Categorical lattice-valued topology Lecture 1: powerset and topological theories, and their models

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1 Categorical lattice-valued topology Lecture 1: powerset and topological theories, and their models Sergejs Solovjovs Department of Mathematics and Statistics, Faculty of Science, Masaryk University Kotlarska 2, Brno, Czech Republic Abstract This lecture introduces the notions of powerset and topological theory, and presents the category of models of the latter, which unifies many approaches to lattice-valued topology, and which appears to be topological over its ground category. We also show a lattice-valued analogue of the adjoint situation between the categories of topological spaces and locales, and its respective concepts of sober topological space and spatial locale. 1. Basics on lattice-valued sets and lattice-valued topology 1.1. Lattice-valued sets Definition 1. Given a set X and a complete lattice L, an L-set in X is a map X α L. Given x X, α(x) is called the degree of membership of x in α. Remark 2. I-sets over the unit interval I = [0, 1] (also called fuzzy sets) were introduced by L. A. Zadeh in 1965 [20]. L-sets over an arbitrary complete lattice L (also called L-fuzzy sets, or lattice-valued sets, or many-valued sets) were introduced by J. A. Goguen in 1967 [6], who also pioneered the use of the term base or basis for the underlying lattice L of L-sets. Example 3. The classical set theory is based in the two-element lattice 2 = {, } (false or true), which is called the crisp approach. Most of the sets in our everyday life though are (implicitly) lattice-valued, e.g., the set of interesting books (from the given ones), the set of clever students (from those available in a classroom), the set of tasty confects (from a box), the set of attractive persons (from a group), etc. Remark 4. A lattice-valued analogue of a crisp concept or property is called a fuzzification of this concept or property, which follows the Fuzzification Principle of J. A. Goguen [6], which states 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).. This lecture course was supported by the ESF Project No. CZ.1.07/2.3.00/ Algebraic methods in Quantum Logic of the Masaryk University in Brno, Czech Republic. address: solovjovs@math.muni.cz (Sergejs Solovjovs) URL: (Sergejs Solovjovs) Preprint submitted to the Masaryk University in Brno April 9, 2013

2 Definition 5. The set of all L-sets in a given set X is called the L-powerset of X and is denoted L X. Given α, β L X, α is said to be a subset of β (denoted α β or α β) provided that α(x) β(x) for every x X. Given a family of L-sets L = (α i ) i I in X, the union (resp. intersection) of L in X is the L-set L or L (resp. L or L) such that ( α)(x) = i I α i(x) (resp. ( α)(x) = i I α i(x)). The whole (resp. empty) L-set in X is the constant map with value L (resp. L ), denoted X L (resp. L ), where L (resp. L ) is the largest (resp. smallest) element of L. If the lattice L is equipped with an order-reversing involution ( ) c, then the complement of an L-set α in X is the L-set α c such that α c (x) = (α(x)) c. Remark 6. Finite intersections of L-sets in a set X are often defined through an additional binary operation on the lattice L. For example, if L = (L,, 1 L ) is a unital quantale, then (α 1 α 2 )(x) = α 1 (x) α 2 (x), and the whole L-set in X is given by the constant map 1 L. In particular, on the unit interval I, the binary operation is often given by a (continuous) t-norm [10]. Example 7. The unit interval I = [0, 1] is equipped with an order-reversing involution I ( )c I, which is given by a c = 1 a for every a I, and also with the product operation I I I (the product t-norm). Definition 