# Étalé spaces of residuated lattices

Saeed Rasouli1∗ and Seyed Naser Hosseini2 and Amin Dehghani3
${}^{1},^{3}$Department of Mathematics, Persian Gulf University, Bushehr, Iran
2Department of mathematics, Shahid Bahonar University, Kerman,Iran
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###### Abstract

This paper explores the concept of étalé spaces associated with residuated
lattices. Notions of bundles and étalés of residuated lattices over a given
topological space are introduced and investigated. For a topological space
$\mathscr{B}$, we establish that the category of étalés of residuated lattices
over $\mathscr{B}$ with morphisms of étalés of residuated lattices is
coreflective in the category of bundles of residuated lattices over
$\mathscr{B}$ along with morphisms of bundles of residuated lattices. We
provide a method for transferring an étalé of residuated lattices over a
topological space to another, utilizing a continuous map. Finally, we define a
contravariant functor, called the section functor, from the category of étalés
of residuated lattices with inverse morphisms to the category of residuated
lattices. 1112020 Mathematics Subject Classification: 06F99,55N30
Key words and phrases: sheaf of residuated lattices; étalé space of residuated
lattice; bundle of residuated lattice.
∗: Corresponding author

##  1 Introduction

In the realm of information processing, particularly inferences derived from
certain information, classical logic serves as a foundational system. However,
to address the challenges posed by uncertainty and fuzzy information, diverse
non-classical logic systems have been extensively proposed and researched
(Turunen and Turunen, 1999; Galatos et al., 2007; Hájek, 2013). These non-
classical logics have proven indispensable in computer science, offering
formal and effective tools for managing uncertain and fuzzy data.

Several logical algebras have been introduced as semantical frameworks to
support non-classical logic systems. Examples of such algebras include BCK-
algebras, residuated lattices, divisible residuated lattices, MTL-algebras,
Girard monoids, BL-algebras, Gödel algebras, MV-algebras, and Heyting
algebras. Residuated lattices, in particular, are fundamental and crucial
algebraic structures that play an essential role in the theory of fuzzy logic.
In the context of residuated lattices, there is an additional binary operation
$\to$ (often called the residual or implication) that relates to the lattice
structure. This operation in a residuated lattice aligns with the idea of
implication in fuzzy logic, where the degree of truth of an implication is
related to the degree of membership of the antecedent and consequent. Indeed,
the connection between residuated lattices and fuzzy sets is based on their
ability to represent and manage uncertainty and reasoning.

Residuated lattices, on the other hand, are not merely important from a
logical standpoint; they also hold intrigue from an algebraic perspective,
boasting noteworthy properties. Their versatility and foundational nature make
them pivotal in reasoning about uncertain information, contributing
significantly to both the logical and algebraic aspects of fuzzy logic theory.

This paper delves into the exploration of étalé spaces associated with
residuated lattices. Historically, the foundational developments leading to
the introduction of étalé spaces can be traced back to the seminal
contributions of the Séminaire Henri Cartan (1948-1964). In collaboration with
eminent mathematicians such as Jean-Pierre Serre and Jean Leray, Henri Cartan
played a pivotal role in pioneering innovative concepts and techniques in
topology and algebraic geometry throughout this seminar. To be precise, it can
be said that the notion of an “étalé space” or “Espace étalé” emanating from
the realm of algebraic geometry, was born by Henri Cartan, commonly ascribed
to his collaborative efforts with Michel Lazard, exactly in Exposé 14 of this
seminar, under the term “faisceau” (Cartan, 1950). According to Gray (1979, p.
7), Lazzard’s faisceau on a regular topological space $X$ is defined by a map
$p$ from a set $F$ to $X$, satisfying two conditions:

  1. 1.

for each $x\in X$, $p^{-1}(x)=F_{x}$, is a $K$-module;

  2. 2.

$F$ has a (non-separated) topology such that the algebraic operations of $F$
are continuous, and $p$ is a local homeomorphism.

Roger Godement, a notable participant in Henri Cartan’s Séminaire, is credited
with introducing the term “Espace étalé” instead of “faisceau” in his
influential work (Godement, 1958). His elegant contribution encompasses a
comprehensive survey of étalé spaces. Godement’s exposition played a pivotal
role in establishing the formalization and examination of étalé spaces,
exerting a substantial influence on the evolution of algebraic geometry.

The notion of étalé spaces has been extensively studied during past decades,
see e.g., (Grothendieck, 1960; Tennison, 1976; Bredon, 1997; MacLane and
Moerdijk, 2012; Brown et al., 2020), and (Rosiak, 2022). For more historical
notes about étalé spaces, the interested reader can be referred to (Gray,
1979; Fasanelli, 1981; Miller, 2000).

Over the years, étalé spaces have evolved into a fundamental component of
modern algebraic geometry, playing a pivotal role in the formulation of
schemes—an essential concept in Grothendieck’s reformulation of algebraic
geometry. Presently, the theory of étalé spaces remains a dynamic and fertile
area of research, establishing crucial connections between algebraic geometry
and topology, and catalyzing advancements across diverse mathematical
disciplines.

The influence of étalé spaces has extended beyond their original domain in
algebraic geometry, reaching into various branches of mathematics, including
number theory and representation theory. Furthermore, the integration of étalé
spaces into Fuzzy Sheaf Theory has proven to be particularly fruitful,
offering valuable insights into challenges related to spatial reasoning,
knowledge representation, and fuzzy image analysis.

Inspired by the foundational notion of étalé spaces, numerous authors have
proposed analogous concepts under different names for various structures
throughout the years. Examples include sheaves of lattices (Brezuleanu and
Diaconescu, 1969), sheaves of rings, $R$-modules, and abelian groups (Hofmann,
1972), sheaf spaces of universal algebras (Davey, 1973), sheaves of chains
over Boolean spaces (Cignoli, 1978), sheaves of normal lattices (Georgescu and
Voiculescu, 1987), sheaves of MV-algebras (Filipoiu and Georgescu, 1995),
sheaves of BL-algebras (Di Nola and Leuştean, 2003), sheaves of Banach spaces
(Hofmann and Keimel, 2006), among others. This diversity of applications
reflects the enduring impact and versatility of the étalé spaces concept
across various mathematical structures and frameworks.

The above discussion motivates us to introduce and investigate the notion of
étalé spaces of residuated lattices in a similar fashion to étalé spaces of
rings and modules. This motivation is to find a common pattern for some facts
about étalé spaces of rings and étalé spaces of residuated lattices which
occur in a much similar way in both cases of rings and of residuated lattices.
Also, as known, étalé spaces arise from the theory of sheaves, a foundational
framework that has significantly advanced our understanding of cohomology and
the topology of algebraic varieties. While the majority of the aforementioned
explorations delve into étalé spaces through the lens of sheaves, our approach
takes a different approach by avoiding their use.

This paper is organized into five sections as follows: In Sec. 2, we revisit
essential definitions, properties, and results related to residuated lattices,
laying the groundwork for subsequent discussions. Sec. 3 delves into the
concept of bundles and étalés of residuated lattices over a given topological
space, recalling specific definitions of étalés of sets that will be utilized
in our exploration. For a topological space $\mathscr{B}$, we establish that
the category of étalés of residuated lattices over $\mathscr{B}$ with
morphisms of étalés of residuated lattices, denoted as $\textbf{RL-
Etale}(\mathscr{B})$, forms a full subcategory of bundles of residuated
lattices over $\mathscr{B}$ along with morphisms of bundles of residuated
lattices, denoted as $\textbf{RL-Bundle}(\mathscr{B})$ (Corollary 3.27). Sect.
4 investigates the interrelation between bundles and étalés of residuated
lattices. As a main result, it sharpens the previous result and demonstrates
that given a topological space $\mathscr{B}$, $\textbf{RL-Etale}(\mathscr{B})$
is coreflective in $\textbf{RL-Bundle}(\mathscr{B})$ (Theorem 4.13). Sect. 5
provides a method for transferring an étalé from one topological space to
another, utilizing a continuous map. At the end of this section, we define a
contravariant functor, called the section functor, from the category of étalés
of residuated lattices with inverse morphisms to the category of residuated
lattices (Theorem 5.8).

##  2 Residuated lattices

In this section, we revisit essential definitions, properties, and results
related to residuated lattices, providing a foundation for subsequent
discussions.

An algebra $\mathfrak{A}=(A;\vee,\wedge,\odot,\rightarrow,0,1)$ is termed a
residuated lattice if $\ell(\mathfrak{A})=(A;\vee,\wedge,0,1)$ is a bounded
lattice, $(A;\odot,1)$ is a commutative monoid, and $(\odot,\rightarrow)$
forms an adjoint pair. A residuated lattice $\mathfrak{A}$ is considered non-
degenerate if $0\neq 1$. For a residuated lattice $\mathfrak{A}$ and $a\in A$,
we define $\neg a:=a\rightarrow 0$, and $a^{n}:=a\odot\ldots\odot a$ ($n$
times) for every integer $n$. The class of residuated lattices, denoted as RL,
is equational and, therefore, a variety. For an in-depth exploration of
residuated lattices, interested readers are directed to Galatos et al. (2007).

######  Example 2.1.

Consider the lattice $A_{4}=\\{0,a,b,1\\}$ with the Hasse diagram shown in
Figure 1. Routine calculations reveal that
$\mathfrak{A}_{4}=(A_{4};\vee,\wedge,\odot,\rightarrow,0,1)$ is a residuated
lattice. The commutative operation $\odot$ is defined by Table 1, and the
operation $\rightarrow$ is articulated by $x\rightarrow y=\bigvee\\{z\in
A_{4}\mid x\odot z\leq y\\}$, for any $x,y\in A_{4}$.

$\odot$ | 0 | a | b | 1
---|---|---|---|---
0 | 0 | 0 | 0 | 0
| a | a | 0 | a
|  | b | b | b
|  |  | 1 | 1
Table 1: Cayley table for “$\odot$” of $\mathfrak{A}_{4}$

0ab1 Figure 1: Hasse diagram of $\mathfrak{A}_{4}$

######  Example 2.2.

Consider the lattice $A_{6}=\\{0,a,b,c,d,1\\}$ with the Hasse diagram shown in
Figure 2. Routine calculations reveal that
$\mathfrak{A}_{6}=(A_{6};\vee,\wedge,\odot,\rightarrow,0,1)$ is a residuated
lattice. The commutative operation $\odot$ is defined by Table 2, and the
operation $\rightarrow$ is articulated by $x\rightarrow y=\bigvee\\{z\in
A_{6}\mid x\odot z\leq y\\}$, for any $x,y\in A_{6}$.

$\odot$ | 0 | a | b | c | d | 1
---|---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | 0 | 0
| a | a | a | 0 | a | a
|  | b | a | 0 | a | b
|  |  | c | c | c | c
|  |  |  | d | d | d
|  |  |  |  | 1 | 1
Table 2: Cayley table for “$\odot$” of $\mathfrak{A}_{6}$

0cabd1 Figure 2: Hasse diagram of $\mathfrak{A}_{6}$

######  Example 2.3.

Consider the lattice $A_{8}=\\{0,a,b,c,d,e,f,1\\}$ with the Hasse diagram
shown in Figure 3. Routine calculations reveal that
$\mathfrak{A}_{8}=(A_{8};\vee,\wedge,\odot,\rightarrow,0,1)$ is a residuated
lattice. The commutative operation $\odot$ is defined by Table 3, and the
operation $\rightarrow$ is articulated by $x\rightarrow y=\bigvee\\{z\in
A_{8}\mid x\odot z\leq y\\}$, for any $x,y\in A_{8}$.

$\odot$ | 0 | a | b | c | d | e | f | 1
---|---|---|---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| a | 0 | a | a | a | a | a | a
|  | b | 0 | 0 | 0 | 0 | b | b
|  |  | c | c | a | c | a | c
|  |  |  | d | a | a | d | d
|  |  |  |  | e | c | d | e
|  |  |  |  |  | f | f | f
|  |  |  |  |  |  | 1 | 1
Table 3: Cayley table for “$\odot$” of $\mathfrak{A}_{8}$

0abcdef1 Figure 3: Hasse diagram of $\mathfrak{A}_{8}$

######  Example 2.4.

Let $n$ be a fixed natural number. By Turunen and Turunen (1999, p. 11,
Example 2) follows that $\L=([0,1];\max,\min,\odot,\rightarrow,0,1)$ is a
residuated lattice, called generalized Łukasiewicz structure, in which the
commutative operation $``\odot"$ is given by $x\odot
y=(\max\\{0,x^{n}+y^{n}-1\\})^{\frac{1}{n}}$, and the operation
$``\rightarrow"$ is given by $x\rightarrow
y=\min\\{1,(1-x^{n}+y^{n})^{\frac{1}{n}}\\}$, for all $x,y\in[0,1]$.

Let $\mathfrak{A}$ be a residuated lattice. A non-void subset $F$ of $A$ is
called a filter of $\mathfrak{A}$ provided that $x,y\in F$ implies $x\odot
y\in F$, and $x\vee y\in F$, for any $x\in F$ and $y\in A$. The set of filters
of $\mathfrak{A}$ is denoted by $\mathscr{F}(\mathfrak{A})$. A filter $F$ of
$\mathfrak{A}$ is called proper if $F\neq A$. Clearly, $F$ is proper iff
$0\notin F$. For any subset $X$ of $A$, the filter of $\mathfrak{A}$ generated
by $X$ is denoted by $\mathscr{F}(X)$. For each $x\in A$, the filter generated
by $\\{x\\}$ is denoted by $\mathscr{F}(x)$ and said to be principal. The set
of principal filters is denoted by $\mathscr{PF}(\mathfrak{A})$. Following
Grätzer (2011, §5.7), a join-complete lattice $\mathfrak{A}$, is called a
frame if it satisfies the join infinite distributive law (JID), i.e., for any
$a\in A$ and $S\subseteq A$, $a\wedge\bigvee S=\bigvee\\{a\wedge s\mid s\in
S\\}$. According to Galatos et al. (2007),
$(\mathscr{F}(\mathfrak{A});\cap,\veebar,\textbf{1},A)$ is a frame in which
$\veebar\mathcal{F}=\mathscr{F}(\cup\mathcal{F})$, for any
$\mathcal{F}\subseteq\mathscr{F}(\mathfrak{A})$. With any filter $F$ of a
residuated lattice $\mathfrak{A}$, we can associate a binary relation
$\equiv_{F}$ on $A$ as follows; $x\equiv_{F}y$ if and only if $x\rightarrow
y,y\rightarrow x\in F$, for any $x,y\in A$. The binary relation $\equiv_{F}$
is a congruence on $\mathfrak{A}$, and called the congruence induced by $F$ on
$\mathfrak{A}$. As usual, the set of all congruences on $\mathfrak{A}$ shall
be denoted by $Con(\mathfrak{A})$. It is well-known that
$\phi:\mathscr{F}(\mathfrak{A})\longrightarrow Con(\mathfrak{A})$, defined by
$\phi(F)=\equiv_{F}$, is a lattice isomorphism, and this implies that
$\mathbf{RL}$ is an ideal determined variety. For a filter $F$ of a residuated
lattice $\mathfrak{A}$, the quotient set $A/\equiv_{F}$ with the natural
operations becomes a residuated lattice which is denoted by $\mathfrak{A}/F$.
Also, for any $a\in A$, the equivalence classes $a/\equiv_{F}$ is denoted by
$a/F$.

