C
ONCEPT OF
F
UZZY
S
ET
Sergejs Solovjovs
Dissertation
zur Erlangung des Grades eines Doktors der Naturwissenschaften
– Dr. rer. nat. –
Vorgelegt im Fachbereich 3 (Mathematik & Informatik) der Universit¨at Bremen
Datum des Promotionskolloquiums: 22. Juni 2007
Erster Gutachter: Prof. Dr. Hans-Eberhard Porst (Universit¨at Bremen) Zweiter Gutachter: Prof. Dr. Alexander ˇSostak (Universit¨at Lettlands)
Abstract v
Acknowledgements vii
Introduction 1
1 Categories of lattice-valued sets as categories of arrows 5
1.1 The category X(A) of A-valued objects . . . . 5
1.1.1 Some examples . . . 7
1.1.2 Basic subcategories . . . 9
1.2 Topological properties of X(A) . . . . 10
1.3 A relation between the functors (−)∗ and (−)◦ . . . 18
2 On a generalization of Goguen’s category Set(L) 21 2.1 The category X(A) and its topological properties . . . . 21
2.2 An example of a non-topological category X(A) . . . . 23
2.3 On concrete cartesian closedness of X(A) . . . . 26
2.4 X(A) is not a topos . . . 28
2.5 On representability of partial morphisms in X(A) . . . . 29
2.6 X(A) is a quasitopos . . . 32
2.6.1 The inner structure of X(A) . . . . 32
2.6.2 A relation between the structures generated by Ω and Δ . . . 36
2.7 Some remarks on representability of partial morphisms in X(A) . . . . 37
3 Aspects of comma categories 41 3.1 Definition of the comma category X∗(A) and its algebraic properties . . . . 41
3.2 A factorization structure for sources on X∗(A) . . . . 47
3.3 Coalgebraic properties of X∗(A) . . . . 50
3.4 A factorization structure for sinks on X∗(A) . . . 54
3.5 Monadic properties of X∗(A) . . . . 56
3.6 Some remarks on the monad T . . . 58
4 On fuzzification of algebraic and topological structures 60
4.1 Fuzzification machinery for algebraic structures . . . 60
4.2 Fuzzification machinery for topological structures . . . 64
5 Quantale modules 67 5.1 Definition of the category Q-Mod of Q-modules . . . 67
5.2 From quantale modules to topological spaces . . . 69
5.3 Q-Mod is a monadic construct . . . . 72
5.4 On some special morphisms in Q-Mod . . . . 75
5.5 Q-Mod is not an abelian category . . . . 79
5.6 Quantale modules do not form a topos . . . 80
5.7 Tensor product of quantale modules . . . 81
5.8 Quantale-valued power-set functors . . . 85
5.9 Factorization structures on Q-Mod . . . . 86
5.10 Completion of partially ordered sets . . . 88
Bibliography 92
The theory of fuzzy structures has been developing rather fast recently. In particular, [22, 23] present different approaches to the foundations of fuzzy sets. By analogy with the aforesaid concept we present the notion of a fuzzy object in a category.
A fuzzy set introduced by Zadeh in [62] is a set equipped with a function to the unit interval [0,1], i.e., just a map X f- [0, 1]. In [18] Goguen replaced the unit interval by an arbitrary (but fixed) partially ordered set L and considered maps of type X f- L called L-fuzzy sets. Later on some authors put a structure on X, e.g., Rosenfeld in [45] used groups. Some obvious generalizations arise, namely:
• consider so-called lattice-valued sets, i.e., allow change of basis L;
• use different lattice-theoretical structures instead of a poset, e.g., quantales or even quantaloids (see, e.g., [46, 47]);
• use different mathematical structures instead of a set.
Bearing the aforesaid ideas in mind we proceed as follows: given a concrete category (A, U ) over X, consider an A-valued object to be an U -structured arrow X f- U A. If A is equipped with the structure of 2-category and the functor U is adjoint one gets a concrete category X(A) of A-valued objects. Our aim is to study properties of this category.
In the first chapter we show the necessary and sufficient conditions for X(A) to be topological. The next two chapters are devoted to the study of two subcategories of X(A). One of them generalizes the Goguen’s category Set(L) of L-fuzzy sets with a fixed basis L (see, e.g., [18]), the other is the comma category (idX ↓ U). The former subcategory gives rise to a fuzzification procedure of algebraic and topological structures considered in the subsequent chapter. The last chapter is devoted to the category of quantale modules (see,
e.g., [46, 47]) as a particular realization of the category A. The chapter is motivated by constructions and results from the category of modules over a ring.
I would like to thank all who helped me in preparation of the thesis. Special thanks are due to my supervisor prof. A. ˇSostaks who was the first man to guide me through my research. I’m also grateful to the team of the University of Bremen, Germany. In the first place to professors H.-E. Porst and H. Herrlich as well as all participants of the seminar ”KatMAT” for many useful suggestions and remarks. Special thanks are to C. Schubert, C. Dzierzon and K. Freund for their patience in answering my questions. An interesting example concerning T0-spaces was suggested by prof. R.-E. Hoffmann when I was participating at his seminar ”Ordered Sets and Lattices”.
During the ”Summer School on General Algebra and Ordered Sets 2005” prof. J. Paseka of Masaryk University in Brno, Chech Republic gave me some good advices on quantale modules as well as sent me his habilitation thesis on this topic.
The financial side of my research was supplied by the European Social Fund (ESF).
Riga, Latvia Sergejs Solovjovs
May 27, 2007
Introduction
It is a well-accepted observation that the real world problems involve different kinds of vague environments. To face the challenge we need new mathematical concepts or, quoting Zadeh [61],
”. . . a radically different kind of mathematics, the mathematics of fuzzy or cloudy quan-tities which are not described in terms of probability distributions.”
This was Zadeh’s main motivation when he started fuzzy sets in [62] as follows:
”A fuzzy set (class) A in [a given set] X is characterized by a membership (characteristic) function fA(x) which associates with each point x in X a real number in the interval [0, 1], with the value of fA(x) at x representing the ”grade of membership” of x in A.”
The next step was done by Goguen in [18] where he wrote:
”A fuzzy set is a set together with a function to a transitive partially ordered set (hereafter called a poset); a fuzzy set is therefore a sort of generalized characteristic function. We habitually denote the poset by L and call the fuzzy set an L-fuzzy set or an L-set.”
By analogy with the category Set of sets Goguen considers the category Set(L) of L-sets with a fixed basis L.
Another important notion of [18] is the Principle of Fuzzification which says that ”a fuzzy (or L-fuzzy or L-) something is an L-set of somethings (i.e., an L-fuzzy set on the set of somethings)”. As a result Rosenfeld in [45] fuzzified the notions of groupoid and group, but Chang [11] and Lowen [32] considered fuzzy topological spaces.
Starting with Hutton [27], Rodabaugh [40], Eklund [15], and others, the following ideas arise
• consider so-called lattice-valued sets, i.e., allow the change of basis L along with the change of set;
• use different lattice-theoretical structures instead of posets, e.g., quantales or even quantaloids (see, e.g., [46, 47]);
• fuzzify different mathematical structures, e.g., modules, fields (in case of the real line R the concept of fuzzy number appears [42]) or categories [59].
This leads to a natural generalization of the notion of fuzzy set, namely, to the notion of fuzzy object in category introduced in this thesis as follows. Start with a concrete category (A, U ) over X.
Definition. A fuzzy (A-fuzzy or A-valued ) object in the category X is an U -structured arrow X f-U A.
Following the standard terminology of the fuzzy community (see, e.g., [25]) we prefer the term A-valued (cf. lattice-valued) object. If A is equipped with the structure of 2-category and the functor U is adjoint one gets the concrete category X(A) of A-valued objects (Definition 1.1.3). The definition allows one to consider different realizations of the category A, for example, such unexpected categories as the category Top of topological spaces or the category Grp of groups (Examples 1.1.9 and 1.1.12). Our aim is to study properties of the category X(A).
In the first chapter we show the necessary and sufficient conditions for X(A) to be topological (Proposition 1.2.24). In a word, the existence of certain functors is required. Since the property of being topological provides the category X(A) with many features of its base category, i.e., the category X× A, one can put the requirements as axioms on X(A). The chapter ends by relations between the aforesaid functors.
The next two chapters are devoted to the study of two subcategories of X(A). The first one generalizes Goguen’s category Set(L) (and therefore is denoted by X(A)), the second one is the comma category (idX ↓ U) (denoted by X∗(A)). We show the necessary and sufficient conditions for X(A) to be a quasitopos (Proposition 2.6.1). It follows that as such the category has an additional rich inner structure (Definitions 2.6.8 – 2.6.16). The main results for the category X∗(A) include the necessary and sufficient conditions to be algebraic (coalgebraic) and monadic (Propositions 3.1.14, 3.3.9 and 3.5.3). We also consider factorization structures on X∗(A) (Propositions 3.2.6 and 3.4.5).