8. Let X f Y be a map, and let L be a complete lattice. Given an L-set α in X, the image of α under f is the L-set fl (α) in Y such that (f L (α))(y) = {α(x) f(x) = y}. Given an L-set β in Y, the preimage of β under f is the L-set fl (β) in X such that (f L (β))(x) = β f(x). The map LX f L L Y (resp. L Y f L L X ) is called the forward (resp. backward) L-powerset operator induced by f. Proposition 9. Given a complete lattice L, and a map X f Y, the map L Y f L L X is a complete lattice homomorphism. Moreover, fl f L, i.e., f L (α) β iff α f L (β) for every α LX and every β L Y. Proof. Given a family of L-sets L = (α i ) i I in X, (f L ( L))(x) = (( L) f)(x) = i I (α i f)(x) = i I (f L (α i))(x) = ( (f L (α i)) i I )(x), and similarly for. The last statement is an easy exercise [18] Lattice-valued topology Definition 10. Given a set X and a complete lattice L, an L-topology on X is a subset τ of L X, which satisfies the following conditions: (1) X L, L τ; (2) if α, β τ, then α β τ; (3) if (α i ) i I τ, then i I α i τ. An L-topology on X is called stratified provided that it has the following property: (4) a τ for every a L, where a denotes the constant map with value a. A pair (X, τ), where τ is a (stratified) L-topology on a set X, is called a (stratified) L-topological space. f A map X 1 X2 between L-topological spaces (X 1, τ 1 ) and (X 2, τ 2 ) is said to be continuous provided that fl (β) τ 1 for every β τ 2. L-Top (resp. L-STop) stands for the category of (resp. stratified) L-topological spaces and continuous maps. Remark 11. I-topological spaces and continuous maps were introduced by C. L. Chang in 1968 [2]. L- topological spaces over cl -monoids L (which are precisely the strictly two-sided quantales) were presented by J. A. Goguen in 1973 [7]. Stratified L-topological spaces go back to 1976 and are due to R. Lowen [11] (motivated by the fact that constant maps between L-topological spaces are not necessarily continuous). Example 12. Both the categories 2-Top and 2-STop are isomorphic to the classical category Top of crisp topological spaces and continuous maps. Theorem 13. The category L-Top (resp. L-STop) is a topological construct. 2

3 Remark 14. There exists an extension of the category L-Top, which uses many lattices instead of one. Definition 15. Given a subcategory C of the dual of the category CLat of complete lattices, C-Top is the concrete category over the product category Set C, whose objects are triples (X, L, τ), where X is a set, L is a C-object, and τ is an L-topology on X, and whose morphisms (X 1, L 1, τ 1 ) (f,ϕ) (X 2, L 2, τ 2 ) contain a map X 1 f X2 and a C-morphism L 1 ϕ L2 such that (f, ϕ) (β) := ϕ op β f τ 1 for every β τ 2. Remark 16. An analogue of the category C-Top was introduced by S. E. Rodabaugh in 1983 in [14]. In 1991, he coined the term point-set lattice-theoretic (poslat) topology for his approach [15]. Remark 17. There exist lattice-valued counterparts of many crisp topological properties. Lattice-valued theories, which are built over one lattice L (e.g., the category L-Top), are called fixed-basis approach [8], and those, which employ many lattices (e.g., the category C-Top), are called variable-basis approach [17] Lattice-valued T 0 -spaces Definition 18. An L-topological space is called L-T 0 provided that for every distinct x, y X, there exists α τ such that α(x) α(y). L-Top 0 denotes the full subcategory of L-Top of L-T 0 L-topological spaces. Example T 0 topological spaces are precisely the crisp T 0 topological spaces (i.e., for every x, y X such that x y, there exists U τ, which contains exactly one of x and y). Theorem 20. L-Top 0 is a reflective subcategory of L-Top. Proof. Given an L-topological space (X, τ), define