######  Example 2.5.

Consider the residuated lattice $\mathfrak{A}_{4}$ from Example 2.1, the
residuated lattice $\mathfrak{A}_{6}$ from Example 2.2, and the residuated
lattice $\mathfrak{A}_{8}$ from Example 2.3. The sets of their filters are
presented in Table 4.

| Filters
---|---
$\mathfrak{A}_{4}$ | $F_{1}=\\{1\\},F_{2}=\\{a,1\\},F_{3}=\\{b,1\\},F_{4}=A_{4}$
$\mathfrak{A}_{6}$ | $F_{1}=\\{1\\},F_{2}=\\{a,b,d,1\\},F_{3}=\\{c,d,1\\},F_{4}=\\{d,1\\},F_{5}=A_{6}$
$\mathfrak{A}_{8}$ | $F_{1}=\\{1\\},F_{2}=\\{a,c,d,e,f,1\\},F_{3}=\\{c,e,1\\},F_{4}=\\{f,1\\},F_{5}=A_{8}$
Table 4: The sets of filters of $\mathfrak{A}_{4}$, $\mathfrak{A}_{6}$, and
$\mathfrak{A}_{8}$

Let $\mathfrak{A}$ and $\mathfrak{B}$ be residuated lattices. A map
$f:A\longrightarrow B$ is called a morphism of residuated lattices, in symbols
$f:\mathfrak{A}\longrightarrow\mathfrak{B}$, if it preserves the fundamental
operations. If $f:\mathfrak{A}\longrightarrow\mathfrak{B}$ is a morphism of
residuated lattices, we put $coker(f)=f^{\leftarrow}(1)$. One can see that $f$
is a injective if and only if $coker(f)=\\{1\\}$.

######  Example 2.6.

Consider the residuated lattice $\mathfrak{A}_{4}$ from Example 2.1, the
residuated lattice $\mathfrak{A}_{6}$ from Example 2.2. One can see that the
map $f:A_{6}\longrightarrow A_{4}$, given by $f(0)=0$, $f(a)=f(b)=a$, $f(c)=b$
and $f(d)=f(1)=1$ is a morphism of residuated lattices.

A proper filter in a residuated lattice $\mathfrak{A}$ is termed maximal if it
is a maximal element within the set of all proper filters of $\mathfrak{A}$.
The collection of maximal filters of $\mathfrak{A}$ is denoted by
$Max(\mathfrak{A})$. Additionally, a proper filter $\mathfrak{p}$ in
$\mathfrak{A}$ is prime if $x\vee y\in\mathfrak{p}$ implies $x\in\mathfrak{p}$
or $y\in\mathfrak{p}$, for any $x,y\in A$. The set of prime filters of
$\mathfrak{A}$ is denoted by $Spec(\mathfrak{A})$. Given the distributive
lattice property of $\mathscr{F}(\mathfrak{A})$, it follows that
$Max(\mathfrak{A})\subseteq Spec(\mathfrak{A})$. Zorn’s lemma establishes that
any proper filter is contained within a maximal filter and, consequently,
within a prime filter. Minimal prime filters, identified as minimal elements
within the set of prime filters, form the set $Min(\mathfrak{A})$. For a
detailed exploration of prime filters in a residuated lattice, refer to
Rasouli (2019).

######  Example 2.7.

Consider the residuated lattice $\mathfrak{A}_{4}$ from Example 2.1, the
residuated lattice $\mathfrak{A}_{6}$ from Example 2.2, and the residuated
lattice $\mathfrak{A}_{8}$ from Example 2.3. The sets of their maximal, prime,
and minimal prime filters are presented in Table 6.

| Prime filters
---|---
| Maximal filters |  | Minimal prime filters
$\mathfrak{A}_{4}$ | $F_{2},F_{3}$ |  | $F_{2},F_{3}$
$\mathfrak{A}_{6}$ | $F_{2},F_{3}$ |  | $F_{1}$
$\mathfrak{A}_{8}$ | $F_{2}$ |  | $F_{3},F_{4}$
Table 5: The sets of maximal, prime, and minimal prime filters of
$\mathfrak{A}_{4}$, $\mathfrak{A}_{6}$, and $\mathfrak{A}_{8}$

Let $\mathfrak{A}$ be a residuated lattice, and $\Pi$ a collection of prime
filters of $\mathfrak{A}$. For a subset $\pi$ of $\Pi$, we define
$k(\pi)=\bigcap\pi$. Additionally, for a subset $X$ of $A$, we set
$h_{\Pi}(X)=\\{P\in\Pi\mid X\subseteq P\\}$ and $d_{\Pi}(X)=\Pi\setminus
h_{\Pi}(X)$.

The collection $\Pi$ can be topologized by considering the collection
$\\{h_{\Pi}(x)\mid x\in A\\}$ as a closed (or open) basis. This topological
structure is referred to as the (dual) hull-kernel topology on $\Pi$ and is
denoted by $\Pi_{h(d)}$. Moreover, the topology generated by
$\tau_{h}\cup\tau_{d}$ on $\text{Spec}(\mathfrak{A})$ is known as the patch
topology and is denoted by $\tau_{p}$.

For a detailed discussion of the (dual) hull-kernel and patch topologies on a
residuated lattice, we refer to Rasouli and Dehghani (2021).

To clarify, we provide some examples of topological spaces associated with
residuated lattices.

######  Example 2.8.

Consider the residuated lattice $\mathfrak{A}_{4}$ from Example 2.1, the
residuated lattice $\mathfrak{A}_{6}$ from Example 2.2, and the residuated
lattice $\mathfrak{A}_{8}$ from Example 2.3. Table 6 presents some topological
spaces related to these structures.

| Topological spaces
---|---
$Spec_{h}(\mathfrak{A}_{4})$ | $\\{\emptyset,\\{F_{2}\\},\\{F_{3}\\},\\{F_{2},F_{3}\\}\\}$
$Max_{d}(\mathfrak{A}_{6})$ | $\\{\emptyset,\\{F_{2}\\},\\{F_{3}\\},\\{F_{2},F_{3}\\}\\}$
$Min_{p}(\mathfrak{A}_{8})$ | $\\{\emptyset,\\{F_{3}\\},\\{F_{4}\\},\\{F_{3},F_{4}\\},\\{F_{2},F_{3},F_{4}\\}\\}$
Table 6: Some topological spaces associated with $\mathfrak{A}_{4}$,
$\mathfrak{A}_{6}$, and $\mathfrak{A}_{8}$

##  3 Étalé spaces of residuated lattices over a topological space

This section introduces and investigates the notion of étalés of residuated
lattices. We recall specific definitions of étalés of sets that will be used
in our exploration. For the sake of completeness, we prove some properties of
étalés of sets.

######  Definition 3.1.

(Engelking, 1989, §4.4) A map $f:\mathscr{T}\longrightarrow\mathscr{B}$ is
called a local homeomorphism provided that for every point in $\mathscr{T}$
there exists a neighbourhood $U$ of this point such that the restriction
$f|_{U}$ is a homeomorphism onto an open set.

######  Lemma 3.2.

A local homeomorphism is continuous.

###### Proof.

Let $f:\mathscr{T}\longrightarrow\mathscr{B}$ be a local homeomorphism and $V$
an open subset of $\mathscr{B}$. Consider $t\in f^{\leftarrow}(V)$. There
exists a neighborhood $U$ of $t$ such that $f|_{U}$ is a homeomorphism.
Evidently, $(f|_{U})^{\leftarrow}(V\cap f(U))$ is a neighborhood of $t$ in
$f^{\leftarrow}(V)$. ∎

######  Lemma 3.3.

A local homeomorphism is open.

###### Proof.

Let $f:\mathscr{T}\longrightarrow\mathscr{B}$ be a local homeomorphism. Assume
that $W$ is an open subset of $\mathscr{T}$. Consider $b\in f(W)$. Pick $t\in
f^{\leftarrow}(b)$. There exists a neighborhood $U$ of $t$ such that $f|_{U}$
is a homeomorphism. Evidently, $f|_{W}(U\cap W)$ is a neighborhood of $y$ in
$f(U)$. ∎

Recall that if the domain of a map $f:\mathscr{T}\longrightarrow Y$ is a
topological space, then the function $f$ is said to be locally injective
provided that for every point in $\mathscr{T}$ there exists a neighbourhood
$U$ of this point such that the restriction $f|_{U}$ is injective.

######  Lemma 3.4.

A local homeomorphism is locally injective.

###### Proof.

There is nothing to prove. ∎

######  Theorem 3.5.

A map is a local homeomorphism if and only if it is continuous, open, and
locally injective. In particular, a bijective local homeomorphism is a
homeomorphism.

###### Proof.

It is an immediate consequence of Lemmas 3.2, 3.3 and 3.4. ∎

######  Proposition 3.6.

Let $f:\mathscr{T}\longrightarrow\mathscr{B}$ be a local homeomorphism. The
family $\\{V\mid
V~{}\textrm{is~{}open~{}in}~{}\mathscr{T},f|_{V}~{}\textrm{is~{}a~{}homeomorphism}\\}$
is a basis for the topology of $\mathscr{T}$.

###### Proof.

Let $U$ be an open set in $\mathscr{T}$. Consider $x\in U$. There exists a
neighborhood $W$ of $x$ such that $f|_{W}$ is a homeomorphism. Let $V=U\cap
W$. It is easy to check that $f|_{V}$ is a homeomorphism and $x\in V\subseteq
U$. ∎

######  Lemma 3.7.

  1. $(1)$

The restriction of a local homeomorphism to a topological subspace is a local
homeomorphism;

  2. $(2)$

The composition of two local homeomorphisms is a local homeomorphism.

###### Proof.

It is evident. ∎

Let $\mathcal{C}$ be a category and $c$ an object of $\mathcal{C}$. Recall
that the slice category (an especial type of comma category (Lane, 1971, §6 of
Chapter II )) of $\mathcal{C}$ over $c$, denoted by $\mathcal{C}/c$, is the
category whose objects are all pairs $(a,f)$, where $a$ is an object of
$\mathcal{C}$ and $f$ is a morphism from $a$ to $c$, and whose morphisms from
$(a,f)$ to $(b,g)$ given by morphisms $h$ of $\mathcal{C}$ such that $gh=f$
(See Fig. 4). If $\mathcal{C}/c$ is a slice category, $c$ is known as the base
space of the category $\mathcal{C}/c$.

${a}$${b}$${c}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{g}$ Figure 4:
Morphism $h$ of the slice category $\mathcal{C}/c$

Let Top ($\textbf{Top}_{\ell}$) denote the category of topological spaces with
continuous maps (local homeomorphisms) as morphisms. Given a topological space
$\mathscr{B}$, the elements of the category
$\mathbf{Bundle}(\mathscr{B})=\textbf{Top}/\mathscr{B}$ are referred to as
bundles over $\mathscr{B}$ (see MacLane and Moerdijk (2012, §2.4)), while the
elements of the category
$\mathbf{Etale}(\mathscr{B})=\textbf{Top}_{\ell}/\mathscr{B}$ are known as
étalés over $\mathscr{B}$. A morphism between bundles (étalés) over
$\mathscr{B}$ is denoted as a morphism of bundles (étalés) over $\mathscr{B}$.

If $(\mathscr{T},\pi)$ is an étalé over $\mathscr{B}$, $\mathscr{T}$ is known
as the total space, and $\pi$ as the étalé projection. For a given
$b\in\mathscr{B}$, the inverse image $\pi^{\leftarrow}(b)$ is referred to as
the stalk of $\mathscr{T}$ at $b$ and denoted by $\mathscr{T}_{b}$. In the
sequel, a bundle (étalé) $(\mathscr{T},\pi)$ over $\mathscr{B}$ shall be
denoted by $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ and simply called
a bundle (an étalé). A morphism of bundles (étalés)
$h:(\mathscr{T}_{1},\pi_{1})\to(\mathscr{T}_{2},\pi_{2})$ over $\mathscr{B}$
shall be denoted by
$h:\mathscr{T}_{1}\overset{\pi_{1}}{\downarrow}\mathscr{B}\to\mathscr{T}_{2}\overset{\pi_{2}}{\downarrow}\mathscr{B}$
and simply called a morphism of bundles (étalés).

The following proposition, which is restated the lemma 3.5 of (Tennison, 1976,
§2) with more detailed proof, establishes that the morphisms of bundles and
étalés over a topological space are the same.

######  Proposition 3.8.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ and
$\mathscr{S}\overset{\phi}{\downarrow}\mathscr{B}$ be two étalés. If
$h:\mathscr{T}\longrightarrow\mathscr{S}$ is a map such that $\phi h=\pi$, the
following are equivalent:

  1. $(1)$

$h$ is a local homeomorphism;

  2. $(2)$

$h$ is open;

  3. $(3)$

$h$ is continuous.

###### Proof.

It is evident by Lemma 3.3.

Applying Proposition 3.6, let $V$ be a basic open set in $\mathscr{S}$ such
that $\phi|_{V}$ is a homeomorphism. Let $x\in h^{\leftarrow}(V)$. So
$\pi(x)\in\phi(V)$. One an find a neighborhood $U$ of $x$ such that $\pi|_{U}$
is a homeomorphism and $\pi(U)\subseteq\phi(V)$. Let
$W=h(U)\cap\phi|^{\leftarrow}_{V}(\pi(U))$. Consider
$t\in\pi|^{\leftarrow}_{U}(\phi|_{V}(W))$. There exist $w\in W$ and $u\in U$
such that $\pi(u)=\phi(h(u))=\phi(w)=\pi(t)$, and this implies that $u=t$.
This states that $t\in h^{\leftarrow}(V)$.

Let $t\in\mathscr{T}$. There exist neighborhoods $U$ and $V$, respectively, of
$t$ and $h(t)$ such that $\pi|_{U}$ and $\phi|_{V}$ are homeomorphisms. By
continuity of $h$, $W=U\cap h^{\leftarrow}(V)$ is a neighborhood of $t$. Let
$h(t_{1})=h(t_{2})$, for some $t_{1},t_{2}\in W$. So we have
$\pi(t_{1})=\phi(h(t_{1}))=\phi(h(t_{2}))=\pi(t_{2})$ and this states that
$t_{1}=t_{2}$, since $\pi|_{U}$ is injective. Let $O$ be an open subset in
$W$. So we have
$h(O)=(\phi|_{V})^{\leftarrow}(\phi(h(O)))=(\phi|_{V})^{\leftarrow}(\pi(h(O)))$
and this shows that $h(O)$ is open. ∎

######  Corollary 3.9.

Given a topological space $\mathscr{B}$, $\mathbf{Etale}(\mathscr{B})$ is a
full subcategory of $\mathbf{Bundle}(\mathscr{B})$.

###### Proof.

Evident by Proposition 3.8. ∎

######  Definition 3.10.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé. A
continuous map $\sigma:X\longrightarrow\mathscr{T}$, where $X$ is a subset of
$\mathscr{B}$, is referred to as a section of $\mathscr{T}$ over $X$ if
$\pi\sigma=1$. The set of all sections of $\mathscr{T}$ over $X$ is denoted by
$\Gamma(X,\mathscr{T})$. Elements of $\Gamma(\mathscr{B},\mathscr{T})$ are
termed global sections of $\mathscr{T}$.

######  Lemma 3.11.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé, and
$t\in\mathscr{T}$. There exist a neighborhood $U$ of $\pi(t)$ and a section
$\sigma$ of $\mathscr{T}$ over $U$ such that $\sigma\pi(t)=t$.

###### Proof.

There exists a neighborhood $V$ of $t$ such that $\pi|_{V}$ is a
homeomorphism. Take $U=\pi(V)$ and define $\sigma:U\longrightarrow\mathscr{T}$
by $\sigma(x)=(\pi|_{V})^{\leftarrow}(x)$, for all $x\in U$. ∎

######  Lemma 3.12.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé, $\sigma$ a
section of $\mathscr{T}$ over an open subset of $\mathscr{B}$, $x$ an element
of the domain $\sigma$, and $V$ a neighborhood of $\sigma(x)$ such that
$\pi|_{V}$ is a homeomorphism. There exists a neighborhood $U$ of $x$ such
that $\sigma(U)\subseteq V$ and $\sigma|_{U}=(\pi|_{V})^{\leftarrow}|_{U}$.

###### Proof.

By hypothesis $U=\sigma^{\leftarrow}(V)$ is a neighborhood of $x$ in
$\mathscr{B}$. Obviously, $\sigma(U)\subseteq V$. Also, we have
$(\pi|_{V})^{\leftarrow}|_{U}(x)=(\pi|_{V})^{\leftarrow}|_{U}(\pi\sigma(x))=(\pi|_{V})^{\leftarrow}(\pi\sigma(x))=\sigma(x)$,
for any $x\in U$. ∎

Let’s note that if $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ is an
étalé, $\sigma:X\longrightarrow\mathscr{T}$ is a section with
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathscr{B}}$
and
$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{X}$
is a subspace of $X$, then $\sigma k:Z\longrightarrow\mathscr{T}$ is a
section. We sometimes write $\sigma|_{Z}$ for the composition $\sigma k$.