3
We continue by considering a generalization of the aforesaid Principle of Fuzzification. The issue has a rich history since the notion of fuzzy set induced fuzzification of differ-ent mathematical structures. We have already mdiffer-entioned L-topological spaces [11, 24, 32] and L-groups [45]. Both approaches are fixed-based and use implicitly Goguen’s category Set(L). A variable-basis approach over the category of semi-quantales (which is good enough since the categories of fuzzified structures are topological over their ground cat-egories) is considered in [39, 41]. With the help of internal category theory (see, e.g., [33, 12, 13, 14, 8]) one gets a fuzzification machinery of algebraic structures in any suffi-ciently good category (existence of some limits is required), e.g., in the aforesaid category Set(L). Sometimes, however, the ground category is supposed to be a topos (see, e.g., [34]) that is not always true for the category Set(L) (see, e.g., Proposition 2.4.3).
With the aforesaid remarks in mind we introduce a fuzzification machinery over the category X(A). Following the historical move we consider the fixed-basis approach and therefore use the category X(A) for a fixed A-object A. The fuzzification procedure is based on the categories of generalized algebraic and topological structures Alg(T ) and Spa(T ) (see, e.g., [2]). As a result we get the categories Bτ(A) and BP(A) of respective
A-valued structures (the meaning of indices τ and P is explained in the thesis). The attentive reader will notice that the former category generalizes the approach used in [45], the latter category, however, falls out of the aforesaid fuzzification schemes.
In the last chapter we present a realization of the category A through the category Q-Mod of quantale modules (see, e.g., [1, 29, 35, 37, 38, 46, 47]) motivated by constructions and results from modules over a ring. The brief history of the category is as follows.
The term quantale was introduced in [35] in connection with certain aspects of C∗ -algebras. On the other hand, the concept was expected to relate to the semantics of non-commutative logics, for example, that of quantum mechanics. It denotes an algebraic structure, say Q, with two properties:
• Q is a complete lattice;
• Q is equipped with a join-preserving associative binary operation.
In particular, each frame (and therefore each complete Boolean algebra) is a quantale. Other examples include the power-set of a semigroup as well as the set of all relations on a set.
While the study of such structures goes back up to the 1930’s, there has recently been much interest in quantales in a variety of contexts. The most topical connection is with Girard’s linear logic (see, e.g., [17]). In particular, [1] enunciates the following slogan: Quantales are to linear logic as frames are to intuitionistic logic.
The first lattice analogy of ring module was introduced in [29] by A. Joyal and M. Tierney in connection with analysis of descent theory. Although they worked with commutative structures, most of their results are also valid for non-commutative case. The idea of quantale module appeared in [1] of S. Abramsky and S. Vickers as the key notion for treatment of process semantics.
It is the purpose of the last chapter to make a further contribution to the theory of quan-tale modules. It presents the collection of results we obtained while studying the category Q-Mod. In particular, we show that the category Q-Mod is monadic (Proposition 5.3.6), semiadditive but (in general) not abelian (Proposition 5.5.2), and monoidal w.r.t. tensor product (Proposition 5.7.15). Since Q-Mod generalizes the category JCPos of complete lattices and join-preserving maps (JCPos and 2-Mod being isomorphic) and there is a natural relation between JCPos and Top, we consider a relation between Q-Mod and the category Q-Top of Q-topological spaces [24] (Proposition 5.2.6). The chapter ends by a generalization of the method of completion of posets through lower-sets (Propositions 5.10.2 and 5.10.14).
Chapter 1
Categories of lattice-valued sets as
categories of arrows
In this chapter we introduce the main subject of our study, i.e., the category X(A) (see, e.g., [54]). It is a generalization of the category Set(JCPos) of lattice-valued subsets of sets (see, e.g., [55]) defined in our master’s thesis as follows (notice that JCPos is the category of complete lattices and join-preserving maps [2]):
Objects: Maps X α- A where X is a set and A is a complete lattice.
Morphisms: Set× JCPos-morphisms (X, α, A) (f, ϕ)- (Y, β, B) such that ϕ◦ α β ◦ f. We replace Set (resp. JCPos) by an abstract category X (resp. A) and try to find such properties of both categories which make X(A) to resemble Set(JCPos). Our main achievement is Proposition 1.2.24 which shows the necessary and sufficient conditions for X(A) to be topological.
1.1
The category X
(A) of A-valued objects
Suppose (A, U ) is a concrete category over X such that the following conditions are fulfilled: 1 A is a 2-category;
2 U is adjoint.
For an introduction into the theory of 2-categories see, e.g., [33, 8]. For shortness sake we write f =τ⇒ g instead of ”there exists a 2-cell f =τ⇒ g”. With the help of 2 we choose an adjoint situation (η, ) : F U : A - X.
Consider the class F of all structured arrows with domain in X, i.e., of all triples (X, α, A) with X ∈ Ob(X), A ∈ Ob(A) and X α- U A∈ Mor(X). From now on we use the following agreements without further reference:
• the elements of F are denoted by X α- U A or just by α alone if the context is clear; • every X-morphism of the form X α-U A is considered as an element of F, in
particular, the set X(X, U A) is considered as a subfamily ofF. We introduce a relation ”” on F.
Definition 1.1.1. Take any twoF elements X α
-β-U A. By 2 we have two A-morphisms
F X α
-β
- A, where α = A◦ F α. Define α β iff α τ
=⇒ β. Below is one property of the defined relation.
Lemma 1.1.2. Suppose X f-Y α
-β
-U A U ϕ- U B are X-morphisms. If α β, then α◦ f β ◦ f and Uϕ ◦ α Uϕ ◦ β.
Proof. α◦ f = A◦ F (α ◦ f) = A◦ F α ◦ F f = α ◦ F f τ =⇒ β ◦ F f = β ◦ f and Uϕ ◦ α = B◦ F (Uϕ ◦ α) = (B◦ F Uϕ) ◦ F α = (ϕ ◦ A)◦ F α = ϕ ◦ α τ =⇒ ϕ ◦ β = Uϕ ◦ β.
Notice that if we replace U A U ϕ- U B by an arbitrary X-morphism U A h-U B, then α β does not imply h ◦ α h ◦ β.
Now the main definition.
Definition 1.1.3. The category X(A) is defined as follows:
Objects: The above-mentioned class F of all structured arrows with domain in X. Morphisms: X× A-morphisms (X, α, A) (f, ϕ)- (Y, β, B) such that U ϕ◦ α β ◦ f.
The fact that X(A)-morphisms are closed under composition can be easily checked with the help of Lemma 1.1.2.
We consider the category X(A) as a concrete category over the product category X×A in the following way.
Definition 1.1.4. Define the forgetful functor X(A) |−|-X× A as follows: |(X, α, A) (f, ϕ)-(Y, β, B)| = (X, A) (f, ϕ)- (Y, B).
7
1.1.1 Some examples
In this subsection we give some examples of the categories of the form X(A).
Example 1.1.6. Any category X is a 2-category in a trivial way with X(X,Y) considered as discrete categories. Since the identity functor X idX- X is adjoint we get a category X(X) which is isomorphic to the comma category (idX↓ idX).
In a similar way one can consider the category A(A) for any 2-category A.
Example 1.1.7. The construct (JCPos, U ) satisfies both properties 1 and 2 since free objects are just power-sets P(X). Moreover, one can easily see that for every two maps X α
-β
-U A, it follows that α β in the usual (pointwise) sense iff α β in the sense of Definition 1.1.1.
Example 1.1.8. Consider the category SetRel of sets and relations. One can easily show that the category is isomorphic to the Kleisli category of the power-set monad. This fact induces the following definition of the forgetful functor SetRel U-Set:
X ρ- Y → P(X) fρ- P(Y ) : S → {y ∈ Y | there exists x ∈ S such that xρy}. Then U is a faithful functor with a left adjoint Set F-SetRel given as follows:
X f-Y → X ρf- Y with xρfy iff f (x) = y.
Notice that F is a non-full embedding. Add natural transformations idSet η- U F with X ηX-U F X : x → {x} as well as F U - idSetRel where F U X X- X is de-fined as follows: SXx iff x ∈ S. You get an adjoint situation (η, ) : F U :
SetRel - Set. Since SetRel is a 2-category you get a category Set(SetRel). Notice that given X α
-β
-U Y it follows that α β iff α(x) ⊆ β(x) for all x ∈ X.
Example 1.1.9. Consider the construct (Top, U ) of topological spaces and continuous maps. U -free objects are just discrete topological spaces. Moreover, the homotopy classes of homotopies as 2-cells make Top a 2-category. Thus, one can consider the category Set(Top).
The construct Top gives rise to one more example.
Example 1.1.10. Consider the category Prost of preordered sets (i.e., sets supplied with a reflexive and transitive relation) and isotone maps. According to [21] there exists a functor Top U- Prostdefined as follows:
U has a full and faithful left adjoint Prost F-Top given as follows:
(X,) f-(Y,) → (X, τ) f- (Y, σ), where τ has{↑ x | x ∈ X} as a subbase. Add natural transformations idProst η- U F with (X,) η(X,)- U F (X,) : x → x as well as F U - idTop with F U (X, τ ) (X,τ)- (X, τ ) : x→ x. You get an adjoint situation (η, ) : F U : Top - Prost. Considering Top as a 2-category as in Example 1.1.9 one gets the category Prost(Top).
