorff extremally disconnected, and
$\mathcal{C}(X)$ be the set of continuous complex valued functions on $X$.
$\mathcal{C}(X)$ is the set of global sections of the sheaf $\mathcal{F}$ of
continuous complex valued functions defined on an open subset of $X$. This
amounts to say that (up to functorial equivalence) $\mathcal{F}$ is exactly
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}$. Notice however that, even though
$\mathcal{F}$ is a topological sheaf on the topological space $X$, when
restricted to
$\mathsf{B}^{+}=\operatorname{\mathsf{RO}}(X)\setminus\left\\{\emptyset\right\\}$
it is not anymore a sheaf on the partial order $\mathsf{B}^{+}$ with coverings
given by the double negation topology (e.g.
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}\restriction\operatorname{\mathsf{RO}}(X)^{+}$
is not a sheaf).

To see this, let $\left\\{U_{n}:n\in\mathbb{Z}\right\\}$ be a maximal
antichain of $\operatorname{\mathsf{RO}}(X)^{+}$ and let
$f_{n}\in\mathcal{F}(U_{n})$ be constant with value $n$. Then
$\left\\{f_{n}:n\in\mathbb{Z}\right\\}$ is a matching family of
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}\restriction\operatorname{\mathsf{RO}}(X)$
which cannot be collated in
$\Gamma^{0}_{p^{0}_{\mathcal{F}}}\restriction\operatorname{\mathsf{RO}}(X)^{+}$
(but can be collated in $\Gamma^{1}_{p^{0}_{\mathcal{F}}}$). The point being
that for $G\in X\setminus\bigcup_{n\in\mathbb{Z}}U_{n}$
$\beta(f):X\to\beta_{0}(\Lambda^{1}_{\mathcal{F}})$ maps $G$ to $\infty$.

Consider now the set

$\mathcal{C}^{+}(X):=\left\\{f:X\to\beta_{0}(\mathbb{C}):f\textit{ is
continuous and }f^{-1}[\\{\infty\\}]\textit{ is nowhere dense}\right\\}.$

It turns out that $\mathcal{C}^{+}(X)$ defines the global sections of the
_stonean sheafification_ $\Gamma^{1}_{p^{1}_{\mathcal{F}}}$ of the sheaf
$\mathcal{F}$ associated to $\mathcal{C}(X)$.

Indeed, we already noticed that
$\Lambda^{1}_{\mathcal{F}}=\Lambda^{0}_{\mathcal{F}}$, since $X$ is
homeomorphic to $\operatorname{St}(\operatorname{\mathsf{RO}}(X))$. Now, if
$s\in\Gamma^{1}_{\Lambda^{0}_{\mathcal{F}}}(X)$, then we can find a family
$\\{V_{i}:i\in I\\}$ of (disjoint) regular open subsets of $X$ such that
$\bigcup_{i\in I}V_{i}$ is open dense in $X$ and $s\restriction
V_{i}=\dot{f}_{i}$ for some $f_{i}\in\mathcal{C}(V_{i})$. Then $\bigcup_{i\in
I}f_{i}\in\mathcal{C}(\bigcup_{i\in I}V_{i})$ and thus
$s\restriction\bigcup_{i\in I}V_{i}=(\bigcup_{i\in
I}f_{i})\restriction\bigcup_{i\in I}V_{i}$. Therefore we can identify $s$ with
the unique continuous extension $\beta(\bigcup_{i\in
I}f_{i}):X=\beta(\bigcup_{i\in I}V_{i})\to\beta_{0}(\mathbb{C})$, which is an
element in $\mathcal{C}^{+}(X)$.

In model theoretic terms (according to the terminology we introduce in Section
4) $\mathcal{C}(X)$ (seen as a $\operatorname{\mathsf{RO}}(X)$-valued model)
does not have the mixing property (see Proposition 5.7), while
$\mathcal{C}^{+}(X)$ does. [10] gives a detailed analysis of the boolean
valued model structure of $\mathcal{C}(X)$ and $\mathcal{C}^{+}(X)$. Other
connections between functional analysis, sheaf theory, and set theory will be
given in Example 5.12.

We end this section with the following corollary, which stresses the fact that
our stonean sheafification is exactly the generalization to topological spaces
of the usual sheafification process for presheaves defined on complete boolean
algebras.

