Stratified categorical fixed-basis fuzzy topological spaces and their duality

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Stratified categorical fixed-basis fuzzy topological spaces and their duality Mustafa Demirci Akdeniz University, Faculty of Sciences, Department of Mathematics, 07058 Antalya, Turkey Received 4 January 2014; received in revised form 29 June 2014; accepted 15 August 2014

Abstract For an abstract category C, a class M of C-monomorphisms and a fixed C-object L, we introduce stratified C-M-L-spaces to be categorical counterparts of stratified fixed-basis fuzzy topological spaces in C, and consider their category SC-M-L-Top. As two main results of this paper, it is shown that SC-M-L-Top is dually adjoint to the comma category L ↓ C, and this adjunction can be restricted to a dual equivalence between the full category of L ↓ C with comma-spatial objects and the full category of SC-M-L-Top with comma-sober objects. The present paper also describes applications and relationships of these results to stratified fixed-basis fuzzy topological spaces. In this respect, a considerable part of this paper is devoted to stratified L-quasi-topological spaces and their duality. © 2014 Published by Elsevier B.V. Keywords: Category theory; Topology; Categorical topology; Adjoint situation; Comma category; Lattice-valued topology; Fuzzy topology; Stratified fuzzy topology

1. Introduction Since R. Lowen’s modification [24] of fuzzy topology satisfying constants condition,1 stratified lattice-valued topological spaces [16,18,29,33,45,46] and their generalizations [34,35,43,44] have been major issues in the field of fuzzy (lattice-valued or many-valued) topology. There are two main approaches to the stratified fixed-basis lattice-valued topological spaces in [16,18]. The first one assumes the constants condition (such spaces are called weakly stratified L-topological spaces in [16,18]), while the second one assumes the truncation condition (such spaces are called stratified L-topological spaces in [16,18]). L-quasi-topological spaces introduced in [32] form a general framework for the fixed-basis lattice-valued topological spaces including L-topological spaces in [16,18] as well. One may extend the notion of (weakly) stratified L-topological space to L-quasi-topological space in an obvious way. Given an abstract category C, C-M-L-spaces, what we mean by categorical fixed-basis fuzzy topological spaces in the title of this paper, have been put forward in [10] to be appropriate categorical counterparts of the fixed-basis fuzzy E-mail address: [email protected]. 1 The term “stratified” for this modification was later coined by P.-M. Pu and Y.-M. Liu [28].

http://dx.doi.org/10.1016/j.fss.2014.08.005 0165-0114/© 2014 Published by Elsevier B.V.

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topological spaces in C that enable to carry over the famous Papert–Papert–Isbell adjunction Top  Loc [20,21,26] to a general adjunction X  Cop in which X is a suitable category of such counterparts. It is shown in [10] that the category C-M-L-Top of C-M-L-spaces provides an answer to the asked category X as well as a categorical framework for the fixed-basis fuzzy topology without needing order theory or algebra. In addition to the adjunction C-M-L-Top  Cop , the dual equivalence (also called duality [27]) between the full subcategory (E, M)-L-Spat-C of C with L-spatial objects and the full subcategory C-M-L-SobTop of C-M-L-Top with L-sober objects is another central result in [10]. The present paper brings the formulation of stratified categorical fixed-basis fuzzy topological spaces and their duality into focus. Firstly, we define stratified C-M-L-spaces (Definition 3.6) together with their category SC-M-L-Top; secondly, we establish an adjunction SC-M-L-Top  (L ↓ C)op (Theorem 3.10) in which L ↓ C is the category of objects under L, also the so-called comma category [25]; thirdly, we refine this adjunction to a duality between the full subcategory (E, M)-CMSpat-L ↓ C of L ↓ C of all comma-spatial objects and the full subcategory CMSob-SC-M-L-Top of SC-M-L-Top of all comma-sober objects (Corollary 3.15). Many categories of (weakly) stratified lattice-valued topological spaces in [16,18,29,33,46] and stratified variety-based spaces in [34] are instances of SC-M-L-Top. Thereby, the duality between (E, M)-CMSpat-L ↓ C and CMSob-SC-M-L-Top is applicable to all of these categories. On the other hand, this duality cannot be employed in the case of the category SL-QTop of stratified L-quasi-topological spaces. To overcome this adverse situation, we invoke the duality between (E, M)-L-Spat-C and C-M-L-SobTop, and formulate a duality for SL-QTop with the category of L∗ -semi-quantales (Corollary 4.38). This paper has been prepared in four sections. After this introductory section, the next section overviews the category C-M-L-Top and the aforementioned adjunction and duality for C-M-L-Top, while Section 3 considers stratified C-M-L-spaces, their category SC-M-L-Top, the adjunction SC-M-L-Top  (L ↓ C)op and the duality between (E, M)-CMSpat-L ↓ C and CMSob-SC-M-L-Top. Final section is devoted to stratified L-quasi-topological spaces and their duality with L∗ -semi-quantales. 2. An overview of categorical fixed-basis fuzzy topological spaces 2.1. Definitions and examples Let C, M and L denote an abstract category with set-indexed products, a class of C-monomorphisms and a fixed C-object, respectively. This will be assumed throughout this paper, unless further assumptions or restrictions are made on C, M and L. As the reference material for categorical notions and facts not given in this paper, we refer to [1,25]. m A C-M-L-space, which can be thought of as a fixed-basis fuzzy topological space in C, is a pair (X, τ → LX ), m consisting of a set X and an M-morphism τ → LX (the so-called C-M-L-topology on X). C-M-L-spaces form a m1

f

m2

category C-M-L-Top with morphisms all (X, τ → LX ) → (Y, ν → LY ), where f : X → Y is a function such that there exists a (necessarily unique) C-morphism rf : ν → τ making ν

rf

m2

LY

(2.1)

τ m1

fL←

LX

commute [8,10]. Here fL← : LY → LX is the unique C-morphism satisfying πx ◦ fL← = πf (x)

(2.2)

for all x ∈ X, where πx : LX → L (πy : LY → L) is the x(y)-th projection morphism for all x ∈ X (y ∈ Y ). Quasivarieties [10] supply a large inventory of examples of C-M-L-Top. Varieties of Ω-algebras [34] and the categories SQuant, SSQuant, USQuant, CGR, CQML, SFrm, Frm of semi-quantales (shortly, s-quantales) [32], strong semi-quantales [7], unital semi-quantales (shortly, us-quantales) [32], complete groupoids [16], complete quasimonoidal lattices [18], semiframes [29,33], frames [21] are known examples of quasivarieties.

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  In this paper, we always use , , , ⊥ for the arbitrary join, arbitrary meet, top element and bottom element in a complete lattice, respectively. Let (L, ≤, ⊗) be an s-quantale (also denoted L), that is, (L, ≤) is a complete lattice equipped with a binary operation ⊗, which may not have any relation to the underlying partial order ≤. Recallfrom [32] that for a given set X, a subset τ of LX is an L-quasi-topology on X iff τ is closed under ⊗ and arbitrary . An L-quasi-topological space is a pair (X, τ ), where τ is an L-quasi-topology on X. L-QTop is a category with objects f