Example 1.1.11. Consider the construct (Cat, U ) of small categories and functors. One can easily check that the functor U is adjoint (see, e.g., Example 18.2 in [2]). Moreover, natural transformations between functors make Cat a 2-category. Thus, one can consider the category Set(Cat).
Example 1.1.12. Consider the construct (Grp, U ) of groups and group homomorphisms. Since U is adjoint it remains to be shown that Grp can be considered as a 2-category that can be done as follows. Given two group homomorphisms A f
-g
-B define a 2-cell f =τ⇒ g as an element τ ∈ B such that for every element a ∈ A, f(a)·τ = τ ·g(a), where · denotes the composition law of the group A. Given 2-cells f =τ⇒ g and h=υ⇒ k define the composition h◦ f =υτ⇒ k ◦ g as an element h(τ) · υ = υ · k(τ).
The next example is in a sense a generalization of Example 1.1.7. It is motivated by [47] and requires two additional definitions.
Definition 1.1.13. A quantaloid is a category Q such that • for A, B ∈ Ob(Q), the hom-set Q(A, B) is a complete lattice; • composition of morphisms in Q preserves joins in both variables.
In other words, a quantaloid is a category enriched in the monoidal category JCPos (see, e.g., [30] as well as Section 5.7 in Chapter 5 of this thesis).
Definition 1.1.14. Let Q and S be quantaloids. A quantaloid homomorphism is a functor Q F- S inducing on hom-sets a join-preserving map Q(A, B) - S(F (A), F (B)). Example 1.1.15. Let Qtlds denote the category of small quantaloids (notice that the assumption of smallness is not crucial here and is used only to avoid dealing with quasicat-egories instead of catquasicat-egories). There exists the obvious forgetful functor Qtlds U- Cat. Given a category X, define the free quantaloid F (X) over X as follows. The objects of F (X) are precisely those of X. Given F (X)-objects X, Y let F (X)(X, Y ) = P(X(X, Y )). Given F (X)-morphisms X S- Y T- Z let T ◦ S = {t ◦ s | t ∈ T, s ∈ S}. This operation preserves unions in each variable and therefore yields a quantaloid. Define
9
Cat F- Qtlds: X G- Y→ F (X) Gˆ- F (Y) with ˆG(X S- Y ) = G(X) G[S]- G(Y ). Add natural transformations idCat η- U F with X ηX- U F X : X f- Y → X {f}- Y and F U -idQtlds with F U Q A- Q : A S- B → A
S
- B. One gets an adjoint situation (η, ) : F U : Qtlds - Cat.
Natural transformations between functors make Qtlds a 2-category and partial order on hom-sets of quantaloids makes it even a 3-category. Thus, one can consider the category Cat(Qtlds).
Example 1.1.15 will be studied more closely in the next chapter. Notice that the 2-categories considered in Examples 1.1.7 and 1.1.8 have thin 2-cells. All other examples have no such property.
1.1.2 Basic subcategories
Below are listed several subcategories of X(A) which will be studied in the next chapters. Example 1.1.16. The category (idX ↓ U) is a nonfull subcategory of X(A). Recall that it is defined as follows:
Objects: X(A)-objects.
Morphisms: X(A)-morphisms (X, α, A) (f, ϕ)- (Y, β, B) such that U ϕ◦ α = β ◦ f. The category (idX↓ U) will be denoted by X∗(A).
Example 1.1.17. Fix an A-object A and define a category X(A) as follows: Objects: X(A)-objects (X, α, A).
Morphisms: X(A)-morphisms (X, α, A) (f, idA-) (Y, β, A).
Since the object A is fixed we will denote objects and morphisms of X(A) by (X, α) and f respectively.
Notice that in case of the category Set(JCPos) the subcategory Set(A) is the category of A-fuzzy subsets of sets defined in [18]. Thus X(A) is a generalization of Set(A).
Example 1.1.18. Fix an X-object X and define a category X(A) as follows: Objects: X(A)-objects (X, α, A).
Morphisms: X(A)-morphisms (X, α, A) (idX, ϕ-) (X, β, B).
Since the object X is fixed we will denote objects and morphisms of X(A) by (α, A) and ϕ respectively. If I is an initial object of X, then I(A) is isomorphic to A.
1.2
Topological properties of X
(A)
In this section we are going to find necessary and sufficient conditions for X(A) to be topological over X× A.
Start with the sufficient ones. The statement that X(A) is topological over X× A can can be proved in two different ways providing either initial or final lifts. We will consider initial ones first.
Introduce the following requirements.
3 For every X × A-object (X, A), (X(X, UA), ) is a complete lattice.
4 For every X-morphism Y f- X and every A-object A it follows that the map X(X, U A) −◦f- X(Y, U A) is meet-preserving.
5 There exists a functor A (−)-∗ Xop : A ϕ- B → B∗ ϕ∗- A∗ with the following properties:
(i) A∗ = U A and B∗ = U B; (ii) idU A ϕ∗◦ Uϕ;
(iii) U ϕ◦ ϕ∗ idU B;
(iv) for every A-morphism B ϕ- A and every X-object X it follows that the map X(X, U A) ϕ∗◦ −- X(X, U B) is order-preserving.
First of all two simple properties of the functor (−)∗. Lemma 1.2.1. 5 determines A (−)-∗ Xop uniquely. Proof. If both A (−)
∗
-(−)- X
op have the required properties and A ϕ- B is an A-morphism,
then ϕ (ϕ∗◦ Uϕ) ◦ ϕ = ϕ∗◦ (Uϕ ◦ ϕ) ϕ∗. Similar ϕ∗ ϕ and therefore ϕ∗ = ϕ. Lemma 1.2.2. Let A (−)-∗ Xop be a function such that (A ϕ-B)∗ = B∗ ϕ∗- A∗. If (−)∗ satisfies (i) – (iv) of 5 , then it is necessarily a functor.
Proof. If A idA- A is an A-identity, then idU A (idA)∗◦ UidA= (idA)∗ = U idA◦ (idA)∗
idU A and therefore idU A = (idA)∗. Thus (−)∗ preserves identities.
If A ϕ- B ψ- C are A-morphisms, then (ψ◦ ϕ)∗ ϕ∗ ◦ Uϕ ◦ (ψ ◦ ϕ)∗ ϕ∗ ◦ ψ∗ ◦ Uψ ◦ Uϕ ◦ (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ∗◦ U(ψ ◦ ϕ) ◦ (ψ ◦ ϕ)∗ ϕ∗◦ ψ∗. On the other hand, ϕ∗◦ψ∗ (ψ◦ϕ)∗◦U(ψ◦ϕ)◦ϕ∗◦ψ∗ = (ψ◦ϕ)∗◦Uψ◦Uϕ◦ϕ∗◦ψ∗ (ψ◦ϕ)∗◦Uψ◦ψ∗ (ψ◦ϕ)∗. Thus, (ψ◦ ϕ)∗ = ϕ∗◦ ψ∗ and therefore (−)∗ preserves composition.
11
Now some examples.
Example 1.2.3. Look at the category JCPos and you will easily find the required functor (−)∗ (see, e.g., Chapter 0.3 in [16]):
JCPos (−)-∗ Setop: A ϕ- B → UB ϕ∗- U A : b→ϕ−1[↓ b],
i.e., ϕ∗ is just the upper adjoint (in the sense of partially ordered sets) of ϕ. Notice that in general ϕ∗ does not preserve arbitrary joins however it does preserve arbitrary meets and therefore is order-preserving.
Example 1.2.4. In case of the category SetRel define the functor SetRel (−)-∗ Setopas follows (notice that power-sets are complete lattices):
X ρ- Y → P(Y ) fρ- P(X) : S → T :=(U ρ)−1[↓ S]. The latter set T can be written as {x ∈ X | {y ∈ Y | xρy} ⊆ S}.
The next proposition is an immediate consequence of assumption 5 .
Proposition 1.2.5. An X×A-morphism |(X, α, A)| (f,ϕ)- |(Y, β, B)| is an X(A)-morphism iff α ϕ∗◦ β ◦ f.
Proof. If U ϕ◦ α β ◦ f, then α ϕ∗ ◦ Uϕ ◦ α ϕ∗◦ β ◦ f. Conversely, α ϕ∗◦ β ◦ f implies U ϕ◦ α Uϕ ◦ ϕ∗◦ β ◦ f β ◦ f.
We are going to show that 3 – 5 guarantee the existence of initial lifts in X(A). The first proposition is immediate from the definition of X(A).
Proposition 1.2.6. The category X(A) is amnestic. Now we can prove the following.
Proposition 1.2.7. The concrete category (X(A), | − |) is topological.
Proof. In view of the preceding proposition it will be enough to show the existence of initial lifts. Let S = ((X, A) (fi, ϕ-i) |(Xi, αi, Ai)|)i∈I be a | − |-structured source. Define
α =
i∈I
[ϕ∗i ◦ αi◦ fi]. Proposition 1.2.5 gives us a lift ˆS of S. We show that the lift is initial.