an equivalence relation on X by x y iff α(x) = α(y) for every α τ. The quotient map (X, τ) (X/, τ/ ) is an L-Top 0 -reflection arrow for (X, τ) Lattice-valued compactness Definition 21. Given a complete lattice L and a set X, a subfamily L L X is called a cover of X provided that L = X L. A subcover of L is a subset M of L such that M = X L. Given an L-topological space (X, τ), a cover L of X is called open provided that L τ. A topological space (X, τ) is said to be L-compact provided that every open cover of X has a finite subcover. Definition 22. Given a complete lattice L and a cardinal κ, L is said to be κ-isolated in L provided that for every subset S L\{ L }, whose cardinality is less or equal than κ, it follows that S < L. Example 23. Given a finite chain k = {0, 1,..., k 1}, k is κ-isolated for every cardinal κ (absolutely isolated). Given the unit interval I, I is κ-isolated for every finite cardinal κ (finitely isolated), but I is not ℵ 0 -isolated, where ℵ 0 is the cardinal of the set of natural numbers (which is countable). Theorem 24. Let L be a strictly two-sided quantale (i.e., L is the unit of the quantale multiplication), and let I be a set, whose cardinality is less or equal than κ. (1) If L is κ-isolated, then every product i I (X i, τ i ) of L-compact L-topological spaces is L-compact. (2) If L is not κ-isolated, then there exists a family ((X i, τ i )) i I of L-compact L-topological spaces such that the product i I (X i, τ i ) is not L-compact. Corollary 25 (Lattice-valued Tychonoff theorem). Given a strictly two-sided quantale L, every product of κ L-compact L-topological spaces is compact iff L is κ-isolated in L. Remark 26. For L = 2, one gets the classical Tychonoff theorem of general topology [5]. 3

4 2. Categorical lattice-valued topology Remark 27. The striking difference in both the underlying lattice-theoretic structures of lattice-valued sets, and the definitions of lattice-valued topological structures themselves, which are currently popular among lattice-valued topologists, motivate a category-theoretic unification of lattice-valued topology. In particular, categorical lattice-valued topology was initiated by P. Eklund in 1986 in [4] Algebras and varieties Remark 28. Categorical lattice-valued topology relies on an extension of the notion of variety of abstract algebras (shortened to variety of algebra) of Universal Algebra, which is done in the sense of, e.g., [12]. Definition 29. Let Ω = (n λ ) λ Λ be a (possibly proper or empty) class of cardinal numbers. An Ω-algebra is a pair (A, (ωλ A) λ Λ), which comprises a set A and a family of maps A n λ ωa λ A (n λ -ary primitive operations on A). An Ω-homomorphism (A, (ωλ A) λ Λ) ϕ (B, (ωλ B) λ Λ) is a map A ϕ B, which makes the diagram A n λ ϕ n λ B n λ ω A λ A ϕ B ω B λ commute for every λ Λ. Alg(Ω) is the construct of Ω-algebras and Ω-homomorphisms. Remark 30. Every concrete category of this lecture is supposed to have the underlying functor to its ground category (e.g., the category Set of sets and maps), the latter mentioned explicitly in each case. Definition 31. Let M (resp. E) be the class of Ω-homomorphisms with injective (resp. surjective) underlying maps. A variety of Ω-algebras is a full subcategory of Alg(Ω), which is closed under the formation of products, M-subobjects (subalgebras) and E-quotients (homomorphic images). The objects (resp. morphisms) of a variety are called algebras (resp. homomorphisms). Definition 32. Given a variety A, a reduct of A is a pair (, B), where B is a variety such that Ω B is a subfamily of Ω A, and A B is a concrete functor (notice the implicit assumption that given an A-algebra A, the primitive operations w.r.t. B on A coincide with their