If $X,Y\subseteq\mathscr{B}$, and $\sigma\in\Gamma(X,\mathscr{T})$ and
$\tau\in\Gamma(Y,\mathscr{T})$, the equalizer of $\sigma|_{X\cap Y}$ and
$\tau|_{X\cap Y}$ in the category Top is denoted, for simplicity, by
$\textsc{Eq}(\sigma,\tau)$. This equalizer is defined as the set $\\{b\in
X\cap Y\mid\sigma(b)=\tau(b)\\}$ equipped with the subspace topology. By a
slight abuse of language, we refer to it as the equalizer of $\sigma$ and
$\tau$.

######  Proposition 3.13.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé. The
following hold:

  1. $(1)$

If $U$ and $V$ are open sets in $\mathscr{B}$,
$\sigma\in\Gamma(U,\mathscr{T})$ and $\tau\in\Gamma(V,\mathscr{T})$, then
$\textsc{Eq}(\sigma,\tau)$ is open;

  2. $(2)$

if $\mathscr{T}$ is Hausdorff, $U$ and $V$ are open sets in $\mathscr{B}$,
$\sigma\in\Gamma(U,\mathscr{T})$ and $\tau\in\Gamma(V,\mathscr{T})$, then
$\textsc{Eq}(\sigma,\tau)$ is clopen;

  3. $(3)$

Each section of $\mathscr{T}$, defined over an open set in $\mathscr{B}$, is
open.

###### Proof.

Let $b\in\textsc{Eq}(\sigma,\tau)$. Assume that $W$ is a neighborhood of
$\sigma(b)$ such that $\pi|_{W}$ is a homeomorphism. Applying Lemma 3.12,
there exist neighborhoods $U_{0}$ and $V_{0}$ of $b$ such that
$\sigma(U_{0}),\tau(V_{0})\subseteq W$. One can see that $U_{0}\cap V_{0}$ is
a neighborhood of $b$ contained in $\textsc{Eq}(\sigma,\tau)$.

It follows by $(1)$ and Willard (2012, Theorem 13.13).

Let $U$ be an open set in $\mathscr{B}$ and $\sigma\in\Gamma(U,\mathscr{T})$.
Consider $t\in\sigma(U)$. So $t=\sigma(b)$, for some $b\in U$. Let $W$ be a
neighborhood $t$ such that $\pi|_{W}$ is a homeomorphism. By Lemma 3.12, there
exists a neighborhood $V$ of $b$ such that $\sigma(V)\subseteq W$ and
$\sigma|_{V}=(\pi|_{W})^{\leftarrow}|_{V}$. So
$\sigma(V)=(\pi|_{W})^{\leftarrow}(V)$ is a neighborhood of $t$ contained in
$\sigma(U)$. ∎

######  Proposition 3.14.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé. The family

$\\{\sigma(U)\mid
U~{}\textrm{is~{}open~{}in}~{}\mathscr{B},\sigma\in\Gamma(U,\mathscr{T})\\}$

is a basis for the topology of $\mathscr{T}$.

###### Proof.

Let $V$ be an open set in $\mathscr{T}$ such that $\pi|_{V}$ is a
homeomorphism. Take $U=\pi(V)$ and define $\sigma:U\longrightarrow\mathscr{T}$
by $\sigma(x)=(\pi|_{V})^{\leftarrow}(x)$, for all $x\in U$. It is evident
that $\sigma\in\Gamma(U,\mathscr{T})$ and $V=\sigma(U)$. So $\\{V\mid
V~{}\textrm{is~{}Open~{}in}~{}\mathscr{T},\pi|_{V}~{}\textrm{is~{}a~{}homeomorphism}\\}\subseteq\\{\sigma(U)\mid
U~{}\textrm{is~{}Open~{}in}~{}\mathscr{B},\sigma\in\Gamma(U,\mathscr{T})\\}$
and so the result holds due to Proposition 3.6. ∎

Let $X$ be a set, $\\{Y_{i}\\}_{i\in I}$ be a collection of topological
spaces, and $\\{f_{i}:Y_{i}\longrightarrow X\\}_{i\in I}$ be a collection of
maps. The final topology induced by $\\{f_{i}\\}_{i\in I}$ on $X$ is the
finest topology on $X$ which makes $f_{i}:Y_{i}\longrightarrow X$ continuous
for all $i\in I$. It is obvious that a subset $U$ of $X$ is open in the final
topology if and only if $f_{i}^{\leftarrow}(U)$ is open in $Y_{i}$, for each
$i\in I$.

######  Proposition 3.15.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé. The
following hold:

  1. $(1)$

The topology of $\mathscr{T}$ coincides with the final topology induced by the
set of sections of $\mathscr{T}$;

  2. $(2)$

the topology induced on each stalk of $\mathscr{T}$ is the discrete topology.

###### Proof.

Let $V$ be a subset of $\mathscr{T}$. If $V$ is open in the topology of
$\mathscr{T}$, for every section $\sigma$ of $\mathscr{T}$,
$\sigma^{\leftarrow}(V)$ is an open subset of the domain $\sigma$. Thus $V$ is
open in the final topology. Conversely, let $V$ be open in the final topology.
Let $x\in V$. Applying Lemma 3.11, there exists a section $\sigma$ such that
$\sigma\pi(x)=x$. So by Proposition 3.13$(3)$, $\sigma\sigma^{\leftarrow}(V)$
is a neighborhood of $x$ contained in $V$. Thus $V$ is open in the topology of
$\mathscr{T}$.

Let $x\in\mathscr{B}$. By Lemma 3.11, it follows that for any
$y\in\mathscr{T}_{x}$ there exists a section $\sigma$ such that
$\sigma\pi(y)=y$. One can see that $\sigma(U)\cap\mathscr{T}_{x}=\\{y\\}$. ∎

######  Definition 3.16.

Let $f:X\to Y$ be a function.

  * a)

The kernel pair of $f$, denoted as $\kappa_{f}$, is given by the following
pullback:

$\textstyle{\kappa_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

  * b)

A map $\rho:\kappa_{f}\to X$ is said to be $f$-proper provided that the
following square commutes:

$\textstyle{\kappa_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{\rho}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

######  Proposition 3.17.

Let $f:X\to Y$ be a function. With $X_{y}=f^{\leftarrow}(y)$ being the stalk
at $y\in Y$, there is a one to one correspondence between,

  * $(1)$

the collections of binary operations $\\{\overset{f}{\diamond}_{y}:X_{y}\times
X_{y}\to X_{y}\mid y\in Y\\}$ and $f$-proper maps
$\overset{f}{\diamond}:\kappa_{f}\to X$.

  * $(2)$

the collections of nullary operations $\\{\overset{f}{\diamond}_{y}:\\{y\\}\to
X_{y}\mid y\in Y\\}$ and global sections $\overset{f}{\diamond}:Y\to X$.

###### Proof.

Given $\\{\overset{f}{\diamond}_{y}:X_{y}\times X_{y}\to X_{y}\mid y\in Y\\}$,
define $\overset{f}{\diamond}:\kappa_{f}\to X$ by
$\overset{f}{\diamond}(x_{1},x_{2})=\overset{f}{\diamond}_{f(x_{1})}(x_{1},x_{2})$.
It follows easily that $\overset{f}{\diamond}$ is $f$-proper. Conversely,
given an $f$-proper map $\overset{f}{\diamond}:\kappa_{f}\to X$, define
$\overset{f}{\diamond}_{y}(x_{1},x_{2})=\overset{f}{\diamond}(x_{1},x_{2})$.
The commutativity of the square shows that
$\overset{f}{\diamond}(x_{1},x_{2})\in X_{y}$. It can be easily verified that
the correspondence is indeed one to one.

Obvious. ∎

######  Definition 3.18.

Let $f:X\to Y$ be a map. Given $f$-proper functions
$\overset{f}{\vee},\overset{f}{\wedge},\overset{f}{\odot},\overset{f}{\to}:\kappa_{f}\to
X$ and global sections $\overset{f}{0},\overset{f}{1}:Y\to X$, we say
$(\kappa_{f};\overset{f}{\vee},\overset{f}{\wedge},\overset{f}{\odot},\overset{f}{\to},\overset{f}{0},\overset{f}{1})$
is a stalk residuated lattice, provided that
$(X_{y},\overset{f}{\vee}_{y},\overset{f}{\wedge}_{y},\overset{f}{\odot}_{y},\overset{f}{\to}_{y},\overset{f}{0}_{y},\overset{f}{1}_{y})$
is a residuated lattice, for any $y\in Y$.

Next, we introduce the notion of bundle and étalé of residuated lattices,
inspired by the one obtained for universal algebras (Davey, 1973).

######  Definition 3.19.

A bundle (an étalé) $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ is said
to be a bundle (an étalé) of residuated lattices provided that the following
assertions hold:

  * $(1)$

there are $\pi$-proper functions
$\overset{\pi}{\vee},\overset{\pi}{\wedge},\overset{\pi}{\odot},\overset{\pi}{\to}$
and global sections $\overset{\pi}{0},\overset{\pi}{1}$ making $\kappa_{\pi}$
a stalk residuated lattice;

  * $(2)$

the functions
$\overset{\pi}{\vee},\overset{\pi}{\wedge},\overset{\pi}{\odot},\overset{\pi}{\to},\overset{\pi}{0}$
and $\overset{\pi}{1}$ are continuous.

######  Remark 1.

The existence of the global section $\overset{f}{0}$ (or $\overset{f}{1}$)
makes a bundle (an étalé) of residuated lattices a retraction in Top.

To clarify the discussion, in the following, we will give some examples of
étalé of residuated lattices over a topological space.

######  Example 3.20.

Consider the residuated lattice $\mathfrak{A}_{4}$ from Example 2.1 and the
topological space $Spec_{h}(\mathfrak{A}_{4})$ from Example 2.8. Let
$\mathscr{T}=\mathfrak{A}_{2}\coprod\mathfrak{A}_{2}=\\{0_{1},1_{1},0_{2},1_{2}\\}$
equipped with the discrete topology. Define $\pi:\mathscr{T}\to
Spec(\mathfrak{A}_{4})$ by $\pi(0_{1})=\pi(1_{1})=F_{2}$ and
$\pi(0_{2})=\pi(1_{2})=F_{3}$. One can see that
$\mathscr{T}\overset{\pi}{\downarrow}Spec(\mathfrak{A}_{4})$ is an étalé of
residuated lattices.

######  Example 3.21.

Consider the residuated lattice $\mathfrak{A}_{6}$ from Example 2.2 and the
topological space $Max_{d}(\mathfrak{A}_{6})$ from Example 2.8. Let
$\mathscr{T}=\mathfrak{A}_{2}\coprod\mathfrak{A}_{3}=\\{0_{1},1_{1},0_{2},a_{2},1_{2}\\}$
equipped with the discrete topology. Define $\pi:\mathscr{T}\to
Spec(\mathfrak{A}_{4})$ by $\pi(0_{1})=\pi(1_{1})=F_{2}$ and
$\pi(0_{2})=\pi(a_{2})=\pi(1_{2})=F_{3}$. One can see that
$\mathscr{T}\overset{\pi}{\downarrow}Max_{d}(\mathfrak{A}_{6})$ is an étalé of
residuated lattices.

######  Example 3.22.

Consider the residuated lattice $\mathfrak{A}_{8}$ from Example 2.3 and the
topological space $Min_{p}(\mathfrak{A}_{8})$ from Example 2.8. Let
$\mathscr{T}=\mathfrak{A}_{3}\coprod\mathfrak{A}_{4}=\\{0_{1},a_{1},1_{1},0_{2},a_{2},b_{2},1_{2}\\}$
equipped with the discrete topology. Define $\pi:\mathscr{T}\to
Spec(\mathfrak{A}_{4})$ by $\pi(0_{1})=\pi(1_{1})=F_{2}$ and
$\pi(0_{2})=\pi(a_{2})=\pi(1_{2})=F_{3}$. One can see that
$\mathscr{T}\overset{\pi}{\downarrow}Min_{p}(\mathfrak{A}_{8})$ is an étalé of
residuated lattices.

######  Proposition 3.23.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be a bundle such that
$\kappa_{f}$ is a stalk residuated lattice. Consider the following statements:

  * $(1)$

$\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ is a bundle of residuated
lattices.

  * $(2)$

$\Gamma(X,\mathscr{T})$ forms a residuated lattice under pointwise operations,
for any subset $X$ of $\mathscr{B}$.

  * $(3)$

$\Gamma(U,\mathscr{T})$ forms a residuated lattice under pointwise operations,
for any open subset $U$ of $B$.

We have $(1)$$\Rightarrow$$(2)$$\Rightarrow$$(3)$ and if
$\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ is an étalé of residuated
lattices, then $(3)$$\Rightarrow$$(1)$ as well.

###### Proof.

Let
$\overset{\pi}{\diamond}\in\\{\overset{\pi}{\vee},\overset{\pi}{\wedge},\overset{\pi}{\odot},\overset{\pi}{\to}\\}$.
The corresponding map
$\diamond:\Gamma(X,\mathscr{T})\times\Gamma(X,\mathscr{T})\to\Gamma(X,\mathscr{T})$
is defined by
$(\sigma\diamond\tau)(x)=\sigma(x)\overset{\pi}{\diamond}_{x}\tau(x)$, for any
$x\in X$. We need to show for continuous sections
$\sigma,\tau:X\to\mathscr{T}$, $\sigma\diamond\tau:X\to\mathscr{T}$ is
continuous. Since $\pi\sigma=1=\pi\tau$, there is a unique map, that we denote
by $((\sigma,\tau))$, rendering commutative the two triangles as shown below.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{((\sigma,\tau))}$$\scriptstyle{\tau}$$\textstyle{\kappa_{\pi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathscr{B}}$

One can easily verify that
$\sigma\diamond\tau=\overset{\pi}{\diamond}\circ((\sigma,\tau))$. Since
$((\sigma,\tau))$ and $\overset{\pi}{\diamond}$ are continuous, so is their
composition and thus $\sigma\diamond\tau$ is continuous.

Now let $\overset{\pi}{\diamond}\in\\{\overset{\pi}{0},\overset{\pi}{1}\\}$.
The corresponding section over $X$ is $\diamond:X\to\mathscr{T}$ defined by
$\diamond=\overset{\pi}{\diamond}\circ i_{X}$, where $i_{X}:X\to\mathscr{B}$
is the inclusion. Since $\overset{\pi}{\diamond}$ and $i_{X}$ are continuous,
so is $\diamond$. It follows that $\Gamma(X,\mathscr{T})$ is a residuated
lattice under the pointwise operations.

Obvious.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ be an étalé of
residuated lattices. Using Proposition 3.14, let $\sigma(U)$ be a basic open
set of $\mathscr{T}$, for some open $U$ in $\mathscr{B}$ and section $\sigma$
in $\Gamma(U,\mathscr{T})$. Let
$(x,y)\in\overset{\pi}{\diamond}^{\leftarrow}(\sigma(U))$. So
$x\overset{\pi}{\diamond}_{b}y=\sigma(b)$, where $\pi(x)=b\in U$. Applying
Lemma 3.11, there exists a neighborhood $W$ of $b$ and sections
$\sigma_{x},\sigma_{y}\in\Gamma(W,\mathscr{T})$ such that $\sigma_{x}(b)=x$
and $\sigma_{y}(b)=y$. By Proposition 3.13$(1)$, one can see that
$V=\textsc{Eq}(\sigma,\sigma_{x}\diamond\sigma_{y})$ is a neighborhood of $b$.
Since
$(x,y)\in\sigma_{x}(V)\times\sigma_{y}(V)\cap\kappa_{\pi}\subseteq\overset{\pi}{\diamond}^{\leftarrow}(\sigma(U))$,
so the result holds by Proposition 3.13$(3)$. ∎

######  Definition 3.24.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ and
$\mathscr{S}\overset{\phi}{\downarrow}\mathscr{B}$ be two bundles (étalés) of
residuated lattices. A bundle (an étalé) morphism
$h:\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\to\mathscr{S}\overset{\phi}{\downarrow}\mathscr{B}$
is said to be a morphism of bundles (étalés) of residuated lattices provided
that the following square commutes:

$\textstyle{\kappa_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((hp_{1},hp_{2}))}$$\scriptstyle{\overset{\phi}{\diamond}}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\kappa_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overset{\psi}{\diamond}}$$\textstyle{\mathscr{S}}$

######  Lemma 3.25.

Let $\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}$ and
$\mathscr{S}\overset{\phi}{\downarrow}\mathscr{B}$ be two bundles (étalés) of
residuated lattices and
$h:\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\to\mathscr{S}\overset{\phi}{\downarrow}\mathscr{B}$
a bundle (an étalé) morphism. The following are equivalent:

  1. $(1)$

$h$ is a morphism of bundles (étalés) of residuated lattices;

  2. $(2)$

the restriction $h|_{\mathscr{T}_{b}}$, denoted as $h|_{b}$, is a morphism of
residuated lattices, for each $b\in\mathscr{B}$.