all L-quasi-topological spaces and morphisms all arrows (X, τ ) → (Y, ν), where f : X → Y is a function such that for all μ ∈ ν, fL← (μ) = μ ◦ f ∈ τ [32]. Example 2.1. Let C, MC and L stand for a construct with the forgetful functor | | : C → Set, the class of all C-morphisms m : A → B such that |A| ⊆ |B| and |m| : |A| → |B| is the inclusion map, and a fixed object of C, respectively. (i) For C = Frm, C-MC -2-Top is isomorphic to the category Top of topological spaces in the usual sense, while C-MC -L-Top is isomorphic to the category L-TopZL of L-topological spaces in [46]. (ii) For C = SFrm, C-MC -L-Top is isomorphic to the category L-TopP R of L-topological spaces in [29,33]. (iii) For C = CGR, C-MC -L-Top is isomorphic to the category L-TOPH of L-topological spaces in [16]. (iv) For C = CQML, C-MC -L-Top is isomorphic to the category L-TOPH S of L-topological spaces in [18]. (v) For C = SQuant, C-MC -L-Top is isomorphic to L-QTop. (vi) For C = SSQuant, C-MC -L-Top is isomorphic to the full subcategory L-SQTop of L-QTop of strong L-quasi-topological spaces (an L-QTop-object (X, τ ) is strong if the constant map X : X → L with value belongs to τ [7]). (vii) For C = USQuant, C-MC -L-Top is isomorphic to the category L-TopR of L-topological spaces in [32]. (viii) For C = Alg(Ω), where Alg(Ω) is the category of Ω-algebras and Ω-homomorphisms [34], C-MC -L-Top is isomorphic to the category ASet(Ω) of affine sets over the Ω-algebra L and affine maps in [14]. Here the notion of affine set grows out of Y. Diers’ studies on the corresponding notion in [11–13]. (ix) For C = V, where V is a quasivariety [10] (resp. a variety [34]), C-MC -L-Top is isomorphic to the category L-TopV of L-spaces in [10] (resp. [34]). Note in Example 2.1 that the categories of L-topological spaces with subscripts (e.g. L-TopZL , L-TopP R and L-TOPH ) do not include such subscripts in their original labelling. Moreover, all of them are instances of L-TopV as well as they are full subcategories of L-QTop for some appropriate selection of L. For example, if L is taken to be a complete groupoid, then L-TopCGR = L-TOPH = L-SQTop. There are also many other constructs different from quasivarieties (e.g. the category Cat(Q) of Q-categories and the category P of augmented posets [9,10]) that produce interesting cases of C-M-L-Top. In this paper, we add one more example to these constructs, and return back to this issue in Section 4. 2.2. Dual adjunction between C-M-L-Top and C Along this section, we assume C to be essentially (E, M)-structured for some class E of C-morphisms, i.e. C fulfills the following two conditions: • C has (E, M)-factorizations, i.e. each C-morphism f has an (E, M)-factorization pair (e, m), i.e. f = m ◦ e with e ∈ E and m ∈ M. • C has the unique (E, M)-diagonalization property, i.e. for every e ∈ E , m ∈ M and every f, g ∈ Mor(C) such that g ◦ e = m ◦ f , there exists a (necessarily unique) C-morphism d fulfilling d ◦ e = f and m ◦ d = g. h

A

Each C-object A determines a unique C-morphism A : A → LC(A,L) with the property that A → L = A → πh eA mA C(A,L) L → L for each h ∈ C(A, L), where C(A, L) denotes the hom-set from A to L. Let A → τA → LC(A,L) be an op arbitrarily fixed (E, M)-factorization of A for each C-object A. LPtM : C → C-M-L-Top, defined by   mA LPtM (A) = C(A, L), τA → LC(A,L) and LPtM (g)(u) = u ◦ g op

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for each A ∈ Ob(C), A1 → A2 ∈ Mor(Cop ) and u ∈ C(A1 , L), is a functor. It should be noted that if the (E, M)-factorization of A is not fixed for each C-object A, then the definition of LPtM does not determine a unique functor, but a collection of essentially unique functors in the sense that any two of such functors are naturally isomorphic. As a result, LPtM does not essentially depend on how the (E, M)-factorizations are selected. In the reverse direction, there is a functor LΩM : C-M-L-Top → Cop , defined by rf   f   m1 m2 LΩM X, τ → LX → Y, ν → LY = τ → ν. op

Recall that an adjoint situation (, φ) : F  G : A → B is a pair of functors G : A → B and F : B → A with the unit 

φ

idB → G ◦ F and co-unit F ◦ G → idA satisfying G(φA ) ◦ G(A) = idG(A) and φF (B) ◦ F (B ) = idF (B) for all A in A and B in B [1]. We say that B is adjoint to A, or B  A is an adjunction iff there exists an adjoint situation (, φ) : F  G : A → B for some F , G,  and φ. m

Theorem 2.2. (See [10].) For each C-M-L-Top-object W = (X, τ → LX ), let the map ηW : X → C(τ, L) be defined by ηW (x) = πx ◦ m for each x ∈ X. Furthermore, for each C-object A, let εA stand for the Cop -morphism op eA : τA → A. Then, for η = (ηW )W ∈Ob(C-M-L-Top) and ε = (εA )A∈Ob(C) , (η, ε) : LΩM  LPtM : Cop → C-M-L-Top is an adjoint situation. Definition 2.3. (See [10].) (i) A C-object A is (E, M)-L-spatial, or shortly L-spatial iff for some (E, M)-factorization A

pair (e, m) of A → LC(A,L) , e is an isomorphism in C. m (ii) A C-M-L-Top-object W = (X, τ → LX ) is L-sober iff for all h ∈ C(τ, L), there exists a unique x ∈ X such that h = πx ◦ m. It is worthwhile to mention the fact that the term “some” in the definition of L-spatiality can be replaced with “all”. Thus, L-spatiality is independent from the choice of the (E, M)-factorization pair (e, m) (see [10, Remark 5.2(i)]). op

Proposition 2.4. (See [10].) (i) A C-object A is L-spatial iff εA is an isomorphism in C. (ii) A C-M-L-Top-object W is L-sober iff ηW is an isomorphism in C-M-L-Top. Corollary 2.5. (See [10].) The full subcategory (E, M)-L-Spat-C of C of all L-spatial objects is dually equivalent to the full subcategory C-M-L-SobTop of C-M-L-Top of all L-sober objects. 3. Stratified C-M-L-spaces and their duality 3.1. Motivation and definition To explain the motivating idea of stratified C-M-L-spaces, let us recall that for a frame L, an L-TopZL -object,  i.e. a Zhang–Liu’s L-topological space (X, τ ), where X is a set and τ is a subset of LX closed under finite and  arbitrary , is stratified [46] iff the following constants condition is satisfied: (CNS) For every a ∈ L, the constant map aX : X → L with the value a belongs to τ . If we consider the frame morphism PX (L) : L → LX defined by [PX (L)](a) = aX , then the condition (CNS) can s

iτ

PX (L)

be equivalently stated as the existence of a frame morphism s : L → τ satisfying L → τ → LX = L → LX . This description of (CNS) motivates its categorical counterpart as follows. m

m

Definition 3.6. A C-M-L-topology τ → LX , or a C-M-L-space (X, τ → LX ) is said to be stratified iff there exists a (necessarily unique) C-morphism sm : L → τ making the triangle

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L

sm

PX (L)

5

(3.3)

τ m

LX commute, where PX (L) : L → LX is the unique C-morphism for which L

PX (L)

idL

LX πx

L commutes for all x ∈ X. We denote by SC-M-L-Top the full subcategory of C-M-L-Top that has stratified C-M-L-spaces as objects. In case C is a quasivariety V or particularly the category Frm, the C-morphism PX (L) : L → LX in Definition 3.6 is precisely the set map a ∈ L → aX ∈ LX , while the V-morphism sm : L → τ (if exists) is precisely the co-domain restriction of this set map to τ , i.e. sm (a) = aX for every a ∈ L. For a quasivariety V and a V-object L, an object (X, τ ) of L-TopV is called stratified if (CNS) holds. The full subcategory SL-TopV of L-TopV with stratified objects will then be isomorphic to SV-MV -L-Top. Note that for a variety V, SL-TopV is first introduced in [34], and is denoted by SL-Top. In keeping with [16,18], we use the term “weakly stratified” for an L-QTop-object (X, τ ) satisfying (CNS) and the label WSL-QTop (resp. WSL-SQTop, WSL-TopR and WSL-TOPH ) for the full subcategory of L-QTop (resp. L-SQTop, L-TopR and L-TOPH ) comprising all weakly stratified objects. Note that WSL-QTop, WSL-SQTop, WSL-TopR , WSL-TOPH , the full subcategory SL-TopP R of L-TopP R with stratified objects in [33] and the full subcategory SL-TopZL of L-TopZL with stratified objects in [46] are special cases of SL-TopV for V = SQuant, SSQuant, USQuant, CGR, SFrm, Frm, respectively. 3.2. Dual adjunction between SC-M-L-Top and L ↓ C L ↓ C, the so-called comma category [25], is defined to be a category with objects all pairs (uA, A), where A g g is a C-object and uA : L → A is a C-morphism, and with morphisms all (uA , A) → (uB , B), where A → B is a C-morphism satisfying g ◦ uA = uB . This section establishes an adjunction between SC-M-L-Top and (L ↓ C)op , which relies on the existence of set-indexed products and a suitableessential factorization structure for L ↓ C. If (uAi , Ai )i∈Iis a set-indexed family of L ↓ C-objects, then (uAi , i∈I Ai ) is a product of (uAi , Ai )i∈I , where uAi  : L → i∈I Ai is the unique C-morphism such that πi ◦ uAi  = uAi for all i ∈ I . In consequence, L ↓ C has set-indexed products. Particularly, if we designate (idL , L) by L, then for every set X, an X-th power LX of L is given as (PX (L), LX ). Furthermore, every L ↓ C-object (uA , A) determines a unique L ↓ C-morphism (uA , A) : (uA , A) → LL↓C((uA ,A),L) satisfying πh ◦ (uA , A) = h for all h ∈ L ↓ C((uA , A), L). g