corresponding ones on A). Example 33. (1) CSLat( ) is the variety of -semilattices, i.e., partially ordered sets having arbitrary. (2) SQuant is the variety of semi-quantales, i.e., -semilattices, equipped with a binary operation called multiplication. (3) Quant is the variety of quantales, i.e., semi-quantales, whose multiplication is associative and distributes across from both sides. (4) UQuant is the variety of unital quantales, i.e., quantales, whose multiplication has the unit. (5) Frm is the variety of frames, i.e., semi-frames which are quantales. (6) DmFrm is the variety of DeMorgan frames, i.e., frames, equipped with an order-reversing involution. (7) CBAlg is the variety of complete Boolean algebras, i.e., DeMorgan frames, whose involution is the complement. (8) SFrm is the variety of semi-frames, i.e., unital semi-quantales, whose multiplication is given by meets. Remark 34. Taken in the reverse order, starting from item (7), the varieties of the above example provide a sequence of reducts. Moreover, an important reduct of Frm is the variety SFrm 4

5 Definition 35. Given a unital quantale Q, a left unital Q-module A is a -semilattice, which is equipped with a map Q A A (called the action of Q on A), which satisfies the following conditions: (1) q ( S) = s S (q s) for every q Q and every S A; (2) ( S) a = s S (s a) for every S Q and every a A; (3) q 1 (q 2 a) = (q 1 q 2 ) a for every q 1, q 2 Q and every a A; (4) 1 Q a = a for every a A. Definition 36. Given a unital commutative quantale Q (i.e., q 1 q 2 = q 2 q 1 for every q 1, q 1 Q), a unital Q-algebra is a tuple (A,, 1 A, ) such that (1) (A, ) is a Q-module; (2) (A,, 1 A ) is a unital quantale; (3) q (a 1 a 2 ) = (q a 1 ) a 2 = a 1 (q a 2 ) for every a 1, a 2 A and every q Q. UAlg(Q) stands for the variety of unital Q-algebras. Remark 37. Every frame is a unital commutative quantale, and every unital commutative quantale is both a module and an algebra over itself (with action given by quantale multiplication). Proposition 38. Given a unital commutative quantale Q and a set X, every Q-subalgebra A of the Q- algebra Q X (pointwise operations) contains all constant maps. Proof. Since 1 Q A, it follows that q = q 1 Q = q 1 Q A for every q Q. Remark 39. From now on, varieties are denoted A or B, with S standing for the subcategories of their dual categories. The categorical dual of a variety A is denoted A op. The dual category of Frm is already known in the literature as Loc [9] (with the addition of S in front for semi-frames). Given a homomorphism ϕ of a variety A, the corresponding morphism of the dual category is denoted ϕ op and vice versa. Every algebra A of a variety A gives rise to the subcategory S A of A op, whose only morphism is the identity A 1 A A Powerset and topological theories, and their models Remark 40. Categorical lattice-valued topology is based in three cornerstones. The first of them relies on the fact that every map X f f Y induces two operators: image operator P(X) P(Y ), where f (S) = {f(x) x S}, and preimage operator P(Y ) f P(X), where f (T ) = {x f(x) T }. Definition 41. A backward powerset theory (bp-theory) in a category X (ground category of the theory) is a functor X P A op to the dual category of a variety A. Example 42. Given a variety A, every subcategory S of A op induces a functor Set S V=( ) A op defined by ((X 1, A 1 ) (f,ϕ) (X 2, A 2 )) = A X1 ((f,ϕ) ) op 1 A X2 2, where (f, ϕ) (α) = ϕ op α f. The case S = S A is denoted V A = ( ) A and is called fixed-basis approach, whereas all other cases are subsumed under variable-basis approach. In particular, P=( ) (1) Set S 2 2 CBAlg op, where 2 is the two-element Boolean algebra, provides the above-mentioned preimage operator; Z=( ) (2) Set S I I DmLoc, where I is the unit interval [0, 1], equipped with the standard algebraic structure, provides the fuzzy backward powerset operator of L. A. Zadeh [20]; G=( ) (3) Set S L L UQuant op provides the L-fuzzy backward powerset operator of J. A. Goguen [7]; L=( ) (4) Set S I I UAlg(I) op provides the fuzzy backward powerset operator of R. Lowen [11]; 5