Take a source T = ((Y, β, B) (gi, ψ-i) (Xi, αi, Ai))i∈I in X(A) and an X× A-morphism
|(Y, β, B)| (f,ϕ)-|(X, α, A)| such that |T | = | ˆS| ◦ (f, ϕ). Since Uϕ ◦ β ϕ∗
i ◦ Uϕi◦ Uϕ ◦ β =
ϕ∗i ◦ Uψi ◦ β ϕ∗i ◦ αi ◦ gi = ϕ∗i ◦ αi ◦ fi ◦ f implies Uϕ ◦ β i∈I
(ϕ∗i ◦ αi ◦ fi ◦ f) =
i∈I
As an immediate consequence we get that both categories Set(JCPos) and Set(SetRel) are topological.
Now consider final lifts. For the moment we will forget about assumptions 3 – 5 and show the second way to prove that X(A) is topological over X× A.
Start with the following definition.
Definition 1.2.8. We say that a category C is a complete quasilattice provided that the following conditions are fulfilled:
(i) If A f- B and B g- A are C-morphisms, then A = B.
(ii) For every family (Ai)i∈I of C-objects, there exists a (necessarily unique) C-object A
such that
(a) for every i∈ I, there exists a C-morphism Ai fi
-A;
(b) if B is another C-object with property (a), then there exists a C-morphism A f- B.
The object A defined in (ii) will be called the quasijoin of the family (Ai)i∈I and denoted
by
i∈I
Ai.
Notice that each complete quasilattice has the unique initial object, i.e., the quasijoin of the empty family.
A simpler characterization of complete quasilattices is given by the next lemma. Lemma 1.2.9. Given a category C define a relation ”” on the class of its objects as follows: A B iff C(A, B) is not empty. Then C is a complete quasilattice iff (Ob(C), ) is a (possibly large) complete lattice with quasijoins becoming joins.
Consider two simple examples of complete quasilattices.
Example 1.2.10. Take a complete po-monoid (X, +,) (see [7]) with the property that 0 = ⊥, where 0 is the identity of (X, +) and ⊥ is the lower bound of (X, ). Define a category C as follows:
Objects: The set X.
Morphisms: Given C-objects x, y let C(x, y) ={a | a ∈ X, x + a y}.
Composition: Given C-morphisms x a- y and y b- x define b◦ a = a + b. One can easily see that C is a complete quasilattice.
For the concrete realization consider the set N∗ = N{∞} with the extension of the usual operations in the following way: x ∞ and x + ∞ = ∞ + x = ∞ for all x ∈ X. Then N∗ has the required properties and gives rise to the category C.
13
Example 1.2.11. Take a complete lattice (X,). Let (ρi)i∈I be a family of relations on
X such that =
i∈I
ρi. Define a graph G as follows:
Objects: The set X.
Morphisms: GivenG-objects x, y let G(x, y) = {ρi| i ∈ I, xρiy}.
Let C be the ”path category of G”. One can easily see that C is a complete quasilattice. For the concrete realization consider the chain 2. Take I = {1, 2, 3} and define ρ1 = {(0, 0)}, ρ2 = {(0, 1)}, ρ3 = {(1, 1)}. Then (ρi)i∈I has the required properties and gives
rise to the category C.
Introduce new requirements.
6 For every X × A-object (X, A), A(F X, A) is a complete quasilattice.
7 For every A-morphism A ϕ- B and every X-object X it follows that the map A(F X, A) ϕ◦−- A(F X, B) is quasijoin-preserving.
8 There exists a functor X (−)-◦ Aop : X f- Y → Y◦ f◦- X◦ with the following properties: (i) X◦= F X and Y◦= F Y ; (ii) F f◦ f◦ τ=⇒ idF Y; (iii) idF X υ =⇒ f◦◦ F f;
(iv) for every X-morphism X f- Y and every A-object A it follows that the map A(F X, A) −◦f-◦ A(F Y, A) is quasijoin-preserving.
Similar to Lemmas 1.2.1 and 1.2.2 one can prove the following. Lemma 1.2.12. (i) – (iii) of 8 determine X (−)-◦ Aop uniquely.
Lemma 1.2.13. Let X (−)-◦ Aop be a function such that (X f- Y )◦ = Y◦ f◦- X◦. If (−)◦ satisfies (i) – (iii) of 8 , then it is necessarily a functor.
Notice that we do not use condition (iv) of 8 in Lemmas 1.2.12 and 1.2.13. Now some examples.
Example 1.2.14. In case of the category JCPos define the required functor as follows: Set (−)-◦ JCPosop: X f- Y → P(Y ) f−1- P(X) : S → f−1[S].
Example 1.2.15. In case of the category SetRel define the required functor as follows: Set (−)-◦ SetRelop: X f-Y → Y ρ
−1 f
-X. Recall that yρ−1f x iff f (x) = y.
The next proposition is an immediate consequence of assumption 8 .
Proposition 1.2.16. An X× A-morphism |(X, α, A)| (f,ϕ)-|(Y, β, B)| is an X(A)-morp-hism iff ϕ◦ α ◦ f◦ τ=⇒ β.
Proof. If U ϕ◦α β ◦f, then ϕ◦α = Uϕ ◦ α=τ⇒ β ◦ f = β ◦F f and therefore ϕ◦α◦f◦ υ=⇒ β◦F f ◦f◦ =⇒ β. Conversely, ϕ◦α=τ⇒ ϕ◦α ◦f◦◦F f =υ⇒ β ◦F f implies Uϕ◦α β ◦f.
Now we can prove the following.
Proposition 1.2.17. In the category X(A) every| − |-costructured sink has a unique final lift.
Proof. In view of Proposition 1.2.6 it will be enough to show the existence of final lifts. Let S = (|(Xi, αi, Ai)| (fi
, ϕ-i)
(X, A))i∈I be a| − |-costructured sink. Define α =
i∈I
[ϕi◦ αi◦ fi◦].
Proposition 1.2.16 gives us a lift ˆS of S. We show that the lift is final. Take a sink T = ((Xi, αi, Ai) (gi
, ψ-i)
(Y, β, B))i∈I in X(A) and an X× A-morphism
|(X, α, A)| (f, ϕ)- |(Y, β, B)| with |T | = (f, ϕ)◦| ˆS|. Since ϕ◦ϕi◦αi◦fi◦◦f◦= ψi◦αi◦g◦i τi =⇒ β implies ϕ◦ α ◦ f◦= ϕ◦ i∈I (ϕi◦ αi◦ fi◦)◦ f◦ = i∈I (ϕ◦ ϕi◦ αi◦ fi◦◦ f◦) υ =⇒ β, then (f, ϕ) is an X(A)-morphism.
We showed that requirements 6 – 8 independent from requirements 3 – 5 imply that X(A) is topological over X× A.
Notice that 3 – 5 imply the following (see Proposition 21.13 in [2]).
Proposition 1.2.18. An X(A)-morphism (X, α, A) (f, ϕ)- (Y, β, B) is an extremal (regular) monomorphism iff (f, ϕ) is an extremal (regular) monomorphism in X×A and α = ϕ∗◦β◦f. Lemma 1.2.19. An X∗(A)-morphism (X, α, A) (f, ϕ)-(Y, β, B) is an initial X(A)-morp-hism iff α = ϕ∗◦ Uϕ ◦ α.
Proof. The necessity: α = ϕ∗◦ β ◦ f = ϕ∗ ◦ Uϕ ◦ α. The sufficiency: α = ϕ∗ ◦ Uϕ ◦ α = ϕ∗◦ β ◦ f.
Applying Lemma 1.2.19 to the category Set(JCPos) we get that U ϕ◦α = β◦f together with injectivity of ϕ imply initiality of (f, ϕ).
15
Proposition 1.2.20. An X(A)-morphism (X, α, A) (f, ϕ)- (Y, β, B) is an extremal (regular) epimorphism iff (f, ϕ) is an extremal (regular) epimorphism in X× A and ϕ ◦ α ◦ f◦= β. Lemma 1.2.21. An X∗(A)-morphism (X, α, A) (f, ϕ)- (Y, β, B) is a final X(A)-morphism iff β◦ F f ◦ f◦ = β.
Applying Lemma 1.2.21 to the category Set(JCPos) we get that U ϕ◦α = β◦f together with surjectivity of f imply finality of (f, ϕ).
Now we are going to prove that the aforesaid requirements are also necessary for X(A) being topological over X× A.
Proposition 1.2.22. Suppose 1 – 2 hold. If X(A) is topological over X × A, then 3 – 5 hold.