Proposition 3.7. For a given class F of C-morphisms, let F denote the class of L ↓ C-morphisms (uA , A) → (uB , B) g such that A → B ∈ F . If C is essentially (E, M)-structured for some class E of C-morphisms, then L ↓ C is essentially (E, M)-structured. g

Proof. Suppose C is essentially (E, M)-structured. For each L ↓ C-morphism (uA , A) → (uB , B), there exists g e m e m an (E, M)-factorization A → C → B of A → B. It is easy to observe that (uA , A) → (e ◦ uA , C) → (uB , B) g is an (E, M)-factorization of (uA , A) → (uB , B), i.e. L ↓ C has (E, M)-factorizations. To prove that L ↓ C has e m the unique (E, M)-diagonalization property, let us consider (uA , A) → (uB , B) ∈ E , (uC , C) → (uD , D) ∈ M and f

g

e

m

L ↓ C-morphisms (uA , A) → (uC , C), (uB , B) → (uD , D) such that g ◦ e = m ◦ f . Since A → B ∈ E , C → D ∈ M,

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g

A → C, B → D ∈ Mor(C), there exists a unique d : B → C ∈ Mor(C) satisfying d ◦ e = f and m ◦ d = g. It is not difficult to see that d : (uB , B) → (uC , C) is the unique L ↓ C-morphism satisfying d ◦ e = f and m ◦ d = g. 2 As an important result needed in the formulation of the questioned adjunction, we now construct an isomorphism between L ↓ C-M-L-Top and SC-M-L-Top. Theorem 3.8. For some class E of C-morphisms including idL , let C have the unique (E, M)-diagonalization property. The functor T : L ↓ C-M-L-Top → SC-M-L-Top, defined by     m m T X, (uτ , τ ) → LX = X, τ → LX and T (f ) = f, is an isomorphism. Proof. Let us first prove that T : L ↓ C-M-L-Top → SC-M-L-Top is indeed a functor. For each L ↓ m m C-M-L-Top-object (X, (uτ , τ ) → LX ), since LX = (PX (L), LX ) and (uτ , τ ) → LX ∈ M, we have PX (L) ◦ idL = m ◦ uτ .

(3.4)

Since m ∈ M and idL ∈ E , we apply the unique (E, M)-diagonalization property to (3.4), and therefore obtain a m C-morphism sm : L → τ making (3.3) commute, i.e. (X, τ → LX ) is an SC-M-L-Top-object. Thus, T maps the objects of L ↓ C-M-L-Top to the objects of SC-M-L-Top. To see that T is a function from Mor(L ↓ C-M-L-Top) to Mor(SC-M-L-Top), we first show that for every function f : X → Y , f ← = fL← , i.e. fL← : LY → LX is the L underlying C-morphism of f ← : LY → LX . For this purpose, it is enough to confirm that fL← : (PY (L), LY ) → L (PX (L), LX ) is an L ↓ C-morphism, i.e. fL← ◦ PY (L) = PX (L).

(3.5)

We know from the definitions of PX (L) and PY (L) that for all x ∈ X, πf (x) ◦ PY (L) = idL = πx ◦ PX (L), and then observe that

  πx ◦ fL← ◦ PY (L) = πx ◦ fL← ◦ PY (L)   = πf (x) ◦ PY (L) by (2.2) = idL = πx ◦ PX (L). m1

m2

Thus, (3.5) follows since (πx )x∈X is a mono-source. Let f : (X, (uτ , τ ) → LX ) → (Y, (uν , ν) → LY ) be an L ↓ rf

C-M-L-Top-morphism, i.e. there exists an L ↓ C-morphism (uν , ν) → (uτ , τ ) for which (uν , ν)

rf

(uτ , τ )

m2

(3.6)

m1

LY

f← L

LX

commutes. Since f ← = fL← , the commutativity of (3.6) implies the commutativity of (2.1), which means that m1

L

m2

f : (X, τ → LX ) → (Y, ν → LY ) is an SC-M-L-Top-morphism. Because T preserves composition and identities, T will be a functor. m m The surjectivity of T on objects is clear, since for each SC-M-L-Top-object (X, τ → LX ), (X, (sm , τ ) → LX ) m1 is an L ↓ C-M-L-Top-object. To see its injectivity on objects, for two objects W1 = (X, (uτ , τ ) → LX ) and W2 = m2 Y (Y, (uν , ν) → L ) in L ↓ C-M-L-Top, suppose T (W1 ) = T (W2 ). Then, since X = Y , τ = ν and m1 = m2 , we have m1 ◦ uτ = PX (L) = m1 ◦ uν . Therefore, since m1 is a monic, uτ = uν follows, and so does W1 = W2 . Faithfulness of T is evident from its very definition.

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We finally show that T is full, and hence an isomorphism. For this purpose, we pick two objects W1 = m1 m2 m1 m2 (X, (uτ , τ ) → LX ), W2 = (Y, (uν , ν) → LY ) in L ↓ C-M-L-Top, and assume that f : (X, τ → LX ) → (Y, ν → LY ) rf

is an SC-M-L-Top-morphism, i.e. (2.1) commutes for a C-morphism ν → τ . Our aim is to prove that f : W1 → W2 rf

is an L ↓ C-M-L-Top-morphism, i.e. (3.6) commutes. If we show that (uν , ν) → (uτ , τ ) is an L ↓ C-morphism, then, m1 since f ← = fL← , the commutativity of (3.6) follows from the commutativity of (2.1). Since (uτ , τ ) → (PX (L), LX ) L

m2

and (uν , ν) → (PY (L), LY ) are L ↓ C-morphisms, we may write m1 ◦ uτ = PX (L),

(3.7)

m2 ◦ uν = PY (L).

(3.8)

Then, since

 by the commutativity of (2.1)   = fL← ◦ PY (L) by (3.8)   = PX (L) by (3.5)   by (3.7) = m1 ◦ uτ

m1 ◦ rf ◦ uν = fL← ◦ m2 ◦ uν



rf

and m1 is a monic, we obtain rf ◦ uν = uτ , i.e. (uν , ν) → (uτ , τ ) is an L ↓ C-morphism.

2

Proposition 3.9. The following properties hold for every L ↓ C-object (uA , A): (i) The hom-set L ↓ C((uA , A), L) is the same as the set I(uA ) of all left-inverses of uA , i.e. all C-morphisms h : A → L such that h ◦ uA = idL . (ii) The unique C-morphism AI(uA ) : A → LI(uA ) , with the property that πh ◦ AI(uA ) = h for all h ∈ I(uA ), is the underlying C-morphism of the L ↓ C-morphism (uA , A) : (uA , A) → LL↓C((uA ,A),L) . Proof. (i) is evident. (ii) We have from (i) LL↓C((uA ,A),L) = (PI(uA ) (L), LI(uA ) ). Using this equality together with the definitions of AI(uA ) and (uA , A), it is sufficient to see that AI(uA ) : (uA , A) → (PI(uA ) (L), LI(uA ) ) is an L ↓ C-morphism, i.e. AI(uA ) ◦ uA = PI(uA ) (L). Since πh ◦ AI(uA ) ◦ uA = h ◦ uA = idL = πh ◦ PI(uA ) (L) is valid for all h ∈ I(uA ), AI(uA ) ◦ uA = PI(uA ) (L) follows from the fact that (πh )h∈I(uA ) is a mono-source.

2

Henceforth, we assume that C is essentially (E, M)-structured for some class E of C-morphisms with idL ∈ E , e(uA ,A)

AI(uA )

m(uA ,A)

and fix an (E, M)-factorization A → t(uA ,A) → LI(uA ) of A → LI(uA ) for every L ↓ C-object (uA , A). In the formulation of the adjunction SC-M-L-Top  L ↓ C, the following definitions will be used: m(uA ,A)     LSM (g) (h) = h ◦ g op , LSM (uA , A) = I(uA ), t(uA ,A) → LI(uA ) , σW : X → I(sm )

by σW (x) = πx ◦ m g

for every L ↓ C-object (uA , A), (L ↓ C)op -morphism (uA , A) → (uB , B), h ∈ I(uA ) and every SC-M-L-Top-object m W = (X, τ → LX ). Theorem 3.10. The identities rf   f   m1 m2 LCM X, τ → LX → Y, ν → LY = (sm1 , τ ) → (sm2 , ν),   g LSM (g) LSM (uA , A) → (uB , B) = LSM (uA , A) → LSM (uB , B),   σ = (σW )W ∈Ob(SC-M-L-Top) and δ = (e(uA ,A) )op (u ,A)∈Ob(L↓C) A op

define an adjoint situation (σ, δ) : LCM  LSM : (L ↓ C)op → SC-M-L-Top.