######  Corollary 3.32.

Let $\mathsf{B}=\operatorname{\mathsf{RO}}(\operatorname{St}(\mathsf{B}))$ be
a complete boolean algebra. Then the functor
$\textrm{Presh}(\mathsf{B},\mathsf{Set})\to\textrm{Sh}(\mathsf{B},\mathsf{Set})$
such that
$\mathcal{F}\mapsto(\Gamma^{1/2}\circ\Lambda^{1})(\mathrm{ext}(\mathcal{F}))$
is the left adjoint of the inclusion of $\textrm{Sh}(\mathsf{B},\mathsf{Set})$
in $\textrm{Presh}(\mathsf{B},\mathsf{Set})$ (where
$\textrm{Sh}(\mathsf{B},\mathsf{Set})$ is the category of sheaves according to
the sup Grothendieck topology).

##  4\. Boolean valued models

We consider only boolean valued models for relational languages as (at least
for our purposes) there is no lack of generality in doing so, and our proofs
are notationally smoother in this set up.

######  Definition 4.1.

Let $\mathcal{L}=\left\\{R_{i}:i\in I,c_{j}:j\in J\right\\}$ be a relational
language (i.e. without functional symbols; constant symbols are allowed) and
let $\mathsf{B}$ be a boolean algebra. A _$\mathsf{B}$ -valued model_
$\mathcal{M}$ for $\mathcal{L}$ consists of:

  1. (1)

a non-empty set $M$, called the _domain_ of $\mathcal{M}$;

  2. (2)

the boolean value of the equality symbol, which is a function

$\begin{aligned} =^{\mathcal{M}}:&M^{2}\to\mathsf{B}\\\
\langle\sigma,&\tau\rangle\mapsto\left\llbracket\sigma=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\end{aligned};$

  3. (3)

for each $n$-ary relational symbol $R\in\mathcal{L}$, a function

$\begin{aligned} R^{\mathcal{M}}:&M^{n}\to\mathsf{B}\\\
\langle\sigma_{1},\dots,&\sigma_{n}\rangle\mapsto\left\llbracket
R(\sigma_{1},\dots,\sigma_{n})\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\end{aligned};$

  4. (4)

for each constant symbol $c\in\mathcal{L}$, an element $c^{\mathcal{M}}\in M$.

We require the following conditions to hold:

  1. (1)

for every $\sigma,\tau,\pi\in M$,

$\begin{gathered}\left\llbracket\sigma=\sigma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=1_{\mathsf{B}},\\\
\left\llbracket\sigma=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=\left\llbracket\tau=\sigma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}},\\\
\left\llbracket\sigma=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\wedge\left\llbracket\tau=\pi\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\leq\left\llbracket\sigma=\pi\right\rrbracket^{\mathcal{M}}_{\mathsf{B}};\end{gathered}$

  2. (2)

for every $n$-ary relational symbol $R\in\mathcal{L}$ and for every
$\sigma_{1},\dots,\sigma_{n},\tau_{1},\dots,\tau_{n}\in M$,

$\Bigl{(}\bigwedge_{i=1}^{n}\left\llbracket\sigma_{i}=\tau_{i}\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\Bigr{)}\wedge\left\llbracket
R(\sigma_{1},\dots,\sigma_{n})\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\leq\left\llbracket
R(\tau_{1},\dots,\tau_{n})\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}.$

A $\mathsf{B}$-valued model $\mathcal{M}$ is _extensional_ if
$\left\llbracket\sigma=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=1_{\mathsf{B}}$
entails $\sigma=\tau$ for all $\sigma,\tau\in M$.

If no confusion can arise, we avoid the supscript $\mathcal{M}$ and the
subscript $\mathsf{B}$. Moreover, we write equivalently $\sigma\in M$ or
$\sigma\in\mathcal{M}$, identifying a boolean valued model $\mathcal{M}$ with
its underlying set $M$.
Monro does not require in [8] that
$\left\llbracket\sigma=\sigma\right\rrbracket=1_{\mathsf{B}}$ for every
$\sigma\in M$; he calls _global elements_ the ones satisfying this condition.