Proof. 3 : Let T (X, A) be the fibre of (X, A). Then the map X(X, UA) h-T (X, A) : α→ α is bijective. Since the following are equivalent:
(i) α β;
(ii) |(X, α, A)| (idX,idA-) |(X, β, A)| is an X(A)-morphism; (iii) (X, α, A) (X, β, A);
and, moreover, (T (X, A),) is a complete lattice, then (X(X, UA), ) must be also. 4 : Let Y f- X be an X-morphism. Consider a subset S = (X αi- U A)i∈I of
X(X, U A). We show that
i∈I
(αi ◦ f) (
i∈I
αi)◦ f. Since X(A) is topological, the source
S = ((X,
i∈I
αi, A) (idX ,idA-)
(X, αi, A))i∈I is initial. Since
|(Y, i∈I (αi◦ f), A)| (f,idA-) |(X, i∈I αi, A) (idX ,idA-) (X, αi, A)|
is an X(A)-morphism for every i∈ I, then |(Y,
i∈I
(αi◦ f), A)| (f,idA-) |(X, i∈I
αi, A)| must
be also and therefore
i∈I
(αi◦ f) (
i∈I
αi)◦ f. The converse inclusion follows immediately
from Lemma 1.1.2.
5 : Let A ϕ- B be an A-morphism. Then (U B, A) (idUB,ϕ-) |(UB, idU B, B)| is an
X× A-morphism and therefore has an initial lift (UB, ϕ∗, A) (idUB,ϕ-) (U B, idU B, B) in
X(A). Consider some properties of ϕ∗.
(ii) Since
|(UA, idU A, A)| (Uϕ,idA-) |(UB, ϕ∗, A) (idUB ,ϕ-)
(B, idU B, B)|
is an X(A)-morphism, then |(UA, idU A, A)| (Uϕ,idA-) |(UB, ϕ∗, A)| must be also and
therefore idU A ϕ∗◦ Uϕ.
(iii) Take X α
-β
-U B with α β. Since
|(X, ϕ∗◦ α, A)| (β,idA-) |(B, ϕ∗, A) (idUB,ϕ-)
(U B, idU B, B)|
is an X(A)-morphism, then |(X, ϕ∗ ◦ α, A)| (β,idA-) |(B, ϕ∗, A)| must be also and therefore ϕ∗◦ α ϕ∗◦ β.
Define A (−)-∗ Xop : A ϕ- B → UB ϕ∗- U A. Lemma 1.2.2 implies that (−)∗ is a functor.
Proposition 1.2.23. Suppose 1 – 2 hold. If X(A) is topological over X × A, then 6 – 8 hold.
Proof. 6 : Consider the relational isomorphism (X(X, UA), ) (−)- (Ob(A(F X, A)), ), where ϕ ψ iff ϕ=τ⇒ ψ. Proposition 1.2.22 implies that (X(X, UA), ) is a complete lattice and therefore (Ob(A(F X, A)), ) must be also. Lemma 1.2.9 implies that A(F X, A) is a complete quasilattice.
7 : Let A ϕ- B be an A-morphism. Consider a subset S = (F X αi- A)i∈I of
A(F X, A)-objects. We show that ϕ◦ (
i∈I
αi) =τ⇒ i∈I
(ϕ◦ αi). Proposition 1.2.22 implies
that the source S = ((X, αi, A) (idX ,idA-)
(X,
i∈I
αi, A))i∈I is final. Since
|(X, αi, A) (idX ,idA-) (X, i∈I αi, A)| (idX ,ϕ-) |(X, i∈I (U ϕ◦ αi), B)|
is an X(A)-morphism for every i ∈ I, then |(X,
i∈I αi, A)| (idX ,ϕ-) |(X, i∈I (U ϕ◦ αi), B)|
must be also and therefore U ϕ◦ (
i∈I αi) i∈I (U ϕ◦ αi). Then ϕ◦ ( i∈I αi) = ϕ◦ i∈I αi = U ϕ◦ ( i∈I αi) τ =⇒ i∈I (U ϕ◦ αi) = i∈I U ϕ◦ αi = i∈I
ϕ◦ αi. The converse inclusion follows
immediately from Definition 1.2.8.
8 : Let X f- Y be an X-morphism. Then |(X, ηX, F X)| (f,idF X-) (Y, F X) is an
X× A-morphism and therefore has a final lift (X, ηX, F X) (f,idF X-) (Y, ˆηX, F X) in X(A).
Then F Y ˆηX- F X is an A-morphism. Consider some properties of ˆηX.
(i) Since (f, idF X) is an X(A)-morphism, then ηX ˆηX ◦ f and therefore idF X τ
=⇒ ˆ
17
(ii) Since
|(X, ηX, F X) (f,idF X-) (Y, ˆηX, F X)| (idY ,F f-)
|(Y, ηY, F Y )|
is an X(A)-morphism, then |(Y, ˆηX, F X)| (idY ,F f-)
|(Y, ηY, F Y )| must be also and
therefore U F f◦ ˆηX ηY that implies F f◦ ˆηX τ
=⇒ idF Y.
(iii) Let X f- Y be an X-morphism and let A be an A-object. Consider a subset S = (F X α-i A)i∈I of A(F X, A)-objects. We show that (
i∈I αi)◦ˆηX τ =⇒ i∈I (αi◦ˆηX).
For every j∈ I, it follows that αj Uαj◦ ˆηX◦f
i∈I (U αi◦ ˆηX◦f) i∈I (U αi◦ ˆηX)◦f and therefore i∈I αi i∈I (U αi◦ ˆηX)◦ f. Since |(X, ηX, F X) (f,idF X-) (Y, ˆηX, F X)| (idY, i∈I αi) - |(Y, i∈I (U αi◦ ˆηX), A)|
is an X(A)-morphism, then |(Y, ˆηX, F X)| (idY,
i∈Iα-i)
|(Y,
i∈I
(U αi◦ ˆηX), A)| must be
also and therefore U (
i∈I αi)◦ ˆηX i∈I (U αi◦ ˆηX). Thus ( i∈I αi)◦ ˆηX τ =⇒ i∈I (αi◦ ˆηX).
The converse inclusion follows immediately from Definition 1.2.8.
Define X (−)-◦ Aop: X f-Y → F Y ˆηX- F X. Lemma 1.2.13 implies that (−)◦ is a functor.
The main result of the section is as follows.
Proposition 1.2.24. Suppose 1 – 2 hold. The following are equivalent: (i) X(A) is topological over X× A;
(ii) 3 – 5 hold; (iii) 6 – 8 hold.
Proof. Follows immediately from Propositions 1.2.7, 1.2.17, 1.2.22, 1.2.23. An immediate consequence of Proposition 1.2.24 is the following result.
Proposition 1.2.25. The categories Set(JCPos) and Set(SetRel) are topological. None of the categories Set(Top), Prost(Top), Set(Cat), Set(Grp) is topological.
Proof. For the categories Set(JCPos) and Set(SetRel) consider Examples 1.2.3 and 1.2.4. All other categories do not satisfy requirement 6 .
1.3
A relation between the functors
(−)
∗and
(−)
◦In the previous section we introduced two functors A (−)-∗ Xop and X (−)-◦ Aop. By Proposition 1.2.24 we know that the existence of one of them implies the existence of both. In this section we are going to investigate somewhat deeper the relation between them.
In the following we assume that the category X(A) is topological over X× A. Proposition 1.3.1. X (−)-◦ Aop can be defined through A (−)-∗ Xop as follows:
X f- Y → F Y F X◦F ((F f)∗◦ηY-) F X.
Proof. Let F Y f•- F X be as defined. Consider some properties of f•.
(i) Since U F f◦(F f)∗ idU F Y implies U F f◦(F f)∗◦ηY ηY, then F f◦f• = F f◦F X◦
F ((F f )∗◦ ηY) = F Y ◦ F (UF f ◦ (F f)∗◦ ηY) = U F f◦ (F f)∗◦ ηY τ
=⇒ ηY = idF Y.
(ii) Since idU F X (F f)∗◦ UF f implies ηX (F f)∗◦ UF f ◦ ηX = (F f )∗◦ ηY ◦ f, then
idF X = ηX τ
=⇒ (F f)∗◦ ηY ◦ f = F X◦F ((F f)∗◦ηY ◦f) = F X◦F ((F f)∗◦ηY)◦F f =
f•◦ F f.
Lemma 1.2.13 implies that (−)•is a functor and therefore (−)• = (−)◦by Lemma 1.2.12.
The next proposition gives another property of both functors. Proposition 1.3.2. The diagram
X F- A Aop (−)◦ ? Uop- X op (−)∗ ? commutes.
Proof. Let X f- Y be an X-morphism. Since F f ◦ f◦ =τ⇒ idF Y implies U (F f ◦ f◦)
U idF Y, then U f◦ (F f)∗◦ UF f ◦ Uf◦ (F f)∗. Conversely, idF X τ
=⇒ f◦ ◦ F f implies U idF X U(f◦◦ F f) and therefore (F f)∗ Uf◦◦ UF f ◦ (F f)∗ Uf◦.
Propositions 1.3.1 and 1.3.2 imply the following result.
Proposition 1.3.3. If X f- Y is an X-morphism, then (F f )∗ = U (F X◦F ((F f)∗◦ηY)).
By Proposition 1.3.1 we know that (−)◦ can be defined through (−)∗. What about the converse relation, namely, is it possible to define (−)∗ through (−)◦. The next proposition answers the question.