6 (5) Set S R1=( ) USQuant op, Set S R2=( ) SLoc and Set S R3=( ) Loc provide the variable-basis lattice-valued backward powerset operators of S. E. Rodabaugh [19], [16], [3]. Additionally, the functors X Rq SQuant op and X R USQuant op provide the backward powerset theories of S. E. Rodabaugh [19]. Remark 43. The second cornerstone of lattice-valued categorical topology make topological theories, which rely on powerset theories, and reducts of the varieties involved in the latter. Definition 44. Let X P A op be a bp-theory and let (, B) be a reduct of A. The topological theory (t-theory) in X induced by P and (, B) is the functor X T B op = X P A op op B op. Remark 45. The essential necessity of topological theories is motivated by the simple observation that crisp powersets of sets are complete Boolean algebras, whereas topologies on sets are just frames, which, however, need an access to their overlying structure for the successful development of their respective theory (cf., e.g., the well-known correspondence between open and closed sets of a given topological space [5]). Remark 46. The third cornerstone of categorical lattice-valued topology is the category of topological structures, which are generated by a given topological theory. Definition 47. Let T be a t-theory in a category X. Top(T ) is the concrete category over X, whose objects (T -spaces) are pairs (X, τ), where X is an X-object and τ is a subalgebra of T X (T -topology on X), and whose morphisms (T -continuous X-morphisms) (X 1, τ 1 ) f f (X 2, τ 2 ) are X-morphisms X 1 X2 with the property that ((T f) op ) (τ 2 ) τ 1 (T -continuity). Remark 48. Top(T ) is also called the category of models of the topological theory T. Example 49. The case of the ground category X = Set S, where S is a subcategory of A op for a variety A, is called variety-based topology. In particular, Top((V A, B)) provides the category A B -Top, which is the setting for fixed-basis variety-based topology, whereas Top((V, B)) gives the category (S, B)-Top (the case A = B is shortened to S-Top), which is the setting for variable-basis variety-based topology. More specific, (1) Top((P, Frm)) is isomorphic to the category Top of topological spaces; (2) Top((Z, Frm)) is isomorphic to the category I-Top of fuzzy topological spaces of C. L. Chang; (3) Top((G, UQuant)) is isomorphic to the category L-Top of L-topological spaces of J. A. Goguen; (4) Top((L, UAlg(I))) is isomorphic to the category I-STop of stratified I-topological spaces of R. Lowen; (5) Top((R 1, USQuant)), Top((R 2, SFrm)), Top((R 3, Frm)) are isomorphic to the categories S-Top i, i {1, 2, 3} for variable-basis poslat topology of S. E. Rodabaugh [19], [16], [3]. The categories Top((R q, SQuant)) and Top((R, USQuant)) provide the categories T XRq and T XR of topological structures of S. E. Rodabaugh [19] Morphisms of topological theories Remark 50. Since there exist many different approaches to lattice-valued topology, one needs a way of intercommunication between different topological theories T and their categories of models. Definition 51. TpThr is the quasicategory, whose objects are t-theories X T B op, and whose morphisms (F,Φ,η) F Φ T 1 T 2 (shortened to η) comprise two functors X 1 X2, B 1 B2 and a natural transformation T 2 F η Φ op T 1, or, more specifically, X 1 T 1 B op 1 F X 2 η Φ op 6 B op T 2 2.