###### Proof.

It follows from the commutativity of the following square, for each
$b\in\mathscr{B}$:

$\textstyle{\mathscr{T}_{b}\times\mathscr{T}_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((hp_{1},hp_{2}))}$$\scriptstyle{\overset{\phi}{\diamond}_{b}}$$\textstyle{\mathscr{T}_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\mathscr{S}_{b}\times\mathscr{S}_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overset{\psi}{\diamond}_{b}}$$\textstyle{\mathscr{S}_{b}}$

∎

######  Theorem 3.26.

Let $\mathscr{B}$ be a topological space. The following hold:

  * $(1)$

The class of bundles of residuated lattices over $\mathscr{B}$ with the class
of morphisms of bundles of residuated lattices forms a category, denoted as
$\textbf{RL-Bundle}(\mathscr{B})$.

  * $(2)$

The class of étalés of residuated lattices over $\mathscr{B}$ with the class
of morphisms of étalés of residuated lattices forms a category, denoted as
$\textbf{RL-Etale}(\mathscr{B})$.

###### Proof.

It is straightforward. ∎

The next result, which has a routine verification, demonstrates that for a
given topological space $\mathscr{B}$, the inclusion functor
$i_{\mathscr{B}}:\textbf{Etale}(\mathscr{B})\to\textbf{Bundle}(\mathscr{B})$,
as introduced in Corollary 3.9, can be lifted to residuated lattices.

######  Corollary 3.27.

For a given topological space $\mathscr{B}$, $\textbf{RL-Etale}(\mathscr{B})$
is a full subcategory of $\textbf{RL-Bundle}(\mathscr{B})$.

##  4 Interrelation between bundles and étalés of residuated lattices

This section investigates the interrelation between bundles and étalés of
residuated lattices. In the preceding section, we demonstrated that given a
topological space $\mathscr{B}$, $\textbf{RL-Etale}(\mathscr{B})$ constitutes
a full subcategory of $\textbf{RL-Bundle}(\mathscr{B})$. While in this
section, we refine this result, stating that $\textbf{RL-Etale}(\mathscr{B})$
is coreflective in $\textbf{RL-Bundle}(\mathscr{B})$.

Let $X$ and $Y$ be topological spaces. We recall (Munkres, 2014, p. 285) that
if $C$ is a compact subspace of $X$ and $U$ is an open subset of $Y$, the sets
$S(C,U)=\\{f\in C(X,Y)\mid f(C)\subseteq U\\}$ form a subbasis for a topology
on $C(X,Y)$ that is called the compact-open topology.

In the sequel, for topological spaces $X$ and $Y$, we endow $\mathcal{C}(X,Y)$
with the compact-open topology, and $X\times Y$ with the product topology.

######  Proposition 4.1.

Let $\mathscr{B}$ be a locally compact Hausdorff space. The following functors
are adjoints:

Top$\scriptstyle{\mathcal{C}(\mathscr{B},-)}$Top$\scriptstyle{\mathscr{B}\times-}$

###### Proof.

$\mathcal{C}(\mathscr{B},-)$ sends a continuous $f:X\to Y$ to
$\mathcal{C}(\mathscr{B},f)=f\circ-:\mathcal{C}(\mathscr{B},X)\to\mathcal{C}(\mathscr{B},Y)$,
which is continuous because for a compact $C\subseteq B$ and an open
$V\subseteq Y$, we have
$\mathcal{C}(\mathscr{B},f)^{\leftarrow}(S(C,V))=S(C,f^{\leftarrow}(V))$. It
then easily follows that $\mathcal{C}(\mathscr{B},-)$ is a functor. Also it is
easily verified that $\mathscr{B}\times-$ is a functor.

Now, we need to show that
$\textbf{Top}(\mathscr{B}\times-,-)\cong\textbf{Top}(-,C(\mathscr{B},-))$,
i.e., the two functors are naturally isomorphic. Let $\mathscr{T}$ and
$\mathscr{X}$ be topological spaces. To each
$h:\mathscr{B}\times\mathscr{X}\to\mathscr{T}$ we correspond
$\hat{h}:\mathscr{X}\to\mathcal{C}(\mathscr{B},\mathscr{T})$, defined by
$\hat{h}(x)(b)=h(b,x)$, and to each
$k:\mathscr{X}\to\mathcal{C}(\mathscr{B},\mathscr{T})$ we correspond
$\bar{k}:\mathscr{B}\times\mathscr{X}\to\mathscr{T}$, defined by
$\bar{k}(b,x)=k(x)(b)$. One can easily verify that $\bar{\hat{h}}=h$ and
$\hat{\bar{k}}=k$. Also $h$ is continuous if and only if $\hat{h}$ is, see
(Munkres, 2014, Theorem 46.11). This proves
$\textbf{Top}(\mathscr{B}\times\mathscr{X},\mathscr{T})\cong\textbf{Top}(\mathscr{X},\mathcal{C}(\mathscr{B},\mathscr{T}))$.
Naturality can be shown by some computations. ∎

In the sequel, let $U:\textbf{Top}\to\textbf{Set}$ and
$D:\textbf{Set}\to\textbf{Top}$ be the forgetful and the discrete functors,
respectively.

######  Corollary 4.2.

The following functors are adjoints:

Top$\scriptstyle{U\circ\mathcal{C}(\mathscr{B},-)}$Set$\scriptstyle{(\mathscr{B}\times-)\circ
D}$

###### Proof.

Follows from Proposition 4.1 and the fact that $U$ is a right adjoint of $D$.
A direct proof shows that the assumption that $\mathscr{B}$ is a locally
compact Hausdorff space is not needed. ∎

For a continuous function $\pi:\mathscr{T}\to\mathscr{B}$, putting the
subspace topology on
$\Gamma(\mathscr{B},\mathscr{T})\subseteq\mathcal{C}(\mathscr{B},\mathscr{T})$
and letting $\pi_{1}:\mathscr{B}\times\mathscr{X}\to\mathscr{B}$ be the
natural projection, we have:

######  Proposition 4.3.

Let $\pi:\mathscr{T}\to\mathscr{B}$ be continuous and $\mathscr{X}$ a
topological space. Suppose $h:\mathscr{B}\times\mathscr{X}\to\mathscr{T}$ is a
continuous function and
$\hat{h}:\mathscr{X}\to\mathcal{C}(\mathscr{B},\mathscr{T})$ its corresponding
counterpart. Then in the following diagram, the left triangle commutes if and
only if there is a map $\dot{h}:\mathscr{X}\to\Gamma(\mathscr{B},\mathscr{T})$
making the right triangle commute. Furthermore $\dot{h}$ is continuous.

$\textstyle{\mathscr{B}\times\mathscr{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\pi_{1}}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathscr{B}}$
$\textstyle{\mathscr{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{h}}$$\scriptstyle{\dot{h}}$$\textstyle{\mathcal{C}(\mathscr{B},\mathscr{T})}$$\textstyle{\Gamma(\mathscr{B},\mathscr{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

###### Proof.

Suppose the left triangle commutes. For $x\in\mathscr{X}$, we have
$(\pi\hat{h}(x))(b)=\pi(\hat{h}(x)(b))=\pi h(b,x)=\pi_{1}(b,x)=b$, so that
$\pi\hat{h}(x)=1$. Thus $\hat{h}(x)\in\Gamma(\mathscr{B},\mathscr{T})$. Set
$\dot{h}(x)=\hat{h}(x)$. Commutativity of the right triangle is obvious.
Conversely, suppose there is a map
$\dot{h}:\mathscr{X}\to\Gamma(\mathscr{B},\mathscr{T})$ making the right
triangle commute. For $(b,x)\in\mathscr{B}\times\mathscr{X}$, we have $\pi
h(b,x)=\pi\hat{h}(x)(b)=\pi\dot{h}(x)(b)=1(b)=\pi_{1}(b,x)$. Thus $\pi
h=\pi_{1}$, as desired. Lastly the continuity of $\dot{h}$ follows from the
commutativity of the right triangle. ∎

In the sequel,
$\pi_{\mathscr{B}}=(\mathscr{B}\times-,\pi_{1}):\textbf{Top}\to\mathbf{Bundle}(\mathscr{B})$
is the functor that takes $\mathscr{X}$ to
$\mathscr{B}\times\mathscr{X}\overset{\pi_{1}}{\downarrow}\mathscr{B}$.

######  Theorem 4.4.

Let $\mathscr{B}$ be a locally compact Hausdorff space. Then the following
functors are adjoints.

$\textstyle{\mathbf{Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(\mathscr{B},-)}$Top$\scriptstyle{\pi_{\mathscr{B}}}$

###### Proof.

Follows from Propositions 4.1 and 4.3. ∎

######  Corollary 4.5.

Let $\mathscr{B}$ be a topological space. The following are adjoint functors.

$\textstyle{\mathbf{Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U\circ\Gamma(\mathscr{B},-)}$Set$\scriptstyle{\pi_{\mathscr{B}}\circ
D}$

###### Proof.

Follows from Theorem 4.4. However a direct proof shows that the assumption
that $\mathscr{B}$ is locally compact Hausdorff is not needed. ∎

######  Theorem 4.6.

Let $\mathscr{B}$ be a locally compact Hausdorff space. The following functors
are adjoints,

Top$\scriptstyle{\pi_{\mathscr{B}}}$$\textstyle{\mathbf{Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{\mathscr{B}}}$

where $U_{\mathscr{B}}:\mathbf{Bundle}(\mathscr{B})\to\textbf{Top}$ is the
forgetful functor.

###### Proof.

Given a bundle $(X,f)$ over $\mathscr{B}$ and a topological space $Y$,
$\textbf{Top}(U_{\mathscr{B}}(X,f),Y)\cong\mathbf{Bundle}(\mathscr{B})((X,f),\pi_{\mathscr{B}}(Y))$,
as can be seen by the assignment that sends each continuous function
$g:U_{\mathscr{B}}(X,f)\to Y$ to the continuous function $\langle
f,g\rangle:(X,f)\to\pi_{\mathscr{B}}(Y)$ and each
$k:(X,f)\to\pi_{\mathscr{B}}(Y)$ to $\pi_{2}k$. The naturality follows easily.
∎

The adjunctions of 4.1, 4.4 and 4.6 are related as follows.

######  Theorem 4.7.

Let $\mathscr{B}$ be a locally compact Hausdorff space. In the following
diagram, the upper and lower triangles commute.

Top$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{\mathcal{C}(\mathscr{B},-)}$$\textstyle{\mathbf{Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{\mathscr{B}}}$$\scriptstyle{\Gamma(\mathscr{B},-)}$Top$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{\mathscr{B}\times-}$

Let $\mathfrak{A}=(A;\vee,\wedge,\odot,\rightarrow,0,1)$ be a residuated
lattice and $\tau$ be a topology on $A$. We recall (Rasouli and Dehghani,
2020, Definition 3.4) that the pair $\mathfrak{A}_{\tau}=(\mathfrak{A};\tau)$
is said to be a topological residuated lattice provided that each operation of
$\mathfrak{A}$ is continuous, i.e., the fundamental operation $\sigma$ of
$\mathfrak{A}$ is continuous as a function from $A\times A$ to $A$. The
category of topological residuated lattices shall be denoted by RL-Top.

######  Theorem 4.8.

Let $\mathscr{B}$ be a topological space. The functor
$\mathscr{C}(\mathscr{B},-):\textbf{Top}\to\textbf{Top}$ lifts to a functor
$\mathscr{C}(\mathscr{B},-):\textbf{RL-Top}\to\textbf{RL-Top}$ in the sense
the following square commutes:

RL-Top$\scriptstyle{U}$$\scriptstyle{\mathscr{C}(\mathscr{B},-)}$RL-
Top$\scriptstyle{U}$Top$\scriptstyle{\mathscr{C}(\mathscr{B},-)}$Top

where $U$ is the forgetful functor.

###### Proof.

Let $\mathfrak{A}$ be a topological residuated lattice and $\sigma$ a binary
fundamental operation. We define the operation $\overline{\sigma}$ on
$\mathscr{C}(\mathscr{B},\mathfrak{A})$, pointwisely. To show
$\overline{\sigma}$ is continuous, we observe that it is the composition of
the two maps shown below.

$\textstyle{\mathscr{C}(\mathscr{B},\mathfrak{A})\times\mathscr{C}(\mathscr{B},\mathfrak{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\overline{\sigma}}$$\textstyle{\mathscr{C}(\mathscr{B},\mathfrak{A})}$$\textstyle{\mathscr{C}(\mathscr{B},\mathfrak{A}\times\mathfrak{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{C}(\mathscr{B},\sigma)}}$

$\mathscr{C}(\mathscr{B},\sigma)$ is continuous by Proposition 4.1 and the
fact that $\sigma$ is continuous. So we need to show that $\theta$ is
continuous. Since the collection

$\\{S(C,U\times V)\mid C\hbox{ compact subset of }\mathscr{B}\hbox{ and
}U,V\hbox{open subsets of }A\\}$

constitutes a subbasis for
$\mathscr{C}(\mathscr{B},\mathfrak{A}\times\mathfrak{A})$ as well and
$\theta^{\leftarrow}(S(C,U\times V))=S(C,U)\times S(C,V)$, continuity of
$\theta$ follows. This states that $\mathscr{C}(\mathscr{B},\mathfrak{A})$ is
a topological residuated lattice. One can also verify that
$\mathscr{C}(\mathscr{B},-)$ sends a continuous morphism
$h:\mathfrak{A}\to\mathfrak{B}$ to a continuous morphism
$\mathscr{C}(\mathscr{B},h):\mathscr{C}(\mathscr{B},\mathfrak{A})\to\mathscr{C}(\mathscr{B},\mathfrak{B})$.
Preservation of identities and composition follows from that of
$\mathscr{C}(\mathscr{B},-):\textbf{Top}\to\textbf{Top}$. ∎

######  Theorem 4.9.

Let $\mathscr{B}$ be a topological space. The functor
$U\circ\mathscr{C}(\mathscr{B},-):\textbf{Top}\to\textbf{Set}$ lifts to a
functor $U\circ\mathscr{C}(\mathscr{B},-):\textbf{RL-Top}\to\textbf{RL}$ in
the sense that the following squares commute:

RL-Top$\scriptstyle{U}$$\scriptstyle{\mathscr{C}(\mathscr{B},-)}$RL-
Top$\scriptstyle{U}$$\scriptstyle{U}$RL$\scriptstyle{U}$Top$\scriptstyle{\mathscr{C}(\mathscr{B},-)}$Top$\scriptstyle{U}$Set

where the functors $U$ are the obvious forgetful functors.