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Proof. By making use of Proposition 3.7 and Proposition 3.9(ii), for each L ↓ C-object (uA , A), (uA , A) : (uA , A) → LL↓C((uA ,A),L) can be factorized as e(uA ,A)

m(uA ,A)

(uA , A) → τ(uA ,A) → LL↓C((uA ,A),L) , where τ(uA ,A) = (e(uA ,A) ◦ uA , t(uA ,A) ). Then, since L ↓ C is an essentially (E, M)-structured category with setindexed products, we apply Theorem 2.2, and obtain an adjoint situation (η, ε) : LΩM  LP tM : (L ↓ C)op → L ↓ C-M-L-Top, where LΩM , LP tM , the W -th component ηW of η and the (uA , A)-th component ε(uA ,A) of ε are given by op

LΩM (W ) = (uτ , τ ), LΩM (f ) = rf , m(uA ,A)   LP tM (uA , A) = I(uA ), τ(uA ,A) → LI(uA ) ,   LP tM (g) (h) = h ◦ g op , ηW (x) = πx ◦ m,

op

ε(uA ,A) = e(uA ,A)

m

for each L ↓ C-M-L-Top-object W = (X, (uτ , τ ) → LX ), L ↓ C-M-L-Top-morphism f , L ↓ C-object (uA , A), (L ↓ C)op -morphism g and every x ∈ X. On the other hand, the isomorphism T in Theorem 3.8 provides an adjoint situation (η, ε) : T −1  T : L ↓ C-M-L-Top → SC-M-L-Top, where η = ididSC-M-L-Top and ε = ididL↓C-M-L-Top .

By virtue of [1, 19.13 Proposition], (η, ε) : LΩM  LP tM : (L ↓ C)op → L ↓ C-M-L-Top and (η, ε) : T −1    LPt : (L ↓ T : L ↓ C-M-L-Top → SC-M-L-Top can be composed as an adjoint situation (η , ε  ) : LΩM M op C) → SC-M-L-Top, where η = T ηT −1 ◦ η,

ε  = ε ◦ LΩM εLP tM ,

 LΩM = LΩM ◦ T −1 ,

 = LC  It is easy to check that η = σ , ε  = δ, LΩM M and LPtM = LSM .

LPt M = T ◦ LP tM .

2

As is shown in [10], a quasivariety V is an essentially (Surj(V), MV )-structured category with set-indexed products. This allows us to apply Theorem 3.10 to quasivarieties: Corollary 3.11. For every quasivariety V and every object L of V, there exists an adjoint situation (σ V , δ V ) : LC V  LS V : (L ↓ V)op → SL-TopV . Regarding Theorem 3.10 and the isomorphism between SL-TopV and SV-MV -L-Top, one may explicitly calg culate LC V , LS V , σ V and δ V as follows. For every L ↓ V-object (uA , A), every (L ↓ V)op -morphism (uA , A) → V (uB , B), every b ∈ A and every SL-TopV -object W = (X, τ ), define αb : I(uA ) → L by αb (h) = h(b), t(u = A ,A) V V V V V V {αb | b ∈ A}, e(uA ,A) : A → t(uA ,A) by e(uA ,A) (b) = αb , LS (uA , A) = (I(uA ), t(uA ,A) ), LS (g) : I(uA ) → I(uB ) by V : X → I(s ) by σ V (x)(μ) = μ(x). Next, LS V (g)(h) = h ◦ g op , sτ : L → τ by sτ (a) = aX , σW τ W op (fL← )|ν  LC (X, τ ) → (Y, ν) = (sτ , τ ) → (sν , ν),   g LS V (g) LS V (uA , A) → (uB , B) = LS V (uA , A) → LS V (uB , B), op   V  V σ V = σW and δ V = e(u . W ∈Ob(SL-Top ) (u ,A)∈Ob(L↓V) A ,A)

V



f

V

A

Corollary 3.11 has a large number of implications, including the following ones to the lattice-valued topological spaces. Corollary 3.12. The following adjunctions hold: (i) For an s-quantale L, WSL-QTop  (L ↓ SQuant)op and WSL-SQTop  (L ↓ SSQuant)op . (ii) For a us-quantale L, WSL-TopR  (L ↓ USQuant)op .

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(iii) For a complete groupoid L, WSL-TOPH  (L ↓ CGR)op . (iv) For a complete lattice L, SL-TopP R  (L ↓ SFrm)op . (v) For a frame L, SL-TopZL  (L ↓ Frm)op . Corollary 3.12(v) was first proved by Zhang and Liu [46]. In this regard, Theorem 3.10 generalizes Zhang–Liu’s result in [46] to the present setting. 3.3. Comma-spatiality, comma-sobriety and their duality The dual adjunction between SC-M-L-Top and L ↓ C can be refined to a dual equivalence involving the notions of comma-spatiality and comma-sobriety. To elucidate this equivalence, we begin with the definitions of these notions: Definition 3.13. (i) An L ↓ C-object (uA , A) is (E, M)-comma-spatial, or shortly comma-spatial iff for some e m (E, M)-factorization A → B → LI(uA ) of AI(uA ) : A → LI(uA ) , e is a C-isomorphism. m (ii) An SC-M-L-Top-object (X, τ → LX ) is comma-sober iff for all h ∈ I(sm ), there exists a unique x ∈ X such that h = πx ◦ m. m

Proposition 3.14. For every L ↓ C-object (uA , A) and every SC-M-L-Top-object (X, τ → LX ), the next properties are true: (i) (uA , A) is comma-spatial iff (uA , A) is (E, M)-L-spatial. m m (ii) (X, τ → LX ) is comma-sober iff (X, (sm , τ ) → LX ) is an L-sober object of L ↓ C-M-L-Top. e

m

Proof. (i) Suppose (uA , A) is comma-spatial, i.e. there exists an (E, M)-factorization A → B → LI(uA ) of AI(uA ) such that e is a C-isomorphism. Since AI(uA ) is the underlying C-morphism of the L ↓ C-morphism (uA , A), e m e (uA , A) → (e ◦ uA , B) → LL↓C((uA ,A),L) is obviously an (E, M)-factorization of (uA , A), and (uA , A) → (e ◦ uA , B) is an L ↓ C-isomorphism. Thus, (uA , A) is (E, M)-L-spatial. Conversely, if (uA , A) is (E, M)-L-spatial, e m e i.e. there exists an (E, M)-factorization (uA , A) → (uB , B) → LL↓C((uA ,A),L) of (uA , A) such that (uA , A) → e m (uB , B) is an L ↓ C-isomorphism, then since A → B → LI(uA ) is an (E, M)-factorization of AI(uA ) and e is a C-isomorphism, (uA , A) is comma-spatial. m (ii) If (X, τ → LX ) is comma-sober, then for all h ∈ L ↓ C((sm , τ ), L), since h ∈ I(sm ), there exists a unique x ∈ X such that h = πx ◦ m. The converse implication is similar. 2 Corollary 3.15. The full subcategory (E, M)-CMSpat-L ↓ C of L ↓ C of all comma-spatial objects is dually equivalent to the full subcategory CMSob-SC-M-L-Top of SC-M-L-Top of all comma-sober objects. Proof. By virtue of Corollary 2.5, there exists a dual equivalence between (E, M)-L-Spat-L ↓ C and L ↓ C-M-L-SobTop. In addition to this, Proposition 3.14(i) gives (E, M)-L-Spat-L ↓ C = CMSpat-L ↓ C, while Proposition 3.14(ii) shows that the isomorphism T : L ↓ C-M-L-Top → SC-M-L-Top in Theorem 3.8 can be restricted to an isomorphism between L ↓ C-M-L-SobTop and CMSob-SC-M-L-Top. Hence, the assertion follows. 2 Comma-spatiality and comma-sobriety for quasivarieties can be expressed in a more explicit form: For a quasivariety V and a V-object L, an L ↓ V-object (uA , A) is (Surj(V), MV )-comma-spatial iff for all a, b ∈ A with a = b, there exists an h ∈ I(uA ) such that h(a) = h(b). On the other hand, if we define comma-sobriety of an SL-TopV -object iτ

(X, τ ) to be the comma-sobriety of the SV-MV -L-Top-object (X, τ → LX ), then the comma-sobriety of (X, τ ) V will be tantamount to that σ(X,τ ) : X → I(sτ ) is a bijection. In conclusion, dropping the prefix (Surj(V), MV ) in (Surj(V), MV )-CMSpat-L ↓ V for the sake of notational simplicity, CMSpat-L ↓ V is dually equivalent to the full subcategory CMSob-SL-TopV of SL-TopV of all comma-sober objects. This particularly yields the dual