7 η 1 η 2 η 2 η 1 The composition of two t-theories T 1 T2, T 2 T3 is given by T 3 F 2 F 1 Φ op 2 Φop 1 T η 2F1 1 = T 3 F 2 F 1 Φ op 2 T Φ op 2 2F η1 1 Φ op 2 Φop 1 T 1. The identity on a t-theory T is the identity natural transformation T 1 T T. η Remark 52. One needs a way to convert a t-theory morphism T 1 T2 into a functor Top(T 1 ) Top η Top(T 2 ), with the aim to obtain a correspondence between the quasicategories TpThr and TpSpc, the objects of the latter being categories of the form Top(T ), and morphisms being functors between them. Proposition 53. There exists the correspondence TpThr Top η TpSpc, which is given by Top(T 1 T2 ) = Top(T 1 ) Top η Top(T 2 ), Top η((x, τ) f (Y, σ)) = (F X, (η op X Φe τ ) (Φτ)) F f (F Y, (η op Y Φe σ) (Φσ)), where τ e τ T 1 X and σ e σ T 1 Y are the respective embeddings. Top preserves identities Properties of the category Top(T ) Definition 54. Given a B-algebra B, a subset S B, and a subfamily Ω Ω B, S Ω denotes the smallest Ω-subalgebra of B containing S ( S ΩB is shortened to S ). Given a T -space (X, τ), a subset S T X is said to be an Ω-base of τ provided that τ = S Ω. Ω B -bases are called subbases. Example 55. The category Top gives the classical definition of base (resp. subbase), where elements of the topology are unions (resp. unions of finite intersections) of elements of the base (resp. subbase). Lemma 56. Let B 1 ϕ B2 be a B-homomorphism, and let Ω Ω B. (1) For every Ω-subalgebra C of B 2, ϕ (C) is an Ω-subalgebra of B 1. (2) For every subset S B 1, ϕ ( S Ω ) = ϕ (S) Ω. Proposition 57 (Subbasic continuity). Let T be a t-theory in a category X, and let (X 1, τ 2 ), (X 2, τ 2 ) be T -spaces such that τ 2 = S Ω. An X-morphism X 1 f X2 is T -continuous if and only if ((T f) op ) (S) τ 1. Proof. To show the sufficiency, notice that ((T f) op ) (τ 2 ) = ((T f) op ) ( S Ω ) ( ) = ((T f) op ) (S) Ω τ 1 Ω = τ 1, where ( ) uses Lemma 56 (2). Theorem 58. Given a t-theory T, the category Top(T ) is fibre-small and topological over X. Proof. The first statement is clear. a -structured source S = (X fi For the second one (employing Proposition 57), notice that given (X i, τ i ) ) i I, the initial structure on X w.r.t. f i S is provided by i I ((T f i) op ) (τ i ), and, moreover, given a -costructured sink S = ( (X i, τ i ) X)i I, the final structure on X w.r.t. S is provided by the intersection i I ((T f i) op ) (τ i ). Corollary 59 ([1]). (1) Top(T ) is (co)complete iff X is (co)complete. (2) Top(T ) is (co-)wellpowered iff X is (co-)wellpowered. (3) Top(T ) is extremally (co-)wellpowered iff X is extremally (co-)wellpowered. (4) Top(T ) is (Epi, Mono-Source)-factorizable iff X is (Epi, Mono-Source)-factorizable. (5) Top(T ) has regular factorizations iff X has regular factorizations. (6) Top(T ) has a (co)separator iff X has a (co)separator. Corollary 60 ([1]). Every construct Top(T ) (1) is complete and cocomplete, (2) is wellpowered (resp. co-wellpowered), (3) is an (Epi, Extremal Mono-Source)-category, (4) has regular factorizations, (5) has separators and coseparators. 7

8 3. Sobriety-spatiality equivalence Remark 61. This section considers the bp-theory Set V A=( ) A A op f, which is defined by (X 1 X2 ) A = (f A X1 A )op A X2, where fa (α) = α f, and its respective construct of models Top((V A, B)), which is denoted A B -Top, and whose objects are referred to as A-spaces. Proposition 62. There exists a functor A B -Top O A B op, where O A ((X 1, τ 1 ) f (X 2, τ 2 )) = τ 1 Theorem 63. The functor O A has a right adjoint. (f A )op τ 2. Proof. Given a B-algebra B, define P t A (B) = B(B, A ) (whose elements are denoted p). Let a map B Φ B A P ta(b) be given