###### Proof.

Follows from Corollary 4.2, Theorem 4.8, and the fact that the right square in
the above diagram commutes. ∎

######  Theorem 4.10.

Let $\mathscr{B}$ be a topological space. The functor

  * $(1)$

$\Gamma(\mathscr{B},-):\textbf{Bundle}(\mathscr{B})\to\textbf{Top}$ lifts to a
functor $\Gamma(\mathscr{B},-):\textbf{RL-Bundle}(\mathscr{B})\to\textbf{RL-
Top}$ in the sense that the following square commutes:

$\textstyle{\textbf{RL-
Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\scriptstyle{\Gamma(\mathscr{B},-)}$RL-
Top$\scriptstyle{U}$$\textstyle{\textbf{Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(\mathscr{B},-)}$Top

  * $(2)$

$\pi_{\mathscr{B}}:\textbf{Top}\to\textbf{Bundle}(\mathscr{B})$ lifts to a
functor $\pi_{\mathscr{B}}:\textbf{RL-Top}\to\textbf{RL-Bundle}(\mathscr{B})$
in the sense that the following square commutes:

$\textstyle{\textbf{RL-
Top}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\scriptstyle{\pi_{\mathscr{B}}}$$\textstyle{\textbf{RL-
Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$Top$\scriptstyle{\pi_{\mathscr{B}}}$$\textstyle{\textbf{Bundle}(\mathscr{B})}$

###### Proof.

Given a bundle of residuated lattices
$\mathscr{T}\overset{\phi}{\downarrow}\mathscr{B}$, by Proposition 3.23,
$\Gamma(\mathscr{B},\mathscr{T})$ is a residuated lattice and by Theorem 4.4,
it is a topological space. We need to show that the residuated lattice
operations are continuous. For
$\dot{\diamond}\in\\{\dot{\vee},\dot{\wedge},\dot{\odot},\dot{\rightarrow}\\}$,
by the definition of $\diamond:\Gamma(B,T)\times\Gamma(B,T)\to\Gamma(B,T)$ as
given in the proof of Proposition 3.23, one can verify the commutativity of
the following square:

$\textstyle{\Gamma(\mathscr{B},\mathscr{T})\times\Gamma(\mathscr{B},\mathscr{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\diamond}$$\scriptstyle{((-,-))}$$\textstyle{\Gamma(\mathscr{B},\mathscr{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathscr{C}(\mathscr{B},\kappa_{\pi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{C}(\mathscr{B},\dot{\diamond})}$$\textstyle{\mathscr{C}(\mathscr{B},\kappa_{\pi})}$

in which $((-,-))$ is injection and $i$ is the inclusion. Since $((-,-))$,
$\mathscr{C}(\mathscr{B},\dot{\diamond})$ and $i$ are continuous, and $i$ is
the inclusion, the continuity of $\diamond$ follows. For
$\dot{\diamond}\in\\{\dot{0},\dot{1}\\}$, the corresponding map
$\diamond=\dot{\diamond}$ and is therefore continuous. Hence,
$\Gamma(\mathscr{B},\mathscr{T})$ is a topological residuated lattice.

Next we show that a bundle residuated lattice morphism goes to a topological
residuated lattice morphism. So let
$h:\mathscr{S}\overset{\phi}{\downarrow}\mathscr{B}\to\mathscr{T}\overset{\psi}{\downarrow}\mathscr{B}$
be a morphism in $\textbf{RL-Bundle}(\mathscr{B})$. We have the morphism
$\kappa_{h}=((hp_{1},hp_{2}))$ rendering commutative the triangles in the
following diagram:

$\textstyle{\kappa_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{hp_{1}}$$\scriptstyle{\kappa_{h}}$$\scriptstyle{hp_{2}}$$\textstyle{\kappa_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{2}}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathscr{B}}$

With
$\dot{\diamond}\in\\{\dot{\vee},\dot{\wedge},\dot{\odot},\dot{\rightarrow}\\}$,
the commutativity of the following square can be verified easily:

$\textstyle{\kappa_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\dot{\diamond}}$$\scriptstyle{\kappa_{h}}$$\textstyle{\kappa_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\dot{\diamond}}$$\textstyle{\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\mathscr{T}}$
(1)

Now consider the following diagram:

$\textstyle{\Gamma(\mathscr{B},\mathscr{S})\times\Gamma(\mathscr{B},\mathscr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((-,-))}$$\scriptstyle{\diamond}$$\scriptstyle{\mathscr{C}(\mathscr{B},h)\times\mathscr{C}(\mathscr{B},h)}$$\textstyle{\Gamma(\mathscr{B},T)\times\Gamma(\mathscr{B},\mathscr{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((-,-))}$$\scriptstyle{\diamond}$$\textstyle{\mathscr{C}(\mathscr{B},\kappa_{\phi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{C}(\mathscr{B},\dot{\diamond})}$$\scriptstyle{\mathscr{C}(\mathscr{B},\kappa_{h})}$$\textstyle{\mathscr{C}(\mathscr{B},\kappa_{\psi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{C}(\mathscr{B},\dot{\diamond})}$$\textstyle{\Gamma(\mathscr{B},S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{C}(\mathscr{B},h)}$$\textstyle{\Gamma(\mathscr{B},T)}$

The left and right triangles commute by definition of $\diamond$. The bottom
square commutes by using the commutativity of (1). The commutativity of the
top square follows from the definitions of $((-,-))$ and $\kappa_{h}$. Hence,
the big square in the above diagram commutes, implying the preservation of
$\diamond$ by $\Gamma(\mathscr{B},-)$. For
$\dot{\diamond}\in\\{\dot{0},\dot{1}\\}$, since for all $b\in B$,
$h_{b}\circ\diamond_{b}=\diamond_{b}$, we get
$(h\circ-)(\diamond)=h\circ\diamond=\diamond$. Hence
$\Gamma(\mathscr{B},h)=h\circ-$ is in RL-Top.

Preservation of identities and composition follows from the fact that
$\Gamma(B,-):\textbf{Bundle}(\mathscr{B})\to\textbf{Top}$ is a functor.
Finally, commutativity of the square involving the forgetful functors is seen
easily.

Given a topological resituated lattice $\mathfrak{A}$, we need to show that
$\mathscr{B}\times\mathfrak{A}\overset{\pi_{1}}{\downarrow}\mathscr{B}$ is a
bundle of residuated lattices. For each $b\in B$, we define
$\diamond_{b}:(\\{b\\}\times A)\times(\\{b\\}\times A)\to\\{b\\}\times A$ by
$(b,x)\diamond_{b}(b,y)=(b,x\diamond_{\mathfrak{A}}y)$, where
$\diamond_{\mathfrak{A}}\in\\{\vee_{\mathfrak{A}},\wedge_{\mathfrak{A}},\odot_{\mathfrak{A}},\rightarrow_{\mathfrak{A}}\\}$,
and we define $0_{b}=(b,0_{\mathfrak{A}})$ and $1_{b}=(b,1_{\mathfrak{A}})$.
The kernel pair $\kappa_{\pi_{1}}$ is
$\mathscr{B}\times\mathfrak{A}\times\mathfrak{A}$, and the corresponding map
$\dot{\diamond}:\mathscr{B}\times\mathfrak{A}\times\mathfrak{A}\to\mathscr{B}\times\mathfrak{A}$
is $1_{B}\times\diamond_{\mathfrak{A}}$, which is continuous, because both
$1_{B}$ and $\diamond_{\mathfrak{A}}$ are. Thus
$\mathscr{B}\times\mathfrak{A}\overset{\pi_{1}}{\downarrow}\mathscr{B}$ is in
$\textbf{RL-Bundle}(\mathscr{B})$.

Also a continuous map $f:X\to Y$ goes to $1_{B}\times f:B\times
X\overset{\pi_{1}}{\downarrow}B\to B\times Y\overset{\pi_{1}}{\downarrow}B$,
which can be easily verified to be a bundle residuated lattice morphism. What
remains to show can be easily verified too. ∎

The next result shows that the adjoint functors of Corollary 4.5 both lift to
residuated lattices.

######  Theorem 4.11.

Let $\mathscr{B}$ be a topological residuated lattice. The functors
$U\circ\Gamma(\mathscr{B},-):\textbf{Bundle}(\mathscr{B})\to\textbf{Set}$ and
$\pi_{\mathscr{B}}\circ D:\textbf{Set}\to\textbf{Bundle}(\mathscr{B})$ both
lift to residuated lattices, i.e, the following squares commute:

$\textstyle{\textbf{RL-
Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\scriptstyle{\Gamma(\mathscr{B},-)}$RL-
Top$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{U}$$\scriptstyle{U}$RL$\scriptstyle{D}$$\scriptstyle{U}$$\textstyle{\textbf{Bundle}(\mathscr{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(\mathscr{B},-)}$Top$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{U}$Set$\scriptstyle{D}$

###### Proof.

Follows from Corollary 4.5, Theorem 4.10, and the fact that the right squares
in the above diagram commute with respect to both $U$ and $D$. ∎

Building upon Corollary 3.9, we have established that, for a given topological
space $\mathscr{B}$, $\textbf{Etale}(\mathscr{B})$ forms a full subcategory of
$\textbf{Bundle}(\mathscr{B})$. Now, we aim to improve this result by stating
that $\textbf{Etale}(\mathscr{B})$ is coreflective in
$\textbf{Bundle}(\mathscr{B})$. While MacLane and Moerdijk (2012, §2.6,
Corollary 3) has stated this result using presheaves, we aim to establish it
independently of them.

######  Proposition 4.12.

Given a topological space $\mathscr{B}$, $\textbf{Etale}(\mathscr{B})$ is
coreflective in $\textbf{Bundle}(\mathscr{B})$.

###### Proof.

To prove this, we show the inclusion
$i_{\mathscr{B}}:\textbf{Etale}(\mathscr{B})\to\textbf{Bundle}(\mathscr{B})$
has a right adjoint
$R_{\mathscr{B}}:\textbf{Bundle}(\mathscr{B})\to\textbf{Etale}(\mathscr{B})$.
So let a bundle $X\overset{f}{\downarrow}\mathscr{B}$ be given. For each
$b\in\mathscr{B}$, set $\Gamma(f)_{b}=\bigcup\limits_{U\in N_{b}}\Gamma(U,X)$,
where $N_{b}$ is the set of neighbourhoods of $b$. Define a relation
$\sim_{b}$ on the set $\Gamma(f)_{b}$ as follows; for sections
$s:U\to\mathscr{T}$ and $t:V\to\mathscr{t}$ in $\Gamma(f)_{b}$, $s\sim_{b}t$
if and only if there exists $W\in N_{b}$ contained in $U\cap V$, such that the
following diagram commutes:

---
$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{W}^{U}}$$\scriptstyle{i_{W}^{V}}$$\textstyle{X}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$

It is easy to verify that $\sim_{b}$ is an equivalence relation. For a given
neighbourhood $U$ of $b$, the equivalence class of a section
$s\in\Gamma(U,\mathscr{T})$ shall be denoted by $[s]_{U,b}$, or simply by
$[s]_{b}$ if we are able to forget about the neighbourhood on which $s$ is
defined. Set $\tilde{\Gamma}(f)_{b}=\Gamma(f)_{b}/\sim_{b}$ and
$\tilde{\Gamma}(f)=\coprod\limits_{b\in B}\tilde{\Gamma}(f)_{b}$. For any open
set $U$ of $\mathscr{B}$ and $s\in\Gamma(U,X)$, let the map
$s_{U}:U\longrightarrow\tilde{\Gamma}(f)$ be given by $x\mapsto[s]_{x}$. We
topologize $\tilde{\Gamma}(f)$ by putting on it the final topology coinduced
by the set $\\{s_{U}:U\hbox{ open in }\mathscr{B},s\in\Gamma(U,X)\\}$.

Let $U,V$ be open sets in $\mathscr{B}$, $s\in\Gamma(U,X)$ and
$t\in\Gamma(V,X)$. With a little bit of effort one infers that $b\in
t_{V}^{\leftarrow}(Im(s_{U}))$ if and only if there exists a neighborhood
$W_{b}$ of $b$ contained in $U\cap V$ such that $s|_{W_{b}}=t|_{W_{b}}$. One
can see that $t_{V}^{\leftarrow}(Im(s_{U}))=\bigcup_{b\in
t_{V}^{\leftarrow}(Im(s_{U}))}W_{b}$. Now, let $O$ be an open set in the final
topology. Consider $[s]_{U,x}\in O$. It is evident that
$[s]_{U,x}=[s|_{s_{U}^{\leftarrow}(O)}]_{s_{U}^{\leftarrow}(O),x}\in
Im((s|_{s_{U}^{\leftarrow}(O)})_{s_{U}^{\leftarrow}(O)})\subseteq O$. This
verifies that the set $\\{Im(s_{U})\mid U\hbox{ open in
}\mathscr{B},s\in\Gamma(U,X)\\}$ forms a basis for the final topology on
$\tilde{\Gamma}(f)$. Define
$\tilde{f}:\tilde{\Gamma}(f)\longrightarrow\mathscr{B}$ to take $[s]_{b}$ to
$b$. Continuity of $\tilde{f}$ follows by
$\tilde{f}^{\leftarrow}(U)=\bigcup\\{Im(s_{V})\mid V\hbox{ open subset of
}U,~{}s\in\Gamma(V,X)\\}$. Let $U$ be an open set in $\mathscr{B}$ and
$s\in\Gamma(U,X)$. The openness of $\tilde{f}$ follows by
$\tilde{f}(Im(s_{U}))=U$. Now, consider $[s]_{U,x}\in\tilde{\Gamma}(f)$. It is
easy to see that $\tilde{f}|_{Im(s_{U})}$ is an injection and so $\tilde{f}$
is locally injective. Hence, by Theorem 3.5, $\tilde{f}$ is a local
homeomorphism, and so
$\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}$ is an étalé.

Now, let $h:X\overset{f}{\downarrow}\mathscr{B}\to
Y\overset{g}{\downarrow}\mathscr{B}$ be a bundle morphism. Define the map
$\tilde{h}:\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}\to\tilde{\Gamma}(g)\overset{\tilde{g}}{\downarrow}\mathscr{B}$
by $\tilde{h}([s]_{b})=[hs]_{b}$. Routine computations show that $\tilde{h}$
is well-defined and $\tilde{g}\tilde{h}=\tilde{f}$. Also, for a section
$s:U\to X$, the equality $\tilde{h}(Im(s_{U}))=Im((hs)_{U})$ yields openness
of $\tilde{h}$. Hence by Proposition 3.8, $\tilde{h}$ is an étalé morphism.

Now, define the mapping $R_{\mathscr{B}}$ to take the object
$X\overset{f}{\downarrow}\mathscr{B}$ to
$\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}$, and the
morphism $h:X\overset{f}{\downarrow}\mathscr{B}\to
Y\overset{g}{\downarrow}\mathscr{B}$ to
$\tilde{h}:\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}\to\tilde{\Gamma}(g)\overset{\tilde{g}}{\downarrow}\mathscr{B}$.
Preservation of identities and composition of $R_{\mathscr{B}}$ follows
readily, and so
$R_{\mathscr{B}}:\textbf{Bundle}(\mathscr{B})\to\textbf{Etale}(\mathscr{B})$
is a functor.

To show that $R_{\mathscr{B}}$ is a right adjoint of $i_{\mathscr{B}}$, we
show that for each bundle $X\overset{f}{\downarrow}\mathscr{B}$, there is a
couniversal morphism
$\varepsilon_{f}:\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}\to
X\overset{f}{\downarrow}\mathscr{B}$. Define $\varepsilon_{f}$ by
$\varepsilon_{f}([s]_{b})=s(b)$. Routine computations show that
$\varepsilon_{f}$ is well-defined and $f\varepsilon_{f}=\tilde{f}$.
Injectivity can be verified by Proposition 3.13$(1)$, and openness follows by
Proposition 3.13$(3)$ and the fact that $\varepsilon_{f}(Im(s_{U}))=s(U)$.
Continuity of $\epsilon_{f}$ follows by the fact that
$\epsilon_{f}^{-1}(O)=\bigcup s_{U}(s^{-1}(O))$, where the union runs over the
set $\\{U\hbox{ open subset of }\mathscr{B},s\in\Gamma(U,X)\\}$.