######  Definition 4.2.

Fix a language $\mathcal{L}$. Let $\mathcal{M}_{1}$ be a
$\mathsf{B}_{1}$-valued model for $\mathcal{L}$ and $\mathcal{M}_{2}$ be a
$\mathsf{B}_{2}$-valued model for $\mathcal{L}$, where $\mathsf{B}_{1}$ and
$\mathsf{B}_{2}$ are boolean algebras. A _morphism of boolean valued models
for $\mathcal{L}$_ is a pair $(\Phi,i)$ where:

  * •

$i:\mathsf{B}_{1}\to\mathsf{B}_{2}$ is an adjoint morphism of boolean
algebras;

  * •

$\Phi:M_{1}\to M_{2}$ is an _$i$ -morphism_, that is: for every $n$-ary
relational symbol $R$ and every constant symbol $c$ in the language and for
every $\tau_{1},\dots,\tau_{n}\in M_{1}$,

$\Phi(c^{\mathcal{M}_{1}})=c^{\mathcal{M}_{2}},$ $i(\llbracket
R(\tau_{1},\dots,\tau_{n})\rrbracket^{\mathcal{M}_{1}}_{\mathsf{B}_{1}})\leq\llbracket
R(\Phi(\tau_{1}),\dots,\Phi(\tau_{n}))\rrbracket^{\mathcal{M}_{2}}_{\mathsf{B}_{2}},$
$i(\llbracket\tau_{1}=\tau_{2}\rrbracket^{\mathcal{M}_{1}}_{\mathsf{B}_{1}})\leq\llbracket\Phi(\tau_{1})=\Phi(\tau_{2})\rrbracket^{\mathcal{M}_{2}}_{\mathsf{B}_{2}}.$

If in both these equations equality holds, we call $\Phi$ an _$i$ -embedding_.

If $i$ is an isomorphism, $\Phi$ is an $i$-embedding, and for all $\tau\in
M_{2}$ there is $\sigma\in M_{1}$ such that
$\left\llbracket\Phi(\sigma)=\tau\right\rrbracket_{\mathsf{B}_{2}}=1_{\mathsf{B}_{2}}$,
$\langle\Phi,i\rangle$ is an _isomorphism of boolean valued models_.

Clearly, if $\mathsf{B}_{1}=\mathsf{B}_{2}=2$ and both $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$ are extensional, an $\textrm{Id}_{2}$-morphism and an
$\textrm{Id}_{2}$-embedding are exactly a morphism and an embedding between
two Tarski models, respectively.
It is worth noting that Monro considers as a morphism an equivalence class of
such functions, where $\Phi$ and $\Psi$ are equivalent whenever
$i(\llbracket\tau=\tau\rrbracket)\leq\llbracket\Phi(\tau)=\Psi(\tau)\rrbracket$141414Remember
that Monro does not require $\llbracket\tau=\tau\rrbracket=1_{\mathsf{B}}$.
for every $\tau$ in the first model (see [8, Definition 5.2]).
We now define the boolean valued semantics. By now, we have not required any
condition on the boolean algebra $\mathsf{B}$. For the definition of the
semantic, however, we need to compute some infinite suprema. For this reason,
we will always assume in the sequel that formulae are assigned truth values in
the boolean completion $\operatorname{\mathsf{RO}}(\mathsf{B}^{+})$ of
$\mathsf{B}$ and we identify $\mathsf{B}^{+}$ with a dense subset of the
partial order $\operatorname{\mathsf{RO}}(\mathsf{B}^{+})^{+}$. In most cases
$\mathsf{B}$ is rightaway a complete booolean algebra in which case
$\operatorname{\mathsf{RO}}(\mathsf{B}^{+})=\mathsf{B}$ and this complication
vanishes.