19
Proposition 1.3.4. A (−)-∗ Xop can be defined through X (−)-◦ Aop as A ϕ- B → UB U(A◦(Uϕ)◦)◦ηUB- U A
iff the following holds: (A) for every X α -β- U B∈ F equivalent are (i) B◦ F α τ =⇒ B◦ F β; (ii) A◦ (Uϕ)◦◦ F α υ
=⇒ A◦ (Uϕ)◦◦ F β for every A-morphism A ϕ
-B. Proof. The sufficiency: Let U B ϕ- U A be as defined. Consider some properties of ϕ.
(i) Since F U ϕ◦ (Uϕ)◦ =τ⇒ idF U B implies U ϕ◦ ϕ = B◦ F (Uϕ ◦ ϕ) = ϕ◦ A◦ F ϕ =
ϕ◦ A◦ F (U(A◦ (Uϕ)◦)◦ ηU B) = ϕ◦ A◦ F U A◦ F (U(Uϕ)◦◦ ηU B) = ϕ◦ A◦ (Uϕ)◦◦
F U B◦F ηU B = ϕ◦A◦(Uϕ)◦ = B◦F Uϕ◦(Uϕ)◦ υ
=⇒ B= idU B, then U ϕ◦ϕ idU B.
(ii) Since idF U A=τ⇒ (Uϕ)◦◦F Uϕ implies idU A = A=υ⇒ A◦(Uϕ)◦◦F Uϕ = A◦(Uϕ)◦◦
F U B◦ F (ηU B ◦ Uϕ) = A◦ F U A◦ F (U(Uϕ)◦◦ ηU B◦ Uϕ) = A◦ F (U(A◦ (Uϕ)◦)◦
ηU B◦ Uϕ) = A◦ F (ϕ◦ Uϕ) = ϕ◦ Uϕ, then idU A ϕ◦ Uϕ.
(iii) Let X α
-β- U B with α β and let A ϕ
-B be an A-morphism. Since α β implies B◦ F α τ =⇒ B◦ F β, then ϕ◦ α = A◦ (Uϕ)◦ ◦ F α υ =⇒ A◦ (Uϕ)◦◦ F β = ϕ◦ β and therefore ϕ◦ α ϕ◦ β.
Lemma 1.2.2 implies that (−) is a functor and therefore (−) = (−)∗ by Lemma 1.2.1. The necessity:
(i)⇒ (ii) If α β, then ϕ∗◦ α ϕ∗◦ β implies A◦ (Uϕ)◦◦ F α = ϕ∗◦ α =τ⇒ ϕ∗◦ β =
A◦ (Uϕ)◦◦ F β.
(ii)⇒ (i) Consider ϕ = idB.
Proposition 1.3.4 says that under the assumption of (A) one can define (−)∗ through (−)◦. We are going to show a simpler necessary condition for the possibility of such defini-tion.
Start with a preliminary lemma. Lemma 1.3.5. If A ϕ
-ψ
Proof. Since ϕ =τ⇒ ψ implies Uϕ Uψ, then idU A ϕ∗ ◦ Uϕ ϕ∗◦ Uψ and therefore
ψ∗ ϕ∗◦ Uψ ◦ ψ∗ ϕ∗.
Corollary 1.3.6. Let X be an X-object. If ϕ is an initial object of A(F X, F X), then idU F X ϕ∗.
For shortness sake introduce the following property
(B) For every X-object X, idU F X U(F X ◦ (Uϕ)◦)◦ ηU F X, where ϕ is an initial object
of A(F X, F X).
Proposition 1.3.7. If (−)∗ can be defined through (−)◦ as in Proposition 1.3.4, then (B) holds.
One can easily see that (B) holds neither in the category Set(JCPos) nor in the cate-gory Set(SetRel).
Chapter 2
On a generalization of Goguen’s
category Set
(L)
In this chapter we fix an A-object A and consider the subcategory X(A) of the category X(A) (see, e.g., [57]). Recall that X(A) can be considered as a generalization of the Goguen’s category Set(L) of L-fuzzy sets [18]. We show the necessary and sufficient condi-tions for X(A) to be a quasitopos (Proposition 2.6.1). It follows that as such the category has an additional rich inner structure (Definitions 2.6.8 – 2.6.16).
2.1
The category X
(A) and its topological properties
We begin by recalling the definition of the category X(A) from the previous chapter. Definition 2.1.1. Let A be an A-object. Define the category X(A) as follows: Objects: (X, α) where X α- U A is a structured arrow.
Morphisms: X-morphisms (X, α) f- (Y, β) such that α β ◦ f.
We consider the category X(A) as a concrete category over X in the following way. Definition 2.1.2. Define the forgetful functor X(A) |−|-X as follows:
|(X, α) f- (Y, β)| = X f- Y.
In the previous chapter we found the necessary and sufficient conditions for the cate-gory X(A) to be topological over X× A (see Proposition 1.2.24). It was crucial to have
the functors (−)∗ and (−)◦. In case of the category X(A) the situation is much simpler. Introduce the following weak versions of requirements 3 – 4 .
3 For each X-object X, (X(X, UA), ) is a complete lattice.
4 For every X-morphism Y f- X it follows that the map X(X, U A) −◦f- X(Y, U A) is meet-preserving.
Proposition 2.1.3. Suppose 1 – 2 hold. The following are equivalent: (i) X(A) is topological over X;
(ii) 3 – 4 hold.
Proof. (i) =⇒ (ii): 3 – 4: Proceed as in Proposition 1.2.22.
(ii) =⇒ (i): Let S = (X fi- |(Xi, αi)|)i∈I be a| − |-structured source. To show that
it has an initial lift, define α =[αi◦ fi] and proceed as in Proposition 1.2.7.
Propositions 1.2.24 and 2.1.3 imply the following result. Proposition 2.1.4. Suppose 1 , 2 hold. Equivalent are:
(i) X(A) is topological over X× A;
(ii) every subcategory X(A) of the category X(A) is topological over X and 5 holds. From now one we assume that the category X(A) (and thus by Proposition 2.1.4 every subcategory of the form X(A)) is topological.
Lemma 2.1.5. Let X f- Y be an X-morphism. Then |(X, α)| f- |(Y, β)| is an X(A)-morphism iff |(X, α, A)| (f,idA-) |(Y, β, A)| is an X(A)-morphism.
With the help of Lemma 2.1.5 and Proposition 1.2.17 one can prove the following. Proposition 2.1.6. Let S = (|(Xi, αi)|
fi
-X)i∈I be a | − |-costructured sink. Define
α =
i∈I
[αi◦ fi◦]. Then ˆS = ((Xi, αi) fi
-(X, α))i∈I is a final lift of S in X(A).
Corollary 2.1.7. Let S = (|(X, αi)| idX
-X)i∈I be a | − |-costructured sink. Then ˆS =
((X, αi) idX
-(X,
i∈I
αi)) is a final lift of S.
23
2.2
An example of a non-topological category X
(A)
The previous section implies the following result: given a JCPos-object A, it follows that the category Set(A) is topological over Set. Example 1.1.15 introduces a generalization of the category Set(JCPos) in the form of the category Cat(Qtlds). Is it true that the aforesaid statement holds in the latter category as well? Unfortunately the answer is negative. Start by clarifying the nature of the relation ”” on objects of the category Cat(Qtlds). Given a natural transformation X
H ⇓η
-G
-U Q in Cat one can formally define a
natural transformation F X
H ⇓η
-G
-Q in Qtlds by putting ηX = ηX (recall that both X and
F X have the same objects). The question arises: does η really is a natural transformation? Lemma 2.2.1. Let X
H
-G- U A be Cat(A)-objects. Then H η
-G is a natural transfor-mation iff H η- G is a natural transformation.
Proof. The necessity: if X S- Y is an F X-morphism, then ηY ◦ HS = ηY ◦ (H[S]) =
s∈S (ηY◦Hs) = s∈S (Gs◦ηX) = GS◦ηX. The sufficiency: if X f -Y is an X-morphism, then X {f}- Y is an F X-morphism and therefore ηY◦Hf = ηY◦H{f} = G{f}◦ηX = Gf◦ηX.
Lemma 2.2.2. Let Q, S be quantaloids. Then the hom-set Qtlds(Q, S) is a quantaloid.
Proof. Given a family of natural transformations Q
H ⇓ηi -G -Swith i∈ I, define H η- G by ηX = i∈I ηiX.
Lemmas 2.2.1 and 2.2.2 yield the following result.
Lemma 2.2.3. Suppose X is a small category and Q is a quantaloid. Then the functor HomCat(X, U Q) (−)- U HomQtlds(F X, Q) is an isomorphism.
In other words, the adjoint situation (η, ) : F U : Qtlds - Cat of Exam-ple 1.1.15 is actually a 2-adjunction (see, e.g., Chapter 7 in [8]). By Lemma 2.2.3 it follows that H G in Cat(X, UQ) iff there exists a natural transformation H η- G. One can easily see that the relation ”” is not always antisymmetric and therefore does not induce a complete lattice structure. Thus one can easily construct an example of a non-topological category Cat(Q). Under some restrictions, however, one can show that (Cat(X, U Q),)
still has some good properties. Recall that a prelattice is a preordered set supplied with the structure of a lattice. We use then the terms prejoins(meets) instead of the usual joins and meets and denote them by ”” and ”” respectively.