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equivalence between CMSpat-L ↓ SQuant (resp. CMSpat-L ↓ SSQuant, CMSpat-L ↓ USQuant, CMSpat-L ↓ CGR, CMSpat-L ↓ SFrm and CMSpat-L ↓ Frm) and the full subcategory of WSL-QTop (resp. WSL-SQTop, WSL-TopR , WSL-TOPH , SL-TopP R and SL-TopZL ) of all comma-sober objects. 4. Stratified L-quasi-topological spaces and their duality Let (L, ≤, ⊗) be an s-quantale, and let ∗ be a binary operation (possibly different from ⊗) on L such that L∗ = (L, ≤, ∗) forms an integral commutative cl-monoid [15] (or a complete residuated lattice in the terminology of [2]), i.e. L∗ is an s-quantale having the further properties that (L, ∗, ) is a commutative monoid, and there exists a binary operation → (the so-called residuum operation) such that for all a, b, c ∈ L, a ∗ b ≤ c iff a ≤ b → c. We refer the reader to [2,15] for the properties of the residuum operation. To be in line with the stratified L-topological spaces in [16,18], we call an L-quasi-topology τ , or an L-quasi-topological space (X, τ ) stratified if the following condition (the so-called truncation condition) is satisfied: (TRN) For every a ∈ L and every μ ∈ τ , aX ∗ μ ∈ τ . (TRN) implies (CNS) if (X, τ ) is a strong L-quasi-topological space, and the reverse implication is also true if ⊗ = ∗. Except for these cases, (TRN)2 and (CNS) are unrelated. As a consequence, in contrast to WSL-QTop, Theorem 3.10 is not applicable to the full subcategory SL-QTop of L-QTop with stratified objects. The same problem is also valid for the full subcategory SL-SQTop (resp. SL-TopR and SL-TOPH ) of L-SQTop (resp. L-TopR and L-TOPH ) with stratified objects. Note that the subscript H in SL-TOPH is not used in [16,18]. Despite the fact that Theorem 3.10 does not work for SL-QTop, we will reveal, by means of Theorem 2.2, that SL-QTop is adjoint to the opposite of the category L∗ -SQuant of L∗ -s-quantales. For this reason, we are first interested in L∗ -SQuant and some of its properties in the next subsection. 4.1. The category of L∗ -s-quantales An L∗ -s-quantale is a generalization of the s-quantale fuzzifying the ordering relation underneath, and uses the notion of L-ordered set in the sense of [23,41,42]. By definition, an L-ordered set (X, R) consists of a set X and an L-order R on X, i.e. R : X × X → L is a map having the properties (i) for all x ∈ X, R(x, x) = , (ii) for all x, y, z ∈ X, R(x, y) ∗ R(y, z) ≤ R(x, z), (iii) for all x, y ∈ X, R(x, y) = R(y, x) = implies x = y. In an L-ordered set (X, R), an L-set μ ∈ LX is said to have a join (resp. meet)  w iff w is an element of X with the properties that μ(x) ≤ R(x, w) (resp. R(w, x) ≤ μ(x)) for all x ∈ X, and x∈X (μ(x) → R(x, y)) ≤ R(w, y) (resp.  X has a join (resp. meet), then it is unique, and is denoted (μ(x) → R(y, x)) ≤ R(y, w)) for all y ∈ X. If μ ∈ L x∈X by μ (resp. μ). An L-ordered set (X, R) is called a complete L-lattice iff each μ ∈ LX has both μ and μ, or equivalently, each μ ∈ LX has μ [23,41,42]. As is observed in [23,42], an L-ordered set (resp. a complete L-lattice) is equivalent to an L-ordered set (resp. a completely lattice L-ordered set) in [2], which is a special L-E-partially ordered set (resp. complete L-E-lattice) in [5,6]. Proposition 4.16. (See [22,23,38,41,42,47].) Every complete L-lattice (X, R) has the following properties: (i) (X, ≤R ) is a complete lattice, where ≤R is a binary  relation on X (called the underlying order of (X, R)) defined by x ≤R y iff R(x, y) = . Furthermore, the join A of any subset A of X in (X, ≤R ) is the same as χA , where χA : X → L is the characteristic function of A. 2 Further discussion and use of (TRN) can also be found in [3,4,44].

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(ii) For every a ∈ L and every x ∈ X, there is an element a  x ∈ X, called the tensor of a with x, such that for every y ∈ X, a → R(x, y) = R(a  x, y). Furthermore, a  x = ax , where ax ∈ LX is defined by ax (x) = a and ax (y) = ⊥ if y = x.   (iii) For every μ ∈ LX , μ = x∈X μ(x)  x, where refers to the join in (X, ≤R ). (iv) For every x ∈ X, ⊥  x is the bottom element of (X, ≤R ). Example 4.17. (See [23].) (i) (L, →) is a complete L-lattice, with the tensor  = ∗, whose underlying order is the partial order on L. (ii) For every set X and the map subX : LX × LX → L, defined by   μ1 (x) → μ2 (x) , subX (μ1 , μ2 ) = x∈X

(LX , subX ) is a complete L-lattice, with the tensor given by a  μ = aX ∗ μ, whose underlying order is the pointwise order on LX . Proposition 4.18. Let (X, R) and (Y, S) be complete L-lattices, and f : X → Y a function. Consider the following sentences: , i.e. f (μ) = fL→ (μ) for all μ ∈ LX , where fL→ : LX → LY is defined by (i) f preserves arbitrary  → [fL (μ)](y) = f (x)=y μ(x). (ii) f preserves tensors in the sense that f (a  x) = a  f (x) and f preserves arbitrary joins in the underlying orders. (iii) f : (X, R) → (Y, S) is L-order-preserving, i.e. R(x, y) ≤ S(f (x), f (y)) for all x, y ∈ X. Then, (i) ⇔ (ii) and (i) ⇒ (iii) hold. Proof. (i) ⇔ (ii) follows from [41, Theorem 3.5(1)] and [23, Proposition 3.13(1)], while (i) ⇒ (iii) is immediate from [41, Theorem 3.5(1)]. 2 Definition 4.19. A triple X = (X, RX , ) is called an L∗ -s-quantale iff (X, RX ) is a complete L-lattice and  is a binary operation on X. Definition 4.20. L∗ -SQuant is a construct whose objects are all L∗ -s-quantales, and whose morphisms are all h : X → Y such that h : X → Y is a function preserving  and arbitrary , i.e. (i) h(x  y) = h(x)  h(x) for all x, y ∈ X, X (ii) h(μ) = h→ L (μ) for all μ ∈ L . L∗ -s-quantale fuzzifies the notion of s-quantale allowing a fuzzily defined complete lattice as its underlying order structure. If (X, R, ) is an L∗ -s-quantale, then (X, ≤R , ) is an s-quantale. In case L∗ = 2, every s-quantale (X, ≤, ) can be identified as a 2-s-quantale, and moreover, 2-SQuant = SQuant. Quantaloid-enriched categories (alias Q-categories) provide an enriched category approach to fuzzy preordered sets [17,37,39]. Quantale preordered sets in [17], quantale-valued preordered sets in [40] and Ω-categories in [22,44] are three interesting cases of Q-categories including L-ordered sets. It is natural to think of getting one of these kinds of preordered sets instead of L-ordered sets in the definition of L∗ -s-quantales. Although this replacement suggests a more general approach to the fuzzification of the notion of s-quantale, further work is needed to investigate in which extent the results in this section can be extended to this new setting. We will not go to this direction, and restrict our attention to only L-ordered sets in this paper. Definition 4.21. Let X be an L∗ -s-quantale. A subset Y of X is an L∗ -subs-quantale of X iff Y is closed under  and arbitrary , i.e. the following conditions are satisfied:

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(a) For all x, y ∈ Y , x  y ∈ Y . (b) For all μ ∈ LY , μ ∈ Y , where μ : X → L is the extension of μ : Y → L to X, defined by μ(x) = μ(x) if x ∈ Y , and μ(x) = ⊥ otherwise. As a comparison of subs-quantale in [7] with L∗ -subs-quantale, the latter notion is a generalization of the former one to the present setting. Definition 4.22. (See [23].) A subset Y of a complete L-lattice (X, R) is a fuzzy opening system iff the following two conditions are satisfied: (OS1) (OS2)

For every subset A of Y , the join of A in (X, ≤R ) belongs to Y . For every x ∈ Y and every a ∈ L, the tensor a  x belongs to Y .