######  Definition 4.3.

Let $\mathcal{M}=\langle M,=^{\mathcal{M}},R^{\mathcal{M}}_{i}:i\in I\rangle$
be a $\mathsf{B}$-valued model for the relational language
$\mathcal{L}=\left\\{R_{i}:i\in I\right\\}$. We evaluate the formulae without
free variables in the extended language
$\mathcal{L}_{\mathcal{M}}:=\mathcal{L}\cup\left\\{c_{\sigma}:\sigma\in
M\right\\}$ by maps with values in the boolean completion
$\operatorname{\mathsf{RO}}(\mathsf{B}^{+})$ of $\mathsf{B}$ as follows:

  * •

$\left\llbracket
c_{\sigma}=c_{\tau}\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}:=\left\llbracket\sigma=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}$
and $\left\llbracket
R(c_{\sigma_{1}},\dots,c_{\sigma_{n}})\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}:=\left\llbracket
R(\sigma_{1},\dots,\sigma_{n})\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}$;

  * •

$\left\llbracket\varphi\wedge\psi\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}:=\left\llbracket\varphi\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}\wedge\left\llbracket\psi\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}$;

  * •

$\left\llbracket\neg\varphi\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}:=\neg\left\llbracket\varphi\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}$;

  * •

$\left\llbracket\exists
x\varphi(x,c_{\sigma_{1}},\dots,c_{\sigma_{n}})\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}:=\bigvee_{\tau\in
M}\left\llbracket\varphi(c_{\tau},c_{\sigma_{1}},\dots
c_{\sigma_{n}})\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}.$

If $\varphi(x_{1},\dots,x_{n})$ is any $\mathcal{L}$-formula with free
variables $x_{1},\dots,x_{n}$ and $\nu$ is an assignment, we define
$\left\llbracket\nu(\varphi(x_{1},\dots,x_{n}))\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}:=\left\llbracket\varphi(c_{\nu(x_{1})},\dots,c_{\nu(x_{n})})\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}$.

We will often write $\varphi(\sigma_{1},\dots,\sigma_{n})$ rather than
$\varphi(c_{\sigma_{1}},\dots,c_{\sigma_{n}})$ and
$\left\llbracket\varphi(\tau_{1},\dots,\tau_{n})\right\rrbracket$ rather than
$\left\llbracket\varphi(\tau_{1},\dots,\tau_{n})\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B}^{+})}$
if no confusion can arise.

By induction on the complexity of $\mathcal{L}$-formulae, it is possible to
show the following:

######  Fact 4.4.

For every $\sigma_{1},\dots,\sigma_{n},\tau_{1},\dots,\tau_{n}\in M$ and for
every $\mathcal{L}$-formula $\varphi(x_{1},\dots,x_{n})$ with displayed free
variables, the following holds:

(1)
$\Bigl{(}\bigwedge_{i=1}^{n}\left\llbracket\sigma_{i}=\tau_{i}\right\rrbracket\Bigr{)}\wedge\left\llbracket\varphi(\sigma_{1},\dots,\sigma_{n})\right\rrbracket\leq\left\llbracket\varphi(\tau_{1},\dots,\tau_{n})\right\rrbracket.$

######  Remark 4.5.

Given a signature $\mathcal{L}$, and a morphism $(\Theta,i)$ of a
$\mathsf{B}$-valued model $\mathcal{M}$ for $\mathcal{L}$ into a
$\mathsf{C}$-valued model $\mathcal{N}$ for $\mathcal{L}$ it is a natural
request that of imposing $i$ to be an adjoint homomorphism: otherwise the
truth value of $i(\left\llbracket\exists
x\psi(x)\right\rrbracket^{\mathcal{M}}_{\mathsf{B}})$ might not be
$\bigvee_{\mathsf{C}}\left\\{i(\left\llbracket\psi(\sigma)\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}):\sigma\in\mathcal{M}\right\\}$,
in which case the morphism would not preserve the expected meaning of
quantifiers.

The following is the correct generalization to boolean valued semantics of the
notion of elementary embedding between Tarski structures:

######  Definition 4.6.

Given a signature $\mathcal{L}$, a morphism $(\Theta,i)$ of a
$\mathsf{B}$-valued model $\mathcal{M}$ for $\mathcal{L}$ into a
$\mathsf{C}$-valued model $\mathcal{N}$ for $\mathcal{L}$ is elementary if for
all $a_{1},\dots,a_{n}\in\mathcal{M}$ and $\mathcal{L}$-formulae
$\phi(x_{1},\dots,x_{n})$

$i(\left\llbracket\phi(a_{1},\dots,a_{n}\right\rrbracket^{\mathcal{M}}_{\mathsf{B}})=\left\llbracket\phi(\Theta(a_{1}),\dots,\Theta(a_{n})\right\rrbracket^{\mathcal{N}}_{\mathsf{C}}.$

######  Definition 4.7.

A sentence $\varphi$ in the language $\mathcal{L}$ is _valid_ in a
$\mathsf{B}$-valued model $\mathcal{M}$ for $\mathcal{L}$ if
$\left\llbracket\varphi\right\rrbracket=1_{\operatorname{\mathsf{RO}}(\mathsf{B})}$.