For couniversality, suppose a bundle morphism
$h:\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\to
X\overset{f}{\downarrow}\mathscr{B}$ is given, with $\pi$ an étalé. For a
given $y\in\mathscr{T}$, by Lemma 3.11, there exists a neighborhood $U$ of
$\pi(y)$ and a section $s:U\to\mathscr{T}$ such that $s\pi(y)=y$. Define
$\bar{h}:\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\to\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}B$
by $\bar{h}(y)=[hs]_{\pi(y)}$. Some computations show that
$\tilde{f}\bar{h}=\pi$ and that the following triangle commutes.

|  |
---|---|---
$\textstyle{\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{f}}$$\textstyle{X\overset{f}{\downarrow}B}$$\textstyle{\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{h}}$$\scriptstyle{h}$

Using injectivity of $\epsilon_{f}$, for any subset $O$ of $\mathscr{T}$, we
have
$\bar{h}^{-1}(O)=\bar{h}^{-1}\epsilon^{-1}_{f}\epsilon_{f}(O)=h^{-1}\epsilon_{f}(O)$.
Thus continuity of $\bar{h}$ follows by openness of $\varepsilon_{f}$ and
continuity of $h$. Uniqueness of $\bar{h}$ follows from the fact that
$\varepsilon_{f}$ is injective. ∎

So far we have shown that, given a topological space $\mathscr{B}$,
$\textbf{RL-Etale}(\mathscr{B})$ is a full subcategory of $\textbf{RL-
Bundle}(\mathscr{B})$ (Corollary 3.27). Now we want to sharpen this result and
say that $\textbf{RL-Etale}(\mathscr{B})$ is coreflective in $\textbf{RL-
Bundle}(\mathscr{B})$.

######  Theorem 4.13.

Given a topological space $\mathscr{B}$, $\textbf{RL-Etale}(\mathscr{B})$ is
coreflective in $\textbf{RL-Bundle}(\mathscr{B})$.

###### Proof.

We show the inclusion $i_{\mathscr{B}}:\textbf{RL-
Etale}(\mathscr{B})\to\textbf{RL-Bundle}(\mathscr{B})$ has a right adjoint
$R_{\mathscr{B}}:\textbf{RL-Bundle}(\mathscr{B})\to\textbf{RL-
Etale}(\mathscr{B})$. Let a bundle of residuated lattices
$X\overset{f}{\downarrow}\mathscr{B}$ be given. By Proposition 4.12, we have
an étalé $\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}$. To
show this is an étalé of residuated lattices, by Proposition 3.23, for each
$U\in\mathscr{O(B)}$, $\Gamma(U,X)$ is a residuated lattice under the
pointwise operations. Now, for each $b\in\mathscr{B}$, $U\in N_{b}$, and each
operation $\diamond_{U}\in\\{\vee_{U},\wedge_{U},\odot_{U},\rightarrow_{U}\\}$
on $\Gamma(U,X)$, define the operation $\tilde{\diamond}_{b}$ on
$\tilde{\Gamma}(f)_{b}$, by $[s]_{b}\ \tilde{\diamond}_{b}\
[t]_{b}=[s_{|_{U\cap V}}]_{b}\ \diamond_{U\cap V}\ [t_{|_{U\cap V}}]_{b}$,
where $s:U\to X$ and $t:V\to X$ are sections over $X$. Also, for the nullary
operations $0_{U}$ and $1_{U}$ on $\Gamma(U,X)$, define $\tilde{0}_{b}$ and
$\tilde{1}_{b}$ in $\tilde{\Gamma}(f)_{b}$ by $\tilde{0}_{b}=[0_{U}]_{b}$ and
$\tilde{1}_{b}=[1_{U}]_{b}$. The corresponding $\tilde{f}$-proper operations
make $\kappa_{\tilde{f}}$ a stalk residuated lattice.

To prove the continuity of the binary operations, consider the following
commutative square,

$\textstyle{\kappa_{\tilde{f}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\dot{\tilde{\diamond}}}$$\scriptstyle{((\varepsilon_{f},\varepsilon_{f}))}$$\textstyle{\kappa_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\dot{\diamond}}$$\textstyle{\tilde{\Gamma}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{f}}$$\textstyle{X}$

where $\varepsilon_{f}$ is given in the proof of Proposition 4.12. The
continuity of $\dot{\tilde{\diamond}}$ now follows from the injectivity and
openness of $\varepsilon_{f}$ and the continuity of $\dot{\diamond}$ and
$((\varepsilon_{f},\varepsilon_{f}))$. The continuity of the sections
$\tilde{0}$ and $\tilde{1}$ is easily verified. This proves that
$\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}\mathscr{B}$ is an étalé of
residuated lattices.

By Proposition 4.12, we know a bundle morphism $h:X\overset{f}{\downarrow}B\to
Y\overset{g}{\downarrow}B$ goes to the étalé morphism
$\tilde{h}:\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}B\to\tilde{\Gamma}(g)\overset{\tilde{g}}{\downarrow}B$,
where $\tilde{h}([s]_{b})=[hs]_{b}$.

To show $\tilde{h}$ preserves operations, we have:

$\begin{array}[]{ll}\tilde{h}([s]_{b}\tilde{\diamond}_{b}[t]_{b})&=\tilde{h}([s_{|_{U\cap
V}}]_{b}\ \diamond_{U\cap V}\ [t_{|_{U\cap V}}]_{b})\\\
&=\tilde{h}([s_{|_{U\cap V}}\diamond_{b}t_{|_{U\cap V}}]_{b})\\\
&=[h(s_{|_{U\cap V}}\diamond_{b}t_{|_{U\cap V}})]_{b}\\\ &=[h(s_{|_{U\cap
V}})\diamond_{b}h(t_{|_{U\cap V}})]_{b}\\\ &=[hs_{|_{U\cap
V}}]_{b}\diamond_{b}[ht_{|_{U\cap V}}]_{b}\\\ &=\tilde{h}([s_{|_{U\cap
V}}]_{b})\diamond_{b}\tilde{h}([t_{|_{U\cap V}}]_{b})\\\ \end{array}$

The preservation of nullary operations can be easily verified. This proves
that $\tilde{h}$ is a morphism of etales of residuated lattices. It follows
that the functor $R_{\mathscr{B}}$ of $X\overset{f}{\downarrow}\mathscr{B}$
lifts to $R_{\mathscr{B}}:\textbf{RL-Bundle}(\mathscr{B})\to\textbf{RL-
Etale}(\mathscr{B})$.

To show that $R_{\mathscr{B}}$ is a right adjoint of $i_{\mathscr{B}}$, let
$X\overset{f}{\downarrow}\mathscr{B}$ be a bundle of residuated lattices. We
show that the morphism $\varepsilon_{f}$ defined in Proposition 4.12 preserves
the operations and is thus a morphism in $\textbf{RL-Bundle}(\mathscr{B})$. We
have:

$\begin{array}[]{ll}\varepsilon_{f}([s]_{b}\tilde{\diamond}_{b}[t]_{b})&=\varepsilon_{f}([s_{|_{U\cap
V}}]_{b}\ \diamond_{U\cap V}\ [t_{|_{U\cap V}}]_{b})\\\
&=\varepsilon_{f}([s_{|_{U\cap V}}\diamond_{b}t_{|_{U\cap V}}]_{b})\\\
&=(s_{|_{U\cap V}}\diamond_{b}t_{|_{U\cap V}})(b)\\\ &=s_{|_{U\cap
V}}(b)\diamond_{b}t_{|_{U\cap V}}(b)\\\
&=\varepsilon_{f}([s]_{b})\diamond_{b}\varepsilon_{f}([t]_{b})\\\ \end{array}$

Next we show $\varepsilon_{f}$ is couniversal. Given a bundle of residuated
lattices morphism $h:\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\to
X\overset{f}{\downarrow}\mathscr{B}$, with $\pi$ an étalé of residuated
lattices, by Proposition 4.12, there is a unique etale-morphism
$\bar{h}:\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\to\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}B$
rendering commutative the following triangle.

|  |
---|---|---
$\textstyle{\tilde{\Gamma}(f)\overset{\tilde{f}}{\downarrow}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{f}}$$\textstyle{X\overset{f}{\downarrow}B}$$\textstyle{\mathscr{T}\overset{\pi}{\downarrow}\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{h}}$$\scriptstyle{h}$

The commutativity of the above triangle, the preservation of the operations by
$\varepsilon_{f}$ and $h$ and the fact that $\varepsilon_{f}$ is injective,
imply that $\bar{h}$ preserves the operations. Thus $\bar{h}$ is a morphism in
$\textbf{RL-Etale}(\mathscr{B})$ concluding the proof. ∎

##  5 Étalés of residuated lattices

So far, we have only been considering morphisms of étalés of residuated
lattices defined over the same space. However, this section presents a method
for transferring an étalé of residuated lattices from one topological space to
another, utilizing a continuous map. At the end, we define a contravariant
functor, called the section functor, from the category of étalés of residuated
lattices with inverse morphisms, denoted as $\textbf{RLE}_{inv}$, to the
category of residuated lattices.

The next proposition shows that étalés over a topological space are pullback
stable in Top.

######  Proposition 5.1.

Let $f:\mathscr{B}\to\mathscr{C}$ be a continuous map.

  1. $(1)$

The pullback of an étalé over $\mathscr{C}$ along $f$ is an étalé over
$\mathscr{B}$;

  2. $(2)$

the pullback of an étalé morphism over $\mathscr{C}$ along $f$ is an étalé
morphism over $\mathscr{B}$.

###### Proof.

Let $\mathscr{S}\overset{\phi}{\downarrow}\mathscr{C}$ be an étalé. Consider
the following pullback square in the category Top.

${f^{*}\mathscr{S}}$${\mathscr{S}}$${\mathscr{B}}$${\mathscr{C}}$$\scriptstyle{f^{*}\phi}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\phi}$$\scriptstyle{f}$

We know that $f^{*}\mathscr{S}$ is the set
$\\{(b,s)\in\mathscr{B}\times\mathscr{S}\mid f(b)=\phi(s)\\}$ endowed with the
subspace topology induced by the product topology on
$\mathscr{B}\times\mathscr{S}$, and that $f^{*}\phi$ and $f^{\prime}$ are
natural projections. Given $(b,s)\in f^{*}\mathscr{S}$, let $U$ be an open
neighborhood of $s$ such that $\phi|_{U}$ is a homeomorphism onto the open set
$\phi(U)$. Thus $f^{\prime\leftarrow}(U)$ is an open neighborhood of $(b,s)$
and $\phi^{\prime}(f^{\prime\leftarrow}(U))=f^{\leftarrow}(\phi(U))$ is open.
One can easily verify that $f^{*}\phi|_{f^{\prime\leftarrow}(U)}$ is a
homeomorphism.

Let
$h:\mathscr{S}_{1}\overset{\phi_{1}}{\downarrow}\mathscr{C}\to\mathscr{S}_{2}\overset{\phi_{2}}{\downarrow}\mathscr{C}$
be a morphism of étalés. To get the pullback $f^{*}h$ of $h$ along $f$, we
take the pullbacks of $\phi_{1}$ and $\phi_{2}$ along $f$ to get the bottom
and big squares in the following diagram. Using the equality
$\phi_{2}h=\phi_{1}$, we get $f^{*}h$ as the unique morphism rendering
commutative the top square and the left triangle.

${f^{*}\mathscr{S}_{1}}$${\mathscr{S}_{1}}$${f^{*}\mathscr{S}_{2}}$${\mathscr{S}_{2}}$${\mathscr{B}}$${\mathscr{C}}$$\scriptstyle{f_{1}}$$\scriptstyle{f^{*}h}$$\scriptstyle{f^{*}\phi_{1}}$$\scriptstyle{h}$$\scriptstyle{\phi_{1}}$$\scriptstyle{f^{*}\phi_{2}}$$\scriptstyle{f_{2}}$$\scriptstyle{\phi_{2}}$$\scriptstyle{f}$

The commutativity of the left triangle makes
$f^{*}h:f^{*}\mathscr{S}_{1}\overset{f^{*}\phi_{1}}{\downarrow}\mathscr{B}\to
f^{*}\mathscr{S}_{2}\overset{f^{*}\phi_{2}}{\downarrow}\mathscr{B}$ a morphism
of étalés. ∎

######  Definition 5.2.

Let $\mathscr{S}\overset{\phi}{\downarrow}\mathscr{C}$ be an étalé and
$f:\mathscr{B}\longrightarrow\mathscr{C}$ a continuous map. The étalé
$f^{*}\mathscr{S}\overset{f^{*}\phi}{\downarrow}\mathscr{B}$ is called the
inverse image of $\mathscr{S}\overset{\phi}{\downarrow}\mathscr{C}$ along $f$.

######  Theorem 5.3.

Let $f:\mathscr{B}\longrightarrow\mathscr{C}$ be a continuous map. Pulling
back along $f$ yields a functor
$f^{*}:\mathbf{Etale}(\mathscr{C})\longrightarrow\mathbf{Etale}(\mathscr{B})$.

###### Proof.

Follows from Proposition 5.1 and Johnstone (1977, p. 35). ∎

######  Proposition 5.4.

Let $f:\mathscr{B}\to\mathscr{C}$ be a continuous map.

  1. $(1)$

The pullback of an étalé of residuated lattices over $\mathscr{C}$ along $f$
is an étalé of residuated lattices over $\mathscr{B}$;

  2. $(2)$

the pullback of an étalé of residuated lattices morphism over $\mathscr{C}$
along $f$ is an étalé of residuated lattices morphism over $\mathscr{B}$.

###### Proof.

Let $\mathscr{S}\overset{\phi}{\downarrow}\mathscr{C}$ be an étalé of
residuated lattices. By Proposition 5.1$(1)$,
$f^{*}\mathscr{S}\overset{f^{*}\phi}{\downarrow}\mathscr{B}$ is an étalé. We
have the following pullbacks:

$\textstyle{f^{*}\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f^{*}\phi}$$\textstyle{\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\kappa_{f^{*}\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\textstyle{f^{*}\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathscr{C}}$$\textstyle{f^{*}\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi}$$\textstyle{\mathscr{C}}$

The commutativity of the above pullbacks yields the commutativity of the outer
square in the following diagram. Since the inner square is a pullback, there
is a unique morphism $((f^{\prime}q_{1},f^{\prime}q_{2}))$ making the
triangles commute.

$\textstyle{\kappa_{f^{*}\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}q_{1}}$$\scriptstyle{((f^{\prime}q_{1},f^{\prime}q_{2}))}$$\scriptstyle{f^{\prime}q_{2}}$$\textstyle{\kappa_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathscr{C}}$

Let $\diamond$ be a $\phi$-proper map. So, there is a unique map
$\overset{f^{*}\phi}{\diamond}$, making the triangles in the following diagram
commute.

$\textstyle{\kappa_{f^{*}\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}q_{1}}$$\scriptstyle{\overset{f^{*}\phi}{\diamond}}$$\scriptstyle{\diamond((f^{\prime}q_{1},f^{\prime}q_{2}))}$$\textstyle{f^{*}\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi}$$\scriptstyle{f^{\prime}}$$\textstyle{\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathscr{C}}$

The commutativity of the lower triangle shows that
$\overset{f^{*}\phi}{\diamond}$ is $f^{*}\phi$-proper and the commutativity of
the upper triangle give the following commutative square.

$\textstyle{\kappa_{f^{*}\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((f^{\prime}q_{1},f^{\prime}q_{2}))}$$\scriptstyle{\overset{f^{*}\phi}{\diamond}}$$\textstyle{f^{*}\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{\kappa_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\diamond}$$\textstyle{\mathscr{S}}$

Now, the continuity of $((f^{\prime}q_{1},f^{\prime}q_{2}))$, $\diamond$ and
$f^{\prime}$, and the fact that the topology on $f^{*}\mathscr{S}$ is the
induced topology by
$\textstyle{f^{*}\mathscr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f^{*}\phi,f^{\prime})}$$\textstyle{\mathscr{B}\times\mathscr{S}}$,
proves the continuity of $\overset{f^{*}\phi}{\diamond}$. It follows that
$f^{*}\mathscr{S}\overset{f^{*}\phi}{\downarrow}\mathscr{B}$ is an étalé of
residuated lattices.