Owing to the following proposition, we may state an L∗ -subs-quantale of X as a fuzzy opening system in (X, RX ) closed under . Proposition 4.23. A subset Y of a complete L-lattice (X, R) is a fuzzy opening system iff the condition (b) in Definition 4.21 is satisfied. Proof. By Proposition 4.16(iii) and (iv),



μ = μ(x)  x = μ(x)  x x∈X

(4.9)

x∈Y

Y for every  μ ∈ L . If Y is a fuzzy opening system, then Yby (OS2), μ(x) x ∈ Y for every x ∈ Y , and by (OS1) and (4.9), μ = x∈Y μ(x)  x ∈ Y . If μ ∈ Y for every μ ∈ L , then it is easy to get (OS1) from (4.9) and Proposition 4.16(i), and similarly (OS2) from (4.9) and Proposition 4.16(ii). 2

For every fuzzy opening system Y in a complete L-lattice (X, R), (Y, R|Y ×Y ) is a complete L-lattice in which the join of μ ∈ LY is performed to be the join of μ ∈ LX in (X, R) [23]. This is the essence of the following remark: Remark 4.24. If Y is an L∗ -subs-quantale of X, then (Y, RX|Y ×Y , ) is an L∗ -s-quantale with the property that for all μ ∈ LY , μ = μ. Proposition 4.25. Let Memb be the class of all L∗ -SQuant-embeddings iY : Y → X such that Y ⊆ X and iY : Y → X is the inclusion map. iY : Y → X ∈ Memb iff Y is an L∗ -subs-quantale of X and RY = RX|Y ×Y . Proof. Let iY : Y → X ∈ Memb . Since iY preserves , x  y = iY (x)  iY (y) = iY (x  y) = x  y ∈ Y for all x, y ∈ Y , i.e. Y is closed under . For all μ ∈ LY , since μ = (iY )→ L (μ) and iY preserves arbitrary , μ = (iY )→ L (μ) = iY (μ) = μ ∈ Y, i.e. Y is closed under arbitrary . So, Y is an L∗ -subs-quantale of X. Then, by Remark 4.24, (Y, RX|Y ×Y , ) is an L∗ -s-quantale. Furthermore, it is easy to see that g : (Y, RX|Y ×Y , ) → X, defined by g(x) = x, is an L∗ -SQuant-morphism with the property that g = iY ◦ idY . Thus, since iY : Y → X is an initial morphism in L∗ -SQuant, idY : (Y, RX|Y ×Y , ) → Y will be an L∗ -SQuant-morphism. Then, by Proposition 4.18, id Y is L-orderpreserving w.r.t. RX|Y ×Y and RY , i.e. RX (x, y) = RX|Y ×Y (x, y) ≤ RY (x, y) for all x, y ∈ Y . Further, since iY : Y → X is an L∗ -SQuant-morphism, we also have from Proposition 4.18 that iY is L-order-preserving w.r.t. RY and RX , i.e. RY (x, y) ≤ RX (x, y) for all x, y ∈ Y , i.e. RY = RX|Y ×Y . Conversely, if Y is an L∗ -subs-quantale of X and RY = RX|Y ×Y , then we know that iY = g : Y → X is an L∗ -SQuant-morphism. Thus, to complete the proof, there remains only to be confirmed the initiality of iY . For an

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L∗ -s-quantale W, an L∗ -SQuant-morphism h : W → X and a function f : W → Y such that iY ◦ f = h, we observe that (i) for all x, y ∈ W , f (x  y) = iY ◦ f (x  y) = h(x  y) = h(x)  h(y) = (iY ◦ f )(x)  (iY ◦ f )(y) = f (x)  f (y), (ii) for all μ ∈ LW ,

  → → f (μ) = iY ◦ f (μ) = h(μ) = h→ L (μ) =  (iY )L ◦ fL (μ)  →  → → = (iY )→ L fL (μ) = fL (μ) = fL (μ),

i.e. f : W → Y is an L∗ -SQuant-morphism, proving the initiality of iY .

2

 Given a family of L∗ -SQuant-objects (Xi = (Xi , Ri , i ))i∈I indexed by the set I , i∈I Ri , ) is the product of (Xi )i∈I in L∗ -SQuant, where    Ri (xi )i∈I , (yi )i∈I = Ri (xi , yi ) and (xi )i∈I  (yi )i∈I = (xi i yi )i∈I i∈I



 i∈I

 Xi = ( i∈I Xi ,

i∈I

for all (xi )i∈I , (yi )i∈I ∈ i∈I Xi . Thus, L∗ -SQuant has set-indexed products. In particular, for every set Z and every L∗ -SQuant-object X, XZ is given as (X Z , RXZ , XZ ), where RXZ : X Z × X Z → L is defined by RXZ (f, g) =  x∈Z R(f (x), g(x)), and X Z is the pointwise extension of , which will also be denoted by . 4.2. Dual adjunction between SL-QTop and L∗ -SQuant To justify the title of this subsection, we show first that SL-QTop is isomorphic to L∗ -SQuant-Memb -L-Top (Theorem 4.28) and then that L∗ -SQuant is essentially (Surj∗ , Memb )-structured (Proposition 4.30), where L = (L, →, ⊗) and Surj∗ is the class of all L∗ -SQuant-morphisms with surjective underlying functions. Proposition 4.26. For a set X and a subset τ of LX , let subτX denote the restriction of subX to τ × τ . Then, τ is a stratified L-quasi-topology on X iff iτ : (τ, subτX , ⊗) → LX ∈ Memb . Proof. Notice first that LX = (LX , subX , ⊗). By virtue of Proposition 4.23 and Remark 4.24, τ is a stratified L-quasitopology on X iff τ is an L∗ -subs-quantale of LX . As a result, the equivalence in question follows immediately from Proposition 4.25. 2 Proposition 4.27. For every function f : X → Y , the map fL← : LY → LX , given by fL← (μ) = μ ◦ f , is the underlying function of fL← : LY → LX . Proof. It is not difficult to see that fL← is an L∗ -SQuant-morphism LY → LX satisfying πx ◦ fL← = πf (x) for all x ∈ X. The assertion thus follows from the uniqueness of fL← . 2 Theorem 4.28. The functor H : SL-QTop → L∗ -SQuant-Memb -L-Top, defined on objects by H (X, τ ) = iτ

(X, (τ, subτX , ⊗) → LX ), and on morphisms by H (f ) = f , is an isomorphism. Proof. By making use of Proposition 4.26, H is a bijective function between Ob(SL-QTop) and Ob(L∗ SQuant-Memb -L-Top). Secondly, we wish to see that H maps SL-QTop-morphisms to L∗ -SQuant-Memb L-Top-morphisms. Let f : (X, τ ) → (Y, ν) be an SL-QTop-morphism. Then, (fL← )|ν : ν → τ is a function. For an arbitrary μ ∈ Lν , we may write  ← →  fL | (μ) = fL← (μ) = fL← (μ) =  fL← L (μ). ν

Furthermore, one can easily verify that

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fL←

 → |ν L

→  (μ) = fL← L (μ),

and then observe that  ← →   →  →   fL | (μ) =  fL← L (μ) =  fL← | L (μ) =  fL← | L (μ), ν

ν

ν

which is why (fL← )|ν preserves arbitrary . Since (fL← )|ν obviously preserves ⊗, it follows that (ν, subνY , ⊗) → (τ, subτX , ⊗) is an L∗ -SQuant-morphism. Then, since fL← = fL← by Proposition 4.27, (ν, subνY , ⊗)

(fL← )|ν

(τ, subτX , ⊗)

iν

LY

(fL← )|ν :

iτ fL←

LX

commutes. Hence, f : H (X, τ ) → H (Y, ν) is an L∗ -SQuant-Memb -L-Top-morphism, which is the desired result. Since H preserves composition and identities, we conclude that H is, in fact, a functor. Given SL-QTop-objects (X, τ ) and (Y, ν), if f : H (X, τ ) → H (Y, ν) is an L∗ -SQuant-Memb -L-Top-morphism, then, since (fL← )|ν : (ν, subνY , ⊗) → (τ, subτX , ⊗) is an L∗ -SQuant-morphism, we have that (fL← )|ν : ν → τ is a function, and f : (X, τ ) → (Y, ν) is therefore an SL-QTop-morphism. This gives the fullness of H . In the last step, H is faithful from its definition, and hence an isomorphism. 2 Lemma 4.29. Let a construct (C, | |) have (E, M)-factorizations such that E ⊆ Surj(C) and M ⊆ Emb(C). Then, C is essentially (E, M)-structured. Proof. We only need to show that C has the unique (E, M)-diagonalization property. Suppose g ◦ e = m ◦ f for e ∈ E , m ∈ M and f, g ∈ Mor(C). It is known that Set has the unique (Surj, Inj)-diagonalization property, where Surj (resp. Inj) is the class of all surjective (resp. injective) functions. Now, by taking the forgetful functor | | : C → Set into consideration, since |g| ◦ |e| = |m| ◦ |f |, |e| ∈ Surj, |m| ∈ Inj and |f |, |g| ∈ Mor(Set), there exists a unique function c such that c ◦ |e| = |f | and |m| ◦ c = |g|. Then, since m is an initial C-morphism satisfying the equality |m| ◦ c = |g|, there exists a C-morphism d with |d| = c. Thereby, |d ◦ e| = |d| ◦ |e| = |f | and

|m ◦ d| = |m| ◦ |d| = |g|,

and hence, d ◦ e = f and m ◦ d = g follow from the faithfulness of | |.