Lemma 2.2.4. Let (X, Q) be a Cat× Qtlds-object. If Q has products, then the following hold:
(i) (Cat(X, U Q),) is a complete prelattice;
(ii) if Y H- X is a Cat-morphism, then the map Cat(X, U Q) −◦H- Cat(Y, U Q) is premeet-preserving.
Proof. The first assumption follows from Theorem 18.22 in [20] and the fact that limits in functor categories are computed ”pointwise”. The second assumption then is straightfor-ward since premeets are just products.
As an immediate consequence of Lemma 2.2.4 one gets the following corollary.
Corollary 2.2.5. Let Q be a quantaloid that has products. Then every | − |-structured source in Cat(Q) has a| − |-initial lift.
Proof. Consider the proof of Proposition 2.1.3. Notice that the lift need not be unique. The last corollary yields the following result.
Lemma 2.2.6. If Q is a quantaloid that has products, then the functor Cat(Q) |−|- Cat is (Generating, Initial Source)-factorizable.
Proof. Given a|−|-structured source S = (X H-i |(Xi, Γi)|)i∈I, set Γ =Γiand get a
(Ge-nerating, Initial Source)-factorization X H-i |(Xi, Γi)| = X idX
-|(X, Γ)| |Hi-| |(X
i, Γi)|
for i∈ I.
As an immediate consequences of Lemma 2.2.6 one gets the following corollary (see Definition 25.1 in [2]).
Corollary 2.2.7. If Q is a quantaloid that has products, then the category (Cat(Q),| − |) is topologically algebraic.
The last corollary together with Proposition 25.12 and Corollary 25.15 in [2] imply the following result.
Corollary 2.2.8. If Q is a quantaloid that has products, then the following hold: (i) the functor Cat(Q) |−|- Catis adjoint;
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(ii) Cat(Q) is strongly complete and cocomplete.
Notice that if we consider the full subcategory Qtlds of Qtlds consisting of all those quantaloids whose set of objects is nonempty, the above results can be generalized to the category Set(Qtlds). Start with the following lemma.
Lemma 2.2.9. Let (X, Γ, Q), (Y, Δ, S) be Cat(Qtlds)-objects. Then every Cat× Qtlds-morphism |(X, Γ, Q)| (H, Ψ)- |(Y, Δ, S)| is a Cat(Qtlds)-morphism.
Proof. For every X-object X let (U Ψ◦ Γ)X τX- (Δ◦ H)X be the bottom element of the respective complete lattice. Given an X-morphism X f- Y , it follows that τY◦(UΨ◦Γ)f =
(∅) ◦ (UΨ ◦ Γ)f = (Δ ◦ H)f ◦ (∅) = (Δ ◦ H)f ◦ τX and therefore U Ψ◦ Γ τ
-Δ◦ H is a natural transformation.
As a consequence one immediately gets the following corollary. Corollary 2.2.10. In the category Cat(Qtlds) every source is initial.
Notice that up to now we do not use the category Qtlds. The next lemma however needs the latter category.
Lemma 2.2.11. The functor Cat(Qtlds) |−|- Cat× Qtlds is (Generating, Initial Source)-factorizable.
Proof. Let S = ((X, Q) (Hi,Ψi-) |(Xi, Γi, Qi)|)i∈I be a | − |-structured source. Since Q
is not empty there exists some Q-object Q and therefore one has the constant functor X GQ- Q : f → idQ. Thus one has the unique functor X GQ
,idX- U Q× X making the diagram X U Q ΠUQ GQ U Q× X GQ,idX ? ΠX - X id X
-commute. Define X Γ- U Q = X ΠUQ◦GQ,idX- U Q. By Lemma 2.2.9 and Corollary 2.2.10 (X, Q) (Hi,Ψ-i) |(Xi, Γi, Qi)| = (X, Q)
(idX,idQ-)
|(X, Γ, Q)| |(Hi,Ψi-)|
|(Xi, Γi, Qi)| is a
(Gen-erating, Initial Source)-factorization of S.
From Lemma 2.2.11 one immediately gets the following corollary.
Going back now to the category Cat(Q) with Q being a Qtlds-object one can easily get the following.
Corollary 2.2.13. Every subcategory Cat(Q) of the category Cat(Qtlds) is topologically algebraic.
2.3
On concrete cartesian closedness of X
(A)
In this section we show necessary and sufficient conditions for the category X(A) to be concretely cartesian closed.
Start with the following requirement. 9 X is cartesian closed.
Since X has finite products and X(A) is topological over X, the following holds. Proposition 2.3.1. The category X(A) has finite products which can be constructed as follows. Let S = ((Xi, αi))i∈I be a finite family of X(A)-objects. The source
(( i∈I Xi, ∗ i∈Iαi) πXi -(Xi, αi))i∈I with ∗ i∈Iαi = i∈I αi◦ πXi is a product of S.
Introduce additional requirements.
10 For every X(A)-object (X, α) and every X-object Y it follows that the map
X(Y, U A) α∗−- X(X× Y, UA) is join-preserving.
11 For every X(A)-object (X, α) the functor X(A) (X,α)×−- X(A) preserves final mor-phisms.
For convenience sake introduce the following notion.
Definition 2.3.2. Define a partial map Ob(X(A)) × Mor(X) −−- Ob(X(A)) as fol-lows: given an X(A)-object (X, α) and an X-morphism X f- Y set α f = β, where (X, α) f- (Y, β) is a final lift of the| − |-structured arrow |(X, α)| f-Y .
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In terms of − − 11 can be stated as follows: for every X(A)-objects (X, α), (Y, β) and every X-morphism Y f- Z it follows that α∗ (β f) = (α ∗ β) (idX × f).
Consider two examples.
Example 2.3.3. A subcategory Set(A) of the category Set(JCPos) satisfies 10 iff A is a frame (or complete Heyting algebra), i.e., it satisfies the distributive law: a ∧ (S) =
s∈S
(a∧ s). 11 follows from 10 .
Example 2.3.4. Since each power-set is a frame, every subcategory Set(A) of the category Set(SetRel) satisfies 10 – 11 .
We are going to show that 9 – 11 guarantee concrete cartesian closedness of X(A). Proposition 2.3.5. If 9 – 11 hold, then X(A) is concretely cartesian closed.
Proof. Fix a functor X(A) (X,α)×−- X(A) and take an X(A)-object (Y, β). By 9 there exists the evaluation morphism X×YX ev- Y in X. Define γ ={YX γi- U A| α∗γ
i β◦ev}. Then α∗ γ = α ∗ ( i γi) = i (α∗ γi) β ◦ ev implies that |(X × YX, α∗ γ)| ev -|(Y, β)| is an X(A)-morphism. Show that ev is an evaluation morphism in X(A).
Let (X, α)×(Z, δ) g- (Y, β) be an X(A)-morphism. Then there exists an X-morphism Z ˆg- YX such that ev◦ (idX × ˆg) = g. Since
|(X × Z, α ∗ δ) (idX׈g)
-(X× YX, (α∗ δ) (idX × ˆg))| ev
-|(Y, β)| is an X(A)-morphism, then|(X × YX, (α∗ δ) (id
X× ˆg))| ev
-|(Y, β)| must be also. Since α∗ (δ ˆg) = (α ∗ δ) (idX× ˆg) β ◦ ev implies δ ˆg γ, then δ (δ ˆg) ◦ ˆg γ ◦ ˆg and
therefore |(Z, δ)| ˆg- |(YX, γ)| is an X(A)-morphism.
Now show that requirements 9 – 11 are also necessary for concrete cartesian closedness of X(A).
Proposition 2.3.6. If X(A) is concretely cartesian closed, then 9 – 11 hold. Proof.
9 : Follows immediately from the definition of concrete cartesian closedness.
11 : By Proposition 27.15 in [2] it follows that for every X(A)-object (X, α) the functor X(A) (X,α)×−- X(A) preserves final sinks and therefore it preserves final morphisms.
10 : Consider a family S = (Y βi- U A)
i∈I. By Corollary 2.1.7 it follows that S =
((Y, βi) idY
-(Y,
i∈I
βi))i∈I is a final sink in X(A) and therefore the sink (X, α) × S =
((X×Y, α∗βi)
idX×id-Y
(X×Y, α∗(
i∈I
βi)))i∈I must be also. Thus α∗( i∈I
βi) = i∈I
Propositions 2.3.5 and 2.3.6 imply the following result. Proposition 2.3.7. The following are equivalent:
(i) X(A) is concretely cartesian closed; (ii) 9 – 11 hold.
From Examples 2.3.3 and 2.3.4 one immediately gets the following consequence of Propo-sition 2.3.7 (cf. PropoPropo-sition 71.4 in [60]).