Let
$h:\mathscr{S}_{1}\overset{\phi_{1}}{\downarrow}\mathscr{C}\to\mathscr{S}_{2}\overset{\phi_{2}}{\downarrow}\mathscr{C}$
be a morphism of étalés of residuated lattices. By Proposition 5.1$(2)$,
$f^{*}h:f^{*}\mathscr{S}_{1}\overset{f^{*}\phi_{1}}{\downarrow}\mathscr{B}\to
f^{*}\mathscr{S}_{2}\overset{f^{*}\phi_{2}}{\downarrow}\mathscr{B}$ is a
morphism of étalés. Using the pullbacks,

$\textstyle{\kappa_{\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\kappa_{\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{f^{*}\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{f^{*}h}$$\scriptstyle{f^{*}\phi_{1}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\phi_{1}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\mathscr{C}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathscr{C}}$$\textstyle{f^{*}\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{f^{*}\phi_{2}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathscr{C}}$

$\textstyle{\kappa_{f^{*}{\phi_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}_{1}}$$\scriptstyle{p^{*}_{2}}$$\textstyle{f^{*}\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi_{1}}$$\textstyle{f^{*}\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi_{1}}$$\textstyle{\mathscr{B}}$
$\textstyle{\kappa_{f^{*}{\phi_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{*}_{1}}$$\scriptstyle{q^{*}_{2}}$$\textstyle{f^{*}\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi_{2}}$$\textstyle{f^{*}\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi_{2}}$$\textstyle{\mathscr{B}}$

we get the unique morphisms making the triangles commute in the following
diagrams.

$\textstyle{\kappa_{\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{((hp_{1},hp_{2}))}}$$\scriptstyle{hp_{2}}$$\scriptstyle{hp_{1}}$$\textstyle{\kappa_{f^{*}\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))}}$$\scriptstyle{f^{*}hp^{*}_{2}}$$\scriptstyle{f^{*}hp^{*}_{1}}$$\textstyle{\kappa_{\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\kappa_{f^{*}\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{*}_{2}}$$\scriptstyle{q^{*}_{1}}$$\textstyle{f^{*}\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi_{2}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathscr{C}}$$\textstyle{f^{*}\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\phi_{2}}$$\textstyle{\mathscr{B}}$$\textstyle{\kappa_{f^{*}\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))}}$$\scriptstyle{f_{1}p^{*}_{2}}$$\scriptstyle{f_{1}p^{*}_{1}}$$\textstyle{\kappa_{f^{*}\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overset{f^{*}\phi_{1}}{\diamond}}$$\scriptstyle{{\overset{\phi_{1}}{\diamond}((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))}}$$\scriptstyle{f^{*}\phi_{1}p^{*}_{1}}$$\textstyle{\kappa_{\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{f^{*}\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{f^{*}\phi_{1}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\mathscr{C}}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathscr{C}}$$\textstyle{\kappa_{f^{*}\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{((f_{2}q^{*}_{1},f_{2}q^{*}_{2}))}}$$\scriptstyle{f_{2}q^{*}_{2}}$$\scriptstyle{f_{2}q^{*}_{1}}$$\textstyle{\kappa_{f^{*}\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overset{f^{*}\phi_{2}}{\diamond}}$$\scriptstyle{{\overset{\phi_{2}}{\diamond}((f_{2}q^{*}_{1},f_{2}q^{*}_{2}))}}$$\scriptstyle{f^{*}\phi_{2}q^{*}_{1}}$$\textstyle{\kappa_{\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{f^{*}\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{f^{*}\phi_{2}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathscr{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathscr{C}}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathscr{C}}$

Since
$h:\mathscr{S}_{1}\overset{\phi_{1}}{\downarrow}\mathscr{C}\to\mathscr{S}_{2}\overset{\phi_{2}}{\downarrow}\mathscr{C}$
is a morphism of étalés of residuated lattices, we have the following
commutative diagram:

$\textstyle{\kappa_{\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((hp_{1},hp_{2}))}$$\scriptstyle{\overset{\phi_{1}}{\diamond}}$$\textstyle{\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\kappa_{\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overset{\phi_{2}}{\diamond}}$$\textstyle{\mathscr{S}_{2}}$

To show that
$f^{*}h:f^{*}\mathscr{S}_{1}\overset{f^{*}\phi_{1}}{\downarrow}\mathscr{B}\to
f^{*}\mathscr{S}_{2}\overset{f^{*}\phi_{2}}{\downarrow}\mathscr{B}$ is a
morphism of étalés of residuated lattices, we need to show that the following
square is commutative:

$\textstyle{\kappa_{f^{*}\phi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))}$$\scriptstyle{\overset{f^{*}\phi_{1}}{\diamond}}$$\textstyle{f^{*}\mathscr{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}h}$$\textstyle{\kappa_{f^{*}\phi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overset{f^{*}\phi_{2}}{\diamond}}$$\textstyle{f^{*}\mathscr{S}_{2}}$

Since the composition of the sides in the above square are to
$f^{*}\mathscr{S}_{2}$, it suffices to show they are equal when composed with
$f^{*}\phi_{2}$ and $f_{2}$. But first we show
$((hp_{1},hp_{2}))((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))$ and
$((f_{2}q^{*}_{1},f_{2}2^{*}_{2}))((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))$ are
equal by showing they are equal when composed with both $q_{1}$ and $q_{2}$.
We have,

$\begin{array}[]{ll}q_{1}((hp_{1},hp_{2}))((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))&=hp_{1}((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))\\\
&=hf_{1}p^{*}_{1}\\\ &=f_{2}f^{*}hp^{*}_{1}\\\
&=f_{2}q^{*}_{1}((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))\\\
&=q_{1}((f_{2}q^{*}_{1},f_{2}2^{*}_{2}))((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2})).\end{array}$

Similarly, we can show that,

$q_{2}((hp_{1},hp_{2}))((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))=q_{2}((f_{2}q^{*}_{1},f_{2}2^{*}_{2}))((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))$

Now we have,

$\begin{array}[]{ll}f^{*}\phi_{2}f^{*}h\overset{f^{*}\phi_{1}}{\diamond}&=f^{*}\phi_{1}\overset{f^{*}\phi_{1}}{\diamond}\\\
&=f^{*}\phi_{1}p^{*}_{1}\\\ &=f^{*}\phi_{2}f^{*}hp^{*}_{1}\\\
&=f^{*}\phi_{2}q^{*}_{1}((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))\\\
&=f^{*}\phi_{2}\overset{f^{*}\phi_{2}}{\diamond}((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2})),\end{array}$

and

$\begin{array}[]{ll}f_{2}f^{*}h\overset{f^{*}\phi_{1}}{\diamond}&=hf_{1}\overset{f^{*}\phi_{1}}{\diamond}\\\
&=h\overset{\phi_{1}}{\diamond}((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))\\\
&=\overset{\phi_{2}}{\diamond}((hp_{1},hp_{2}))((f_{1}p^{*}_{1},f_{1}p^{*}_{2}))\\\
&=\overset{\phi_{2}}{\diamond}((f_{2}q^{*}_{1},f_{2}q^{*}_{2}))((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2}))\\\
&=f_{2}\overset{f^{*}\phi_{2}}{\diamond}((f^{*}hp^{*}_{1},f^{*}hp^{*}_{2})).\end{array}$

and that concludes the proof. ∎

The following result shows that the functor of Theorem 5.3 can be lifted to
residuated lattices.

######  Theorem 5.5.

Let $f:\mathscr{B}\longrightarrow\mathscr{C}$ be a continuous map. Pulling
back along $f$ yields a functor $f^{*}:\textbf{RL-
Etale}(\mathscr{C})\longrightarrow\textbf{RL-Etale}(\mathscr{B})$.

###### Proof.

Follows from Theorem 5.3, Proposition 5.4 and Johnstone (1977, p. 35). ∎

######  Definition 5.6.

Let $\mathscr{B}$ be a topological space. A pair
$(\mathscr{B},\mathscr{T_{B}})$ is called an $RLE$-space provided that
$\mathscr{T_{B}}\overset{\pi_{\mathscr{B}}}{\downarrow}\mathscr{B}$ is an
étalé of residuated lattices, for some local homeomorphism
$\pi_{\mathscr{B}}:\mathscr{T_{B}}\to\mathscr{B}$. An inverse morphism from
$RLE$-space $(\mathscr{B},\mathscr{T_{B}})$ to $RLE$-space
$(\mathscr{C},\mathscr{T_{C}})$, in short an $\textbf{RLE}_{inv}$-morphism, is
a pair $(f,\alpha)$, in symbol
$(f,\alpha):(\mathscr{B},\mathscr{T_{B}})\longrightarrow(\mathscr{C},\mathscr{T_{C}})$,
consisting of a continuous map $f:\mathscr{B}\longrightarrow\mathscr{C}$ and
an $RLE_{\mathscr{B}}$-morphism
$\alpha:f^{*}\mathscr{T_{C}}\longrightarrow\mathscr{T_{B}}$. The composition
of two $\textbf{RLE}_{inv}$-morphisms
$(f,\alpha):(\mathscr{B},\mathscr{T_{B}})\longrightarrow(\mathscr{C},\mathscr{T_{C}})$
and
$(g,\beta):(\mathscr{C},\mathscr{T_{C}})\longrightarrow(\mathscr{D},\mathscr{T_{D}})$
is given by $(gf,\alpha f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}})$, where the
morphism used in the composition is given by Diagram 5, in which all squares
are pullbacks (so that $g^{*}\pi_{\mathscr{D}}=\pi_{\mathscr{C}}\beta$,
$f^{*}\pi_{\mathscr{C}}=\pi_{\mathscr{B}}\alpha$,
$f^{*}g^{*}\pi_{\mathscr{D}}=\pi_{\mathscr{B}}\alpha f^{*}\beta$ and
$(gf)^{*}\pi_{\mathscr{D}}=\pi_{\mathscr{B}}\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}$) and $\lambda_{f,g,\mathscr{T_{D}}}$
is the unique isomorphism between two pullbacks.

${(gf)^{*}\mathscr{T_{D}}}$${f^{*}g^{*}\mathscr{T_{D}}}$${g^{*}\mathscr{T_{D}}}$${\mathscr{T}_{\mathscr{D}}}$${f^{*}\mathscr{T_{C}}}$${\mathscr{T}_{\mathscr{C}}}$${\mathscr{T}_{\mathscr{B}}}$${\mathscr{B}}$${\mathscr{C}}$${\mathscr{D}}$$\scriptstyle{\lambda_{f,g,\mathscr{T_{D}}}}$$\scriptstyle{(gf)^{\prime}}$$\scriptstyle{(gf)^{*}\pi_{\mathscr{D}}}$$\scriptstyle{f^{*}\beta}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{f^{*}g^{*}\pi_{\mathscr{D}}}$$\scriptstyle{\beta}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g^{*}\pi_{\mathscr{D}}}$$\scriptstyle{\pi_{\mathscr{D}}}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f^{*}\pi_{\mathscr{C}}}$$\scriptstyle{\pi_{\mathscr{C}}}$$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{f}$$\scriptstyle{gf}$$\scriptstyle{g}$
Figure 5: The composition of two $\textbf{RLE}_{inv}$-morphisms

In Definition 5.6, $\alpha$, $\beta$ and thus $f^{*}\beta$ are
$\textbf{RLE}_{inv}$-morphisms, and $\lambda_{f,g,\mathscr{T_{D}}}$ can be
easily verified to be an $\textbf{RLE}_{inv}$-morphism. Thus the composition
$\alpha f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}$ is an
$\textbf{RLE}_{inv}$-morphism, guaranteeing the well-definedness of the
composition.

######  Theorem 5.7.

The class of $RLE$-spaces with $\textbf{RLE}_{inv}$-morphisms, forms a
category, denoted by $\textbf{RLE}_{inv}$.

###### Proof.

Letting the identity morphism on $(\mathscr{B},\mathscr{T_{B}})$ to be
$(1_{\mathscr{B}},1_{\mathscr{T}_{\mathscr{B}}})$, we need to show it acts
neutral on both sides. Let
$(f,\alpha):(\mathscr{A},\mathscr{T_{A}})\longrightarrow(\mathscr{B},\mathscr{T_{B}})$
be an $\textbf{RLE}_{inv}$-morphism. Using Theorem 5.5, which implies that
$f^{*}1_{\mathscr{T}_{\mathscr{B}}}=1_{f^{*}\mathscr{T_{B}}}$, and Definition
5.6, it follows that
$(f,\alpha)(1_{\mathscr{B}},1_{\mathscr{T}_{\mathscr{B}}})=(f,\alpha)$ (See
Fig. 6). Analogously, one can show that
$(1_{\mathscr{B}},1_{\mathscr{T}_{\mathscr{B}}})(f,\alpha)=(f,\alpha)$, for
some $\textbf{RLE}_{inv}$-morphism
$(f,\alpha):(\mathscr{B},\mathscr{T_{B}})\longrightarrow(\mathscr{C},\mathscr{T_{C}})$.

${f^{*}\mathscr{T_{B}}}$${f^{*}\mathscr{T_{B}}}$${\mathscr{T_{B}}}$${\mathscr{T}_{\mathscr{B}}}$${f^{*}\mathscr{T_{B}}}$${\mathscr{T_{B}}}$${\mathscr{T}_{\mathscr{A}}}$${\mathscr{A}}$${\mathscr{B}}$${\mathscr{B}}$$\scriptstyle{1_{f^{*}\mathscr{T_{B}}}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f^{*}1_{\mathscr{T}_{\mathscr{B}}}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{1_{\mathscr{T}_{\mathscr{B}}}}$$\scriptstyle{1_{\mathscr{T}_{\mathscr{B}}}}$$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{\pi_{\mathscr{A}}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{1_{\mathscr{B}}}$
Figure 6: The identity morphism of $RLE$-space $(\mathscr{B},\mathscr{T_{B}})$

To show that composition is associative, let the
$\textbf{RLE}_{inv}$-morphisms

$\textstyle{(\mathscr{B},\mathscr{T_{B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f,\alpha)}$$\textstyle{(\mathscr{C},\mathscr{T_{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g,\beta)}$$\textstyle{(\mathscr{D},\mathscr{T_{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(h,\gamma)}$$\textstyle{(\mathscr{E},\mathscr{T_{E}})}$

be given. Following Diagram 7, in which all squares are pullbacks and all
triangles commute, it gives the compositions

$(g,\beta)(f,\alpha)=(gf,\alpha f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}})$ and
$(h,\gamma)(g,\beta)=(hg,\beta g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})$.