2

Proposition 4.30. L∗ -SQuant is essentially (Surj∗ , Memb )-structured. Proof. Due to Lemma 4.29, we only need to prove that L∗ -SQuant has (Surj∗ , Memb )-factorizations. Let us eh(X)

ih(X)

take an L∗ -SQuant-morphism h : X → Y with the aim of factorizing h as X → h(X) → Y, where h(X) = (h(X), RY |h(X)×h(X) , ), eh(X) : X → h(X) is the co-domain restriction of h and ih(X) : h(X) → Y is the inclusion map. We first show that h(X) is an L∗ -subs-quantale of Y. Closedness of h(X) under  is clear. For every μ ∈ Lh(X) , if we consider the extension μ : Y → L of μ, and if we set μh : X → L by μh (x) = μ(h(x)), then μ = h→ L (μh ). Therefore, μ = h→ L (μh ) = h(μh ) ∈ h(X). As a result, h(X) is an L∗ -subs-quantale of Y, and then ih(X) : h(X) → Y ∈ Memb follows from Proposition 4.25. Also, it is easily seen that eh(X) ∈ Surj∗ . Hence, the equality h = ih(X) ◦ eh(X) finishes the proof. 2 The dual adjunction between SL-QTop and L∗ -SQuant in Theorem 4.31 below is constructed by the following sequences of definitions:

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Ψa : L∗ -SQuant(P, L) → L by Ψa (k) = k(a), τP∗ = {Ψa : a ∈ P },   LPt∗ (P) = L∗ -SQuant(P, L), τP∗ , LPt∗ (h)(k) = k ◦ hop ,    ∗ τ ∗ by η(X,τ η(X,τ ) : X → L∗ -SQuant τ, subX , ⊗ , L ) (x)(μ) = μ(x), eP∗ : P → τP∗

by eP∗ (a) = Ψa h

for every L∗ -SQuant-object P, every a ∈ P , every k ∈ L∗ -SQuant(P, L), every (L∗ -SQuant)op -morphism P → Q, every SL-QTop-object (X, τ ) and every μ ∈ τ . Theorem 4.31. (η∗ , ε ∗ ) : LΩ ∗  LPt∗ : (L∗ -SQuant)op → SL-QTop is an adjoint situation, where     (fL← )op  f |ν  LΩ ∗ (X, τ ) → (Y, ν) = τ, subτX , ⊗ → ν, subνY , ⊗ , h

LPt∗ (h)

LPt∗ (P → Q) = LPt∗ (P) → LPt∗ (Q),  ∗  op  and ε∗ = eP∗ . η ∗ = ηW W ∈Ob(SL-QTop) P∈Ob(L∗ -SQuant) Proof. Since L∗ -SQuant is an essentially (Surj∗ , Memb )-structured category with set-indexed products, Theorem 2.2 produces an adjoint situation (η, ε) : LΩMemb  LP tMemb : (L∗ -SQuant)op → L∗ -SQuant-Memb -L-Top. On the other hand, Theorem 4.28 allows us to write an adjoint situation (ididSL-QTop , ididB ) : H  H −1 : L∗ -SQuant-Memb L-Top → SL-QTop. Analogous to Theorem 3.10, the composition of these two adjoint situations yields the adjoint situation asked for. 2 4.3. Sobriety of stratified L-quasi-topological spaces, spatiality of L∗ -s-quantales and their duality Definition 4.32. An L-QTop-object (X, τ ) is called L-sober iff ψ : X → Lpt (τ ), defined by [ψ(x)](μ) = μ(x), is a bijection, where   Lpt (τ ) = SQuant (τ, ≤, ⊗), (L, ≤, ⊗) . For an L-QTop-object (X, τ ), let Lpt∗-mod (τ ) be the set of all h ∈ Lpt (τ ) satisfying h(aX ∗ λ) = a ∗ h(λ) for all a ∈ L and λ ∈ τ such that aX ∗ λ ∈ τ , and let Lpt∗ (τ ) denote the hom-set L∗ -SQuant((τ, subτX , ⊗), L) if (X, τ ) ∈ Ob(SL-QTop). Then, because ψ(X) ⊆ Lpt∗-mod (τ ) and ψ(X) ⊆ Lpt∗ (τ ) (if (X, τ ) ∈ Ob(SL-QTop)), the restriction of ψ to Lpt∗-mod (τ ) results in a map ψ ∗-mod : X → Lpt∗-mod (τ ), while its restriction to Lpt∗ (τ ) ∗ (if (X, τ ) ∈ Ob(SL-QTop)) coincides with η(X,τ ) : X → Lpt∗ (τ ). This suggests two other definitions of sobriety: Definition 4.33. (i) An L-QTop-object (X, τ ) is ∗-modified L-sober iff ψ ∗-mod : X → Lpt∗-mod (τ ) is a bijection. ∗ (ii) An SL-QTop-object (X, τ ) is L∗ -sober iff η(X,τ ) : X → Lpt∗ (τ ) is a bijection. The notion of L-sobriety (resp. ∗-modified L-sobriety) is motivated by the notion of L-sobriety (resp. modified  L-sobriety) in [29,30]. These two notions are generally different from each other. However, for the case ⊗ = ∗ = , L-sobriety (resp. ∗-modified L-sobriety) of a given (X, τ ) ∈ Ob(L-TopP R ) implies its L-sobriety (resp. modified L-sobriety) in [29,30]. Proposition 4.34. An SL-QTop-object (X, τ ) is ∗-modified L-sober iff it is L∗ -sober. Proof. The assertion is a direct consequence of the equality Lpt∗-mod (τ ) = Lpt∗ (τ ) that we easily deduce from the equivalence (i) ⇔ (ii) in Proposition 4.18. 2 Proposition 4.35. The full subcategory ∗-SobSL-QTop of SL-QTop consisting of all L∗ -sober objects is isomorphic to L∗ -SQuant-Memb -L-SobTop.

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Proof. It is clear from the very definition of L∗ -sobriety that an SL-QTop-object (X, τ ) is L∗ -sober iff H (X, τ ) is an L-sober object of L∗ -SQuant-Memb -L-Top. Because of this, the isomorphism H : SL-QTop → L∗ -SQuantMemb -L-Top in Theorem 4.28 restricts to an isomorphism between ∗-SobSL-QTop and L∗ -SQuant-Memb L-SobTop. 2 Definition 4.36. An L∗ -s-quantale P is called L-spatial iff for all a, b ∈ P with a = b, there exists k ∈ L∗ -SQuant(P, L) such that k(a) = k(b), or equivalently eP∗ : P → τP∗ is a bijection. Proposition 4.37. (Surj∗ , Memb )-L-Spat-L∗ -SQuant is precisely the full subcategory Spat-L∗ -SQuant of L∗ -SQuant of L-spatial objects. Proof. The assertion follows from the fact that P is (Surj∗ , Memb )-L-spatial iff P is L-spatial. The proof of this bi-implication primarily uses the fact that eP∗ : P → τP∗ is the underlying function of the L∗ -SQuant-morphism eP : τ∗

P → τP , where τP = LΩMemb (LP tMemb (P)) = (τP∗ , subLP∗ -SQuant(P,L) , ⊗). If P is (Surj∗ , Memb )-L-spatial, i.e. eP : P → τP is an L∗ -SQuant-isomorphism, then, since eP∗ : P → τP∗ is a bijection, P is L-spatial. Conversely, if P is L-spatial, i.e. eP∗ : P → τP∗ is a bijection, then it is easy to check that (eP∗ )−1 is an L∗ -SQuant-morphism τP → P, and so eP∗ will be an L∗ -SQuant-isomorphism P → τP , i.e. P is (Surj∗ , Memb )-L-spatial. 2 Corollary 4.38. Spat-L∗ -SQuant is dually equivalent to ∗-SobSL-QTop. Proof. In view of Corollary 2.5, the equivalence in question follows from Proposition 4.35 and Proposition 4.37.