A theory $T$ is _valid_ in $\mathcal{M}$ (equivalently, $\mathcal{M}$ is a
$\mathsf{B}$-model for $T$) if every axiom of $T$ is valid in $\mathcal{M}$.

######  Theorem 4.8 (Soundness and Completeness).

Let $\mathcal{L}$ be a relational language. An $\mathcal{L}$-formula $\varphi$
is provable syntactically by an $\mathcal{L}$-theory $T$ if and only if, for
every boolean algebra $\mathsf{B}$ and for every $\mathsf{B}$-valued model
$\mathcal{M}$ for $T$,
$\left\llbracket\nu(\varphi)\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B})}=1_{\operatorname{\mathsf{RO}}(\mathsf{B})}$
for every assignment $\nu$ taking values in $\mathcal{M}$.

###### Proof.

See for instance [13, Theorems 4.1.5 and 4.1.8]. ∎

###  4.1. Łoś Theorem for boolean valued models and fullness

######  Definition 4.9.

Let $\mathsf{B}$ be a boolean algebra and let $\mathcal{M}=\langle
M,=^{\mathcal{M}},R^{\mathcal{M}}_{i}:i\in I\rangle$ be a $\mathsf{B}$-valued
model for the relational language $\mathcal{L}=\left\\{R_{i}:i\in I\right\\}$.
Let $F$ be a filter of $\mathsf{B}$. The _quotient_ $\mathcal{M}/_{F}$ is the
$\mathsf{B}/_{F}$-valued model for $\mathcal{L}$ defined as follows:

  * •

the domain is $M/_{F}:=\left\\{[\sigma]_{F}:\sigma\in M\right\\}$, where
$[\sigma]_{F}:=\left\\{\tau\in M:\left\llbracket\tau=\sigma\right\rrbracket\in
F\right\\}$;

  * •

if $R\in\mathcal{L}$ is an $n$-ary relational symbol and
$[\sigma_{1}]_{F},\dots,[\sigma_{n}]_{F}\in M/_{F}$,

$\left\llbracket
R([\sigma_{1}]_{F},\dots,[\sigma_{n}]_{F})\right\rrbracket^{\mathcal{M}/_{F}}_{\mathsf{B}/_{F}}:=\Bigl{[}\left\llbracket
R(\sigma_{1},\dots,\sigma_{n})\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\Bigr{]}_{F}\in\mathsf{B}/_{F}.$

It is easy to check that this quotient is well-defined. In particular, if $G$
is an ultrafilter, the quotient $\mathcal{M}/_{G}$ is a two-valued Tarski
structure for $\mathcal{L}$. On the other hand if $\mathsf{B}$ is a boolean
algebra and $F$ is a filter on $\mathsf{B}$, it can happen that $\mathsf{B}$
is complete while $\mathsf{B}/_{F}$ is not, and conversely. In particular it
is not clear how the semantics of formulae with quantifiers is affected by the
quotient operation. We now address this problem.

######  Definition 4.10.

Given a first order signature $\mathcal{L}$, a $\mathsf{B}$-valued model
$\mathcal{M}$ for $\mathcal{L}$ is

  * •

_well behaved_ if: for all $\mathcal{L}$-formulae $\phi(x_{1},\dots,x_{n})$
and $\tau_{1},\dots,\tau_{n}\in\mathcal{M}$
$\left\llbracket\phi(\tau_{1},\dots,\tau_{n})\right\rrbracket^{\mathcal{M}}_{\operatorname{\mathsf{RO}}(\mathsf{B})}$
is in $\mathsf{B}$;

  * •

_full_ if for all ultrafilters $G$ on $\mathsf{B}$, all $\mathcal{L}$-formulae
$\phi(x_{1},\dots,x_{n})$ and all $\tau_{1},\dots,\tau_{n}\in\mathcal{M}$