${(hg)^{*}\mathscr{T}_{\mathscr{E}}}$${(gf)^{*}\mathscr{T}_{\mathscr{D}}}$${g^{*}h^{*}\mathscr{T}_{\mathscr{E}}}$${h^{*}\mathscr{T}_{\mathscr{E}}}$${\mathscr{T}_{\mathscr{E}}}$${f^{*}g^{*}\mathscr{T}_{\mathscr{D}}}$${g^{*}\mathscr{T}_{\mathscr{D}}}$${\mathscr{T}_{\mathscr{D}}}$${f^{*}\mathscr{T_{B}}}$${\mathscr{T_{C}}}$${\mathscr{T}_{\mathscr{B}}}$${\mathscr{B}}$${\mathscr{C}}$${\mathscr{D}}$${\mathscr{E}}$$\scriptstyle{(hg)^{\prime}}$$\scriptstyle{\lambda_{g,h,\mathscr{T_{E}}}}$$\scriptstyle{\lambda_{f,g,\mathscr{T_{D}}}}$$\scriptstyle{(gf)^{\prime}}$$\scriptstyle{g^{\prime\prime}}$$\scriptstyle{g^{*}\gamma}$$\scriptstyle{\gamma}$$\scriptstyle{h^{\prime}}$$\scriptstyle{\pi_{\mathscr{E}}}$$\scriptstyle{f^{*}\beta}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{\beta}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\pi_{\mathscr{D}}}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi_{\mathscr{C}}}$$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{f}$$\scriptstyle{gf}$$\scriptstyle{g}$$\scriptstyle{hg}$$\scriptstyle{h}$
Figure 7: Associativity of the composition of $\textbf{RLE}_{inv}$-morphisms

Following Diagram 8, it verifies that

$(h,\gamma)((g,\beta)(f,\alpha))=(h(gf),\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}).$

${(h(gf))^{*}\mathscr{T_{E}}}$${(gf)^{*}h^{*}\mathscr{T_{E}}}$${h^{*}\mathscr{T_{E}}}$${\mathscr{T}_{\mathscr{E}}}$${(gf)^{*}\mathscr{T_{D}}}$${\mathscr{T}_{\mathscr{D}}}$${\mathscr{T}_{\mathscr{B}}}$${\mathscr{B}}$${\mathscr{D}}$${\mathscr{E}}$$\scriptstyle{\lambda_{gf,h,\mathscr{T_{E}}}}$$\scriptstyle{(h(gf))^{\prime}}$$\scriptstyle{(gf)^{*}\gamma}$$\scriptstyle{(gf)^{\prime\prime}}$$\scriptstyle{\gamma}$$\scriptstyle{h^{\prime}}$$\scriptstyle{\pi_{\mathscr{E}}}$$\scriptstyle{\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}}$$\scriptstyle{(gf)^{\prime}}$$\scriptstyle{\pi_{\mathscr{D}}}$$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{gf}$$\scriptstyle{h}$
Figure 8: The composition of two $\textbf{RLE}_{inv}$-morphisms $gf$ and $h$

Following Diagram 9, it verifies that

$((h,\gamma)(g,\beta))(f,\alpha)=((hg)f,\alpha f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}).$

${((hg)f)^{*}\mathscr{T_{E}}}$${f^{*}(hg)^{*}\mathscr{T_{E}}}$${(hg)^{*}\mathscr{T_{E}}}$${\mathscr{T}_{\mathscr{E}}}$${f^{*}\mathscr{T_{C}}}$${\mathscr{T}_{\mathscr{C}}}$${\mathscr{T}_{\mathscr{B}}}$${\mathscr{B}}$${\mathscr{C}}$${\mathscr{E}}$$\scriptstyle{\lambda_{f,hg,\mathscr{T_{E}}}}$$\scriptstyle{((hg)f)^{\prime}}$$\scriptstyle{f^{\prime\prime\prime}}$$\scriptstyle{f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})}$$\scriptstyle{\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}}$$\scriptstyle{(hg)^{\prime}}$$\scriptstyle{\pi_{\mathscr{E}}}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi_{\mathscr{C}}}$$\scriptstyle{\pi_{\mathscr{B}}}$$\scriptstyle{f}$$\scriptstyle{hg}$
Figure 9: The composition of two $\textbf{RLE}_{inv}$-morphisms $f$ and $hg$

To show that
$(h,\gamma)((g,\beta)(f,\alpha))=((h,\gamma)(g,\beta))(f,\alpha)$, we know
$h(gf)=(hg)f$, so we need to show $\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=\alpha
f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}$,
which holds if the equality

$f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}$ (2)

holds. Codomains of two morphisms in the equality 2 are
$f^{*}(\mathscr{T_{B}})$, so by a pullback in Diagram 7, Equation 2 holds if
and only if equalities

$\pi_{\mathscr{B}}\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=\pi_{\mathscr{B}}\alpha
f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}$ (3)

and

$f^{\prime}f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=f^{\prime}f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}$ (4)

hold. For Equation 3 we have,

$\begin{array}[]{ll}\pi_{\mathscr{B}}\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\\\
=(h(gf))^{*}\pi_{\mathscr{E}}&\mbox{ by the outer pullback square in Diagram
\ref{fig:rlespacdeffig1}}\\\ =((hg)f)^{*}\pi_{\mathscr{E}}&\mbox{ by
associativity of morphisms in $\textbf{Top}$}\\\ =\pi_{\mathscr{B}}\alpha
f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}.&\mbox{
by the outer pullback square in Diagram \ref{fig:rlespacdeffig2}}\end{array}$

For Equation 4 the left side is

$\begin{array}[]{ll}f^{\prime}f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\\\
=\beta
f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}},&\mbox{
by the outer pullback square in Diagram \ref{fig:rlespacdeffig}}\end{array}$

and the right side is

$\begin{array}[]{ll}f^{\prime}f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}&\\\
=\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}.&\mbox{
by the outer pullback square in Diagram \ref{fig:rlespacdeffig2}}\end{array}$

So Equation 4 holds if the equality

$f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}$
(5)

holds. Codomains of the two morphisms in Equation 5 are
$g^{*}\mathscr{T_{D}}$, so by a pullback in Diagram 7, Equation 5 holds if and
only if equalities

$\pi_{\mathscr{C}}\beta
f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=\pi_{\mathscr{C}}\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}$
(6)

and

$g^{\prime}f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}=g^{\prime}g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}$
(7)

hold. For Equation 6 we have,

$\begin{array}[]{ll}\pi_{\mathscr{C}}\beta
f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\\\
=f\pi_{\mathscr{B}}\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig}}\\\
=f(h(gf))^{*}\pi_{\mathscr{E}}&\mbox{ by the outer pullback square in Diagram
\ref{fig:rlespacdeffig1}}\\\ =f((hg)f)^{*}\pi_{\mathscr{E}}&\mbox{ by
associativity of morphisms in $\textbf{Top}$}\\\ =f\pi_{\mathscr{B}}\alpha
f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}&\mbox{
by the outer pullback square in Diagram \ref{fig:rlespacdeffig2}}\\\
=\pi_{\mathscr{C}}f^{\prime}f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig}}\\\
=\pi_{\mathscr{C}}\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}.&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig2}}\\\ \end{array}$

For Equation 7 the left side is

$\begin{array}[]{ll}g^{\prime}f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\\\
=(gf)^{\prime}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\mbox{ by a
commutative triangle in Diagram \ref{fig:rlespacdeffig}}\\\
=\gamma(gf)^{\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}},&\mbox{ by a pullback
square in Diagram \ref{fig:rlespacdeffig1}}\end{array}$

and the right side is

$\begin{array}[]{ll}g^{\prime}g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}&\\\
=\gamma
g^{\prime\prime}\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}.&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig}}\end{array}$

So Equation 7 holds if the equality

$(gf)^{\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}=g^{\prime\prime}\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}$
(8)

holds. Codomains of the two morphisms in the equality 8 are
$h^{*}\mathscr{T_{E}}$, so by a pullback in Diagram 7, Equation 8 holds if and
only if the equalities

$\pi_{\mathscr{D}}\gamma(gf)^{\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}=\pi_{\mathscr{D}}\gamma
g^{\prime\prime}\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}$
(9)

and

$h^{\prime}(gf)^{\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}=h^{\prime}g^{\prime\prime}\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}$
(10)

hold. For Equation 9 we have,

$\begin{array}[]{ll}\pi_{\mathscr{D}}\gamma(gf)^{\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}&\\\
=gf\pi_{\mathscr{B}}\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(gf)^{*}\gamma\lambda_{gf,h,\mathscr{T_{E}}}&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig1}}\\\
=gf(h(gf))^{*}\pi_{\mathscr{E}}&\mbox{ by the outer pullback square in Diagram
\ref{fig:rlespacdeffig1}}\\\ =gf((hg)f)^{*}\pi_{\mathscr{E}}&\mbox{ by
associativity of morphisms in $\textbf{Top}$}\\\ =gf\pi_{\mathscr{B}}\alpha
f^{*}(\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}})\lambda_{f,hg,\mathscr{T_{E}}}&\mbox{
by the outer pullback square in Diagram \ref{fig:rlespacdeffig2}}\\\
=g\pi_{\mathscr{C}}f^{\prime}\beta
g^{*}\gamma\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig2}}\\\
=\pi_{\mathscr{D}}\gamma
g^{\prime\prime}\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}.&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig}}\\\ \end{array}$

For Equation 10 we have,

$\begin{array}[]{ll}h^{\prime}(gf)^{\prime\prime}\lambda_{gf,h,\mathscr{T_{E}}}&\\\
=(h(gf))^{\prime}&\mbox{ by a commutative triangle in Diagram
\ref{fig:rlespacdeffig1}}\\\ =((hg)f)^{\prime}&\mbox{ by associativity of
morphisms in $\textbf{Top}$}\\\
=(hg)^{\prime}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}&\mbox{ by a
commutative triangle in Diagram \ref{fig:rlespacdeffig2}}\\\
=h^{\prime}g^{\prime\prime}\lambda_{g,h,\mathscr{T_{E}}}f^{\prime\prime\prime}\lambda_{f,hg,\mathscr{T_{E}}}.&\mbox{
by a commutative triangle in Diagram \ref{fig:rlespacdeffig}}\end{array}$

This concludes the result. ∎

At the end, we define a contravariant functor, called the section functor,
from the category $\textbf{RLE}_{inv}$ to the category of residuated lattices.

######  Theorem 5.8.

The mapping $\mathcal{S}:RLE_{inv}\longrightarrow\textsc{RL}$, sending an
$RLE$-space $(\mathscr{B},\mathscr{T_{B}})$ to the residuated lattice
$\Gamma(\mathscr{B},\mathscr{T_{B}})$ and an $\textbf{RLE}_{inv}$-morphism
$(f,\alpha):(\mathscr{B},\mathscr{T_{B}})\longrightarrow(\mathscr{C},\mathscr{T_{C}})$
to the residuated lattice morphism
$\mathcal{S}(f,\alpha):\Gamma(\mathscr{C},\mathscr{T_{C}})\longrightarrow\Gamma(\mathscr{B},\mathscr{T_{B}})$,
given by $\mathcal{S}(f,\alpha)(\sigma)(b)=\alpha(b,\sigma(f(b)))$, is a
contravariant functor.

###### Proof.

Let $(\mathscr{B},\mathscr{T_{B}})$ be an $RLE$-space. By Definition 3.19
$\Gamma(\mathscr{B},\mathscr{T_{B}})$ is a residuated lattice. Let
$(f,\alpha):(\mathscr{B},\mathscr{T_{B}})\longrightarrow(\mathscr{C},\mathscr{T_{C}})$
be an $\textbf{RLE}_{inv}$-morphism. For a given global section $\sigma$ of
$\mathscr{T_{C}}$, with a little effort, one can see that
$\mathcal{S}(f,\alpha)(\sigma)$ is a global section of $\mathscr{T_{B}}$. Let
$\sigma$ and $\tau$ be two global section of $\mathscr{T_{C}}$. For an
arbitrary $b\in\mathscr{B}$, We have

$\begin{array}[]{ll}\mathcal{S}(f,\alpha)(\sigma\vee\tau)(b)&=\alpha(b,\sigma\vee\tau(f(b)))\\\
&=\alpha(b,\sigma(f(b))\vee_{f(b)}\tau(f(b)))\\\
&=\alpha|_{b}(b,\sigma(f(b))\vee_{f(b)}\tau(f(b)))\\\
&=\alpha|_{b}(b,\sigma(f(b))\vee_{f(b)}\tau(f(b)))\\\
&=\alpha|_{b}((b,\sigma(f(b)))\vee_{b}(b,\tau(f(b))))\\\
&=\alpha|_{b}((b,\sigma(f(b))))\vee_{b}\alpha|_{b}((b,\tau(f(b))))\\\
&=\alpha((b,\sigma(f(b))))\vee_{b}\alpha((b,\tau(f(b))))\\\
&=\mathcal{S}(f,\alpha)(\sigma)(b)\vee_{b}\mathcal{S}(f,\alpha)(\tau)(b)\\\
&=(\mathcal{S}(f,\alpha)(\sigma)\vee\mathcal{S}(f,\alpha)(\tau))(b).\end{array}$

Analogously, we can show that the function $\mathcal{S}(f,\alpha)$ preserves
the other fundamental operations, and so it is an $RL$-morphism. Also, it is
obvious that
$\mathcal{S}(1_{\mathscr{B}},1_{\mathscr{T}_{\mathscr{B}}})=1_{\Gamma(\mathscr{B},\mathscr{T_{B}})}$.
Next we show that $\mathcal{S}$ preserves the reverse composition, let the
$\textbf{RLE}_{inv}$-morphisms

$\textstyle{(\mathscr{B},\mathscr{T_{B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f,\alpha)}$$\textstyle{(\mathscr{C},\mathscr{T_{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g,\beta)}$$\textstyle{(\mathscr{D},\mathscr{T_{D}})}$

be given. To show that
$\mathcal{S}((g,\beta)(f,\alpha))=\mathcal{S}(f,\alpha)\mathcal{S}(g,\beta)$,
we need to show for an arbitrary section $\sigma$ of $\mathscr{T_{D}}$ and
$b\in\mathscr{B}$,

$\mathcal{S}((g,\beta)(f,\alpha))(\sigma)(b)=(\mathcal{S}(f,\alpha)\mathcal{S}(g,\beta))(\sigma)(b)$
(11)

By definition we have,

$\begin{array}[]{ll}\mathcal{S}((g,\beta)(f,\alpha))(\sigma)(b)&=\mathcal{S}(gf,\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}})(\sigma)(b)\\\ &=\alpha
f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))\end{array}$

and

$\begin{array}[]{ll}(\mathcal{S}(f,\alpha)\mathcal{S}(g,\beta))(\sigma)(b)&=\mathcal{S}(f,\alpha)(\mathcal{S}(g,\beta)(\sigma))(b)\\\
&=\alpha(b,(\mathcal{S}(g,\beta)(\sigma)(f(b))))\\\
&=\alpha(b,\beta(f(b),\sigma(gf(b))))\\\ \end{array}$

So the equation 11 holds if the equality

$f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))=(b,\beta(f(b),\sigma(gf(b))))$
(12)

holds. By a pullback in Diagram 5, Equation 12 holds if and only if equalities

$f^{*}\pi_{\mathscr{C}}f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))=f^{*}\pi_{\mathscr{C}}(b,\beta(f(b),\sigma(gf(b))))$
(13)

and

$f^{\prime}f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))=f^{\prime}(b,\beta(f(b),\sigma(gf(b))))$
(14)

hold. For Equation 13 we have,

$\begin{array}[]{ll}f^{*}\pi_{\mathscr{C}}f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))&\\\
=(gf)^{*}\pi_{\mathscr{D}}(b,\sigma(gf(b)))&\mbox{by a commutative triangle in
Diagram \ref{fig:rlespacdeffig0}}\\\ =b&\mbox{$(gf)^{*}\pi_{\mathscr{D}}$ is a
projection on first argument}\\\
=f^{*}\pi_{\mathscr{C}}(b,\beta(f(b),\sigma(gf(b)))).&\mbox{$f^{*}\pi_{\mathscr{C}}$
is a projection on first argument}\end{array}$

For Equation 14 the left side is

$\begin{array}[]{ll}f^{\prime}f^{*}\beta\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))&\\\
=\beta f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b))),&\mbox{
by a pullback square in Diagram \ref{fig:rlespacdeffig0}}\end{array}$

and the right side is

$\begin{array}[]{ll}f^{\prime}(b,\beta(f(b),\sigma(gf(b))))&\\\
=\beta(f(b),\sigma(gf(b))).&\mbox{$f^{\prime}$ is a projection on second
argument}\end{array}$

So Equation 14 holds if the equality

$f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))=(f(b),\sigma(gf(b)))$
(15)

holds. By a pullback in Diagram 5, Equation 15 holds if and only if equalities

$g^{*}\pi_{\mathscr{D}}f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))=g^{*}\pi_{\mathscr{D}}(f(b),\sigma(gf(b)))$
(16)

and

$g^{\prime}f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))=g^{\prime}(f(b),\sigma(gf(b)))$
(17)

hold.

For Equation 16 we have,

$\begin{array}[]{ll}g^{*}\pi_{\mathscr{D}}f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))&\\\
=f(gf)^{*}\pi_{\mathscr{D}}(b,\sigma(gf(b)))&\mbox{by a pullback square in
Diagram \ref{fig:rlespacdeffig0}}\\\ =f(b)&\mbox{$(gf)^{*}\pi_{\mathscr{D}}$
is a projection on the first argument}\\\
=g^{*}\pi_{\mathscr{D}}(f(b),\sigma(gf(b))).&\mbox{$g^{*}\pi_{\mathscr{D}}$ is
a projection on first argument}\end{array}$

For Equation 17 we have

$\begin{array}[]{ll}g^{\prime}f^{\prime\prime}\lambda_{f,g,\mathscr{T_{D}}}(b,\sigma(gf(b)))&\\\
=(gf)^{\prime}(b,\sigma(gf(b)))&\mbox{ by a commutative triangle in Diagram
\ref{fig:rlespacdeffig0}}\\\ =\sigma(gf(b))&\mbox{$(gf)^{\prime}$ is a
projection on the second argument}\\\
=g^{\prime}(f(b),\sigma(gf(b))).&\mbox{$g^{\prime}$ is a projection on the
second argument}\end{array}$

This concludes the result. ∎

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