2

As the last word, following the steps of the preceding two subsections, results analogous to Theorem 4.31 and Corollary 4.38 can be stated for SL-SQTop, SL-TopR and SL-TOPH . We leave the details to the reader. 5. Conclusion This is a follow-up study to our recent paper [10], where categorical fixed-basis fuzzy topological spaces (alias C-M-L-spaces), their category C-M-L-Top, the adjunction C-M-L-Top  Cop and the duality between (E, M)-L-Spat-C and C-M-L-SobTop are presented. In order to unify (weakly) stratified lattice-valued topological spaces in [16,18,29,33,46] and stratified variety-based spaces in [34] under the framework of C-M-L-spaces, we introduced stratified C-M-L-spaces and their category SC-M-L-Top. We then proceeded to prove the adjunction SC-M-L-Top  (L ↓ C)op as an application of C-M-L-Top  Cop , and to set up the duality between the full subcategory (E, M)-CMSpat-L ↓ C of L ↓ C of all comma-spatial objects and the full subcategory CMSob-SC-M-L-Top of SC-M-L-Top of all comma-sober objects by making use of the duality of (E, M)-L-Spat-C with C-M-L-SobTop. Stratified L-topological spaces that were extensively studied in [16,18] have been extended to stratified L-quasitopological spaces. Inasmuch as stratified L-quasi-topological spaces are not instances of stratified C-M-L-spaces, the duality between (E, M)-CMSpat-L ↓ C and CMSob-SC-M-L-Top is not applicable to them. Another significant contribution of this paper is to show a duality between the category of stratified L-quasi-topological spaces and the category of L∗ -semi-quantales. C-M-generalized spaces, C-M-variable-basis spaces and C-M-Hutton spaces were introduced in [8] as categorytheoretic counterparts of variety-based generalized topological spaces [36], variable-basis fuzzy topological spaces [31,32] and Hutton fuzzy topological spaces [19,31], respectively. Fixed-basis nature of the present paper stimulates the question of how stratified C-M-generalized spaces and their duality can be formulated. The same question is also valid for C-M-variable-basis spaces and C-M-Hutton spaces. These questions remain open for future studies. References [1] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories: The Joy of Cats, John Wiley & Sons, New York, 1990. [2] R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Publishers, New York, 2002. [3] P. Chen, D. Zhang, Alexandroff L-co-topological spaces, Fuzzy Sets Syst. 161 (2010) 2505–2514.

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[4] P. Chen, H. Lai, D. Zhang, Coreflective hull of finite strong L-topological spaces, Fuzzy Sets Syst. 182 (2011) 79–92. [5] M. Demirci, A theory of vague lattices based on many-valued equivalence relations—I: general representation results, Fuzzy Sets Syst. 151 (3) (2005) 437–472. [6] M. Demirci, A theory of vague lattices based on many-valued equivalence relations—II: complete lattices, Fuzzy Sets Syst. 151 (3) (2005) 473–489. [7] M. Demirci, Pointed semi-quantales and lattice-valued topological spaces, Fuzzy Sets Syst. 161 (9) (2010) 1224–1241. [8] M. Demirci, Category-theoretic fuzzy topological spaces and their dualities, Fuzzy Sets Syst. 227 (2013) 1–24. [9] M. Demirci, Applications of categorical fixed-basis fuzzy topological spaces to A-valued spaces, in: The 8th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2013, Milan, Italy, September 11–13, 2013. [10] M. Demirci, Fundamental duality of abstract categories and its applications, Fuzzy Sets Syst. (2013), http://dx.doi.org/10.1016/j.fss. 2013.08.015, in press. [11] Y. Diers, Categories of algebraic sets, Appl. Categ. Struct. 4 (1996) 329–341. [12] Y. Diers, Affine algebraic sets relative to an algebraic theory, J. Geom. 65 (1999) 54–76. [13] Y. Diers, Topological geometrical categories, J. Pure Appl. Algebra 168 (2002) 177–187. [14] E. Giuli, D. Hoffman, Affine sets: the structure of complete objects and duality, Topol. Appl. 156 (2009) 2129–2136. [15] U. Höhle, Commutative, residuated l-monoids, in: U. Höhle, E.P. Klement (Eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Boston, Dordrecht, 1995, pp. 53–106. [16] U. Höhle, Many Valued Topology and Its Applications, Kluwer Academic Publishers, Boston, 2001. [17] U. Höhle, T. Kubiak, A non-commutative and non-idempotent theory of quantale sets, Fuzzy Sets Syst. 166 (2011) 1–43. [18] U. Höhle, A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, in: The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, 1999, pp. 123–272. [19] B. Hutton, I. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Sets Syst. 3 (1980) 93–104. [20] J.R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972) 5–32. [21] P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982. [22] H. Lai, D. Zhang, Complete and directed complete Ω-categories, Theor. Comput. Sci. 388 (2007) 1–25. [23] H. Lai, D. Zhang, Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory, Int. J. Approx. Reason. 50 (2009) 695–707. [24] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621–633. [25] S. Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 1971. [26] D. Papert, S. Papert, Sur les treillis des ouverts et les paratopologies, in: Séminaire Ehresmann, in: Topologie et Géométrie Différentielle, vol. 1, 1957/1958, pp. 1–9. [27] H.E. Porst, W. Tholen, Concrete dualities, in: H. Herrlich, H.E. Porst (Eds.), Category Theory at Work, in: Research and Expositions in Mathematics, vol. 18, Heldermann Verlag, 1990, pp. 111–136. [28] P.-M. Pu, Y.-M. Liu, Fuzzy topology II: product and quotient spaces, J. Math. Anal. Appl. 77 (1980) 20–37. [29] A. Pultr, S.E. Rodabaugh, Lattice-valued frames, functor categories, and classes of sober spaces, in: S.E. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003, pp. 153–187 (Chapter 6). [30] A. Pultr, S.E. Rodabaugh, Examples for different sobrieties in fixed-basis topology, in: S.E. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003, pp. 427–440 (Chapter 17). [31] S.E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, in: The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, 1999, pp. 273–388. [32] S.E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics, Int. J. Math. Math. Sci. 2007 (2007) 1–71. [33] S.E. Rodabaugh, Necessity of non-stratified and anti-stratified spaces in lattice-valued topology, Fuzzy Sets Syst. 161 (2010) 1253–1269. [34] S.A. Solovyov, Sobriety and spatiality in varieties of algebras, Fuzzy Sets Syst. 159 (2008) 2567–2585. [35] S.A. Solovyov, From quantale algebroids to topological spaces: fixed-and variable-basis approaches, Fuzzy Sets Syst. 161 (2010) 1270–1287. [36] S.A. Solovyov, Generalized fuzzy topology versus non-commutative topology, Fuzzy Sets Syst. 173 (2011) 100–115. [37] I. Stubbe, Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory Appl. Categ. 14 (2005) 1–45. [38] I. Stubbe, Categorical structures enriched in a quantaloid: tensored and cotensored categories, Theory Appl. Categ. 16 (2006) 283–306. [39] I. Stubbe, An introduction to quantaloid-enriched categories, Fuzzy Sets Syst. (2013), http://dx.doi.org/10.1016/j.fss.2013.08.009, in press. [40] Y. Tao, H. Lai, D. Zhang, Quantale-valued preorders: globalization and cocompleteness, Fuzzy Sets Syst. (2012), http://dx.doi.org/10.1016/ j.fss.2012.09.013, in press. [41] W. Yao, L.-X. Lu, Fuzzy Galois connections on L-ordered sets, Math. Log. Q. 55 (2009) 105–112. [42] W. Yao, F.-G. Shi, Quantitative domains via fuzzy sets: part I: continuity of fuzzy directed complete posets, Fuzzy Sets Syst. 161 (2010) 973–987. [43] Y. Yue, Lattice-valued induced fuzzy topological spaces, Fuzzy Sets Syst. 158 (2007) 1461–1471. [44] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets Syst. 158 (2007) 349–366. [45] D. Zhang, Y.M. Liu, A localic L-fuzzy modification of topological spaces, Fuzzy Sets Syst. 56 (1993) 215–227. [46] D. Zhang, Y.M. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets Syst. 76 (1995) 259–270. [47] H. Zhao, D. Zhang, Many valued lattices and their representations, Fuzzy Sets Syst. 159 (2008) 81–94.