Proposition 2.3.8. A subcategory Set(A) of the category Set(JCPos) is concretely carte-sian closed iff A is a frame. In case of the category Set(SetRel), every subcategory of the form Set(A) is concretely cartesian closed.
2.4
X
(A) is not a topos
In this section we show that X(A) is (almost always) not a topos. (For an introduction into the theory of toposes see, e.g., [19, 10]. Some aspects related to the theory of fuzzy sets can be found in [60]).
Since the category X(A) is topological one can introduce the following notations. Definition 2.4.1. Let X be an X-object. By (resp. ⊥) we denote the initial (resp. the final) structure on X w.r.t. the empty source (resp. sink).
Proposition 2.4.2. Suppose X is a topos. The following are equivalent: (i) X(A) is a topos;
(ii) U A is a terminal object in X.
Proof. (i) =⇒ (ii) : If X(A) is a topos, then it has a subobject classifier (T, ) t- (Ω, θ), where T is a terminal objects in X. For every X-object X the digram
(X,) (X,⊥) idX -id X -(X,) id X -pullback (T,) ? t -(Ω, θ) ?
commutes and therefore (X,) = (X, ⊥) that implies X(X, UA) = {α}.
(ii) =⇒ (i) : If UA is a terminal object in X, then the forgetful functor X(A) |−|- X is an isomorphism and therefore X(A) is a topos.
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As a consequence of Proposition 2.4.2 one immediately gets the following result. Proposition 2.4.3. A subcategory Set(A) of the category Set(JCPos) is a topos iff A = 1. A subcategory Set(A) of the category Set(SetRel) is a topos iff A =∅.
2.5
On representability of partial morphisms in X
(A)
In this section we show the necessary and sufficient conditions for X(A) to have representable Minit-partial morphisms.
Start with the definition of the class Minit.
Definition 2.5.1. Let M be a class of morphisms in X. Define Minit to be the class of all initial X(A)-morphisms m with|m| ∈ M.
Introduce a property of X-morphisms.
Definition 2.5.2. We say that an X-morphism X m-Y satisfies property (B) provided that for every X(A)-object (Z, γ), every pullback diagram of the form
W mˆ- |(Z, γ)| X ˆ f ? m- Y f ? (2.1) in X implies γ◦ F ˆm◦ ˆf◦= γ◦ f◦◦ F m. Consider an example.
Example 2.5.3. If Set(A) is a subcategory of Set(JCPos) (resp. Set(SetRel)), then every Set-map has property (B).
In fact the category Set satisfies a stronger property considered in the following defini-tion.
Definition 2.5.4. Let X be a category. X is said to have pullbacks, which satisfy the Beck-Chevalley Property (BCP) provided that for every pullback diagram
W mˆ- Z X ˆ f ? m- Y f ? in X it follows that F ˆm◦ ˆf◦= f◦◦ F m.
Lemma 2.5.5. If a category X has pullbacks, which satisfy (BCP), then every X-morphism satisfies (B).
Introduce the following requirements.
12 X has representable M-partial morphisms. 13 Every m ∈ M satisfies (B).
Proposition 2.5.6. Let M be a class of X-morphisms. The following are equivalent: (i) 12 , 13 hold;
(ii) X(A) has representable Minit-partial morphisms. Moreover, | − | preserves representations.
Proof. (i) =⇒ (ii) : By Theorem 28.15 in [2] it will be enough to show that final sinks in X(A) are stable under pullbacks along Minit.
Let S = ((Yi, βi) fi
-(Y, β))i∈I be a final sink in X(A), let (X, α) m
-(Y, β) be an element of Minit and let ˆS = ((Xi, αi)
ˆ fi
-(X, α))i∈I be a pullback of S along m. Show
that α = i∈I (αi◦ ˆfi◦). Since α = β◦m, then α = β◦F m = i∈I (βi◦fi◦)◦F m = i∈I (βi◦fi◦◦F m). Pullback stability of initial morphisms in X(A) implies αi = βi◦ mi and therefore αi◦ ˆf◦ = βi◦ F mi◦ ˆfi◦ =
βi◦ fi◦◦ F m. Thus α =
i∈I
(αi◦ ˆfi◦).
(ii) =⇒ (i) : 12 : Follows from Proposition 28.12 in [2].
13 : Take m ∈ M and consider diagram (2.1). Since X(A) is topological over X, there exists a final lift (Z, γ) f- (Y, β) in X(A) of the| − |-costructured sink (|(Z, γ)| f- Y ). Similarly, there exists an initial lift (X, α) m- (Y, β) in X(A) of the| − |-structured source (X m- |(Y, β)|). Finally, there exists a structure δ on W such that (2.1) becomes a pullback in X(A). By Theorem 28.15 in [2] final sinks in X(A) are stable under pullbacks along Minit. Then γ◦ f◦◦ F m = β ◦ F m = α = δ ◦ ˆf◦ = γ◦ F ˆm◦ ˆf◦.
Introduce a stronger version of 13 . 13 X has pullbacks, which satisfy (BCP).
The following proposition follows immediately from Lemma 2.5.5 and Proposition 2.5.6. Proposition 2.5.7. Suppose 13 holds. The following are equivalent:
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(ii) X(A) has representable Minit-partial morphisms.
Corollary 2.5.8. Suppose 13 holds. If X has representable extremal partial morphisms, then X(A) has representable extremal partial morphisms.
Proof. Minit is the class of extremal monomorphisms in X(A).
As a consequence of Corollary 2.5.8 one immediately gets the following.
Proposition 2.5.9. Every subcategory Set(A) of the categories Set(JCPos) as well as Set(SetRel) has representable extremal partial morphisms and | − | preserves these repre-sentations.
Let X have representable M -partial morphisms and let (X, α) be an X(A)-object. Then there exists some X mX- X∗ that represents M -partial morphisms into X. By Proposi-tion 2.5.6 we have some (X, α) mX- (X∗, α∗) that represents Minit-partial morphisms into (X, α). The next proposition shows the nature of α∗.
Proposition 2.5.10. Suppose 12 , 13 hold. Let (X, α) be an X(A)-object, let X mX- X∗ represent M -partial morphisms into X. Define α∗={X∗ α-i U A| α = αi◦ mX}. Then
(X, α) mX- (X∗, α∗) represents Minit-partial morphisms into (X, α). Proof. Let α be as defined. Since m
X ∈ Minit, then α = α∗◦ mX and therefore α∗ α.
Conversely, let α = αi◦ mX. Since ((X∗, αi) mX
(X, α) idX- (X, α)) is an Minit-partial morphism from (X∗, αi) to (X, α), there exists a pullback
(X, α) m-X (X∗, αi) (X, α) idX ? m-X (X ∗, α∗) id∗X ?
in X(A). Since | − | preserves representations, then id∗X = idX∗ and therefore αi α∗.
Thus α α∗.
Consider the following example (cf. Proposition 71.3 in [60]).
Example 2.5.11. Let Set(A) be a subcategory of one of the categories Set(JCPos) or Set(SetRel) and let (X, α) be a Set(A)-object. Set X∗ = X{∞} and get the obvious in-clusion X mX- X∗that represents (extremal) partial morphisms into X in the category Set. Let X∗ α∗- U A be as follows: α∗(x) = α(x) and α(∞) = . Then (X, α) mX- (X∗, α∗) represents extremal partial morphisms into (X, α) in the category X(A).
2.6
X
(A) is a quasitopos
Proposition 2.4.2 shows that for the categories of the form X(A) the concept of a topos is not interesting. However, the concept of a quasitopos is fruitful. In this section we show the necessary and sufficient conditions for X(A) to be a quasitopos.
Start with the following requirement. 14 X is a quasitopos.
Now the main proposition.
Proposition 2.6.1. Suppose 13 holds. The following are equivalent. (i) 10 , 11 , 14 hold;
(ii) (X(A),| − |) is a concrete quasitopos, i.e., X(A) is a quasitopos and | − | preserves power objects and representations of extremal partial morphisms.
Proof. Follows immediately from Propositions 2.3.7 and 2.5.7.
As a consequence of Proposition 2.6.1 one immediately gets the following.
Proposition 2.6.2. A subcategory Set(A) of the category Set(JCPos) is a quasitopos iff A is a frame. In case of the category Set(SetRel), every subcategory of the form Set(A) is a quasitopos.
2.6.1 The inner structure of X(A)
Suppose requirements 10 , 11 , 13, 14 hold. By Proposition 2.6.1 X(A) is a quasitopos. In this subsection we consider the inner structure of X(A).
Start with the standard one (cf. [2] Chapter VII). Recall that X(X, U A) −◦f- X(Y, U A) preserves by 4.
Proposition 2.6.3. Let T t- Ω be an extremal-subobject classifier (ESC) in X. Then (T,) t- (Ω,) is an ESC in X(A).
Proof. Propositions 2.6.1 and 2.5.10 imply that (T,) t- (Ω, θ) represents extremal partial morphisms into (T,), where θ = {Ω αi- U A| = αi ◦ t}. By = ◦ t it
follows that θ =.
Consider the following lemma.