$\mathcal{M}/_{G}\models\phi([\tau_{1}]_{G},\dots,[\tau_{n}]_{G})\qquad\text{
if and only if
}\qquad\left\llbracket\phi(\tau_{1},\dots,\tau_{n})\right\rrbracket^{\mathcal{M}}\in
G.$

Notice that if $\mathsf{B}$ is complete
$\mathsf{B}\cong\operatorname{\mathsf{RO}}(\mathsf{B}^{+})$ hence any
$\mathsf{B}$-valued model $\mathcal{M}$ is automatically well-behaved. But
being well-behaved does not require $\mathsf{B}$ to be complete, just to be
able to compute all the suprema and infima required by the satisfaction
clauses for $\exists,\forall$. On the other hand for atomic
$\mathcal{L}_{\mathcal{M}}$-formulae the condition expressing fullness is
automatic by definition. The unique delicate case occurs for
$\mathcal{L}_{\mathcal{M}}$-formulae in which the principal connective is a
quantifier, for example of the form $\exists x\phi(x)$, in which case the
fullness conditions for the formula amounts to ask that
$\sup_{\sigma\in\mathcal{M}}\left\llbracket\phi(\sigma)\right\rrbracket^{\mathcal{M}}$
is actually a finite supremum, as we will see. We will investigate this
property in many details in the last two sections of this paper. For the
moment letting a $\mathcal{L}_{\mathcal{M}}$-formula $\phi(x_{1},\dots,x_{n})$
be _$F$ -full_ for a filter $F$ if and only if

$\left\llbracket\phi([\tau_{1}]_{G},\dots,[\tau_{n}]_{G})\right\rrbracket^{\mathcal{M}/_{F}}=1_{\mathsf{B}/_{F}}\qquad\Longleftrightarrow\qquad\left\llbracket\phi(\tau_{1},\dots,\tau_{n})\right\rrbracket^{\mathcal{M}}\in
F,$

we automatically have that the family of $F$-full
$\mathcal{L}_{\mathcal{M}}$-formulae includes the atomic formulae and is
closed under conjuctions for any filter $F$, and under boolean combinations
for any ultrafilter $F$. Another key observation is that if $F$ is a filter
and

$\bigvee_{i=1}^{m}\left\llbracket\phi(\sigma_{i})\right\rrbracket^{\mathcal{M}}=\left\llbracket\exists
x\,\phi(x)\right\rrbracket^{\mathcal{M}}=\bigvee_{\tau\in\mathcal{M}}\left\llbracket\phi(\tau)\right\rrbracket^{\mathcal{M}},$

then

$\bigvee_{i=1}^{m}\left\llbracket\phi([\sigma_{i}]_{F})\right\rrbracket^{\mathcal{M}/_{F}}=\left\llbracket\exists
x\,\phi(x)\right\rrbracket^{\mathcal{M}/_{F}},$

since

$\bigvee_{i=1}^{m}[\left\llbracket\phi(\sigma_{i})\right\rrbracket^{\mathcal{M}}]_{F}=\Bigl{[}\bigvee_{i=1}^{m}\left\llbracket\phi(\sigma_{i})\right\rrbracket^{\mathcal{M}}\Bigr{]}_{F}\geq_{F}[\left\llbracket\phi(\tau)\right\rrbracket^{\mathcal{M}}]_{F}$

holds for all $\tau\in\mathcal{M}$.The notion of full $\mathsf{B}$-valued
model displays its full power in the following result:

######  Theorem 4.11 (Łoś Theorem for boolean valued models).

Let $\mathcal{M}$ be a well behaved $\mathsf{B}$-valued model for the
signature $\mathcal{L}$. The following are equivalent:

  * •

$\mathcal{M}$ is full.

  * •

for all $\mathcal{L}_{\mathcal{M}}$-formulae $\phi(x_{0},\dots,x_{n})$ and all
$\tau_{1},\dots,\tau_{n}\in\mathcal{M}$ there exist
$\sigma_{1},\dots,\sigma_{m}\in\mathcal{M}$ such that

$\bigvee_{\tau\in\mathcal{M}}\left\llbracket\phi(\tau,\tau_{1},\dots,\tau_{n})\right\rrbracket=\bigvee_{i=1}^{m}\left\llbracket\phi(\sigma_{i},\tau_{1},\dots,\tau_{n})\right\rrbracket.$

###### Proof.