by (Φ B (b))(p) = p(b), obtaining thereby a B-homomorphism, and let σ B = Φ B (B). One gets an A-space (P t A(B), σ B ) and a B-homomorphism B εop B O(P t A (B), σ B ). To show that ε B is an O A -co-universal arrow for B, notice that given a B op -morphism O A (X, τ) ϕ B, define a map X f P t A (B) by (f(x))(b) = (ϕ op (b))(x). To show its continuity, notice that given b B, since (fa (Φ B(b)))(x) = (Φ B (b)) f(x) = (f(x))(b) = (ϕ op (b))(x) for every x X, fa (Φ B(b)) = ϕ op (b) τ. The condition ε B O A f = ϕ and the uniqueness of f with such property are straightforward computations. Remark 64. Given a B-algebra B, the A-space (P t A (B), σ B ) is called the A-spectrum of B. Remark 65. Theorem 63 is a special instance of the technique of [13, Definition 1.6 and Theorem 1.7]. Corollary 66. There exists an adjunction (η, ε) : O A P t A : B op A B -Top, where P t A (B 1 ϕ B2 ) = (P t A (B 1 ), σ B1 ) (ϕop ) A (P t A (B 2 ), σ B2 ), and (X, τ) η (X,τ) P t A O A (X, τ) is given by (η (X,τ) (x))(α) = α(x). Corollary 67. Given an A-space (X, τ), the map ε op τ is injective, and, moreover, (η (X,τ) ) A Φ τ = 1 τ. Definition 68. A B-algebra B is called A-spatial provided that there exists an A-space (X, τ) such that B is isomorphic to the A-topology τ. Proposition 69. For every B-algebra B, the following are equivalent: (1) B is A-spatial; (2) for every distinct b 1, b 2 B, there exists p P t A (B) such that p(b 1 ) p(b 2 ); (3) ε op B is injective; (4) ε op B is an isomorphism. Proof. For (2) (3) recall the definition of ε op A. For (3) (4) notice that εop A is always surjective. (4) (1) is obvious. (1) (4): If B ϕ τ is a B op -isomorphism, then ε op B = (O A P t A (ϕ)) op ε op τ (ϕ op ) 1. By Corollary 67, ε op τ is injective, and therefore, ε op B is an isomorphism. Corollary 70. For every A-space (X, τ), the A-topology τ is A-spatial. Definition 71. An A-space (X, τ) is called (1) A-T 0 provided that for every distinct x 1, x 2 X, there exists α τ such that α(x 1 ) α(x 2 ); (2) A-sober provided that for every B-homomorphism τ ϕ A, there exists a unique x X such that ϕ(α) = α(x) for every α τ. Proposition 72. Let A B -Top 0 be the full subcategory of A B -Top of A-T 0 A-spaces. Then A B -Top 0 is reflective in A B -Top. 8

9 Proof. Given an A-space (X, τ), define an equivalence relation on X by x y iff α(x) = α(y) for every α τ. The quotient map (X, τ) q (X/, τ/ ), where τ/ = {α A X/ qa (α) τ} provides then an A B -Top 0 -reflection arrow for (X, τ). Indeed, to show that the A-space (X/, τ/ ) is A-T 0, notice that given distinct equivalence classes [x], [y] X/, there exists α τ such that α(x) α(y). The map X/ α/ A given by α/ ([z]) = α(z) is an element of τ/ such that (α/ )([x]) (α/ )([y]). Proposition 73. An A-space (X, τ) is A-T 0 iff η (X,τ) is injective. Proof. Recall the definition of η (X,τ). Proposition 74. For every A-space (X, τ), the following are equivalent: (1) (X, τ) is A-sober; (2) η X is bijective; (3) η X is a homeomorphism. Proof. For (1) (2) recall the definition of η (X,τ). (3) (2) is obvious. (2) (3): To show continuity of P t A O A (X, τ) η 1 (X,τ) (X, τ), choose some α τ. Then (η 1 (X,τ) ) ( ) A (α) = α η 1 (X,τ) = ((η (X,τ) ) A Φ τ )(α) η 1 (X,τ) = Φ τ (α) η (X,τ) η 1 (X,τ) = Φ τ (α), where ( ) uses Corollary 67. Corollary 75. For every B-algebra B, the A-spectrum of B is A-sober. Definition 76. Let A-Sob (resp. A-Spat) be the full subcategory of A B -Top (resp. B op ) consisting of all A-sober A-spaces (resp. A-spatial B-algebras). Theorem 77. The categories A-Sob and A-Spat are equivalent. Proposition 78. The composite P t A O A is a left adjoint to the embedding A-Sob E A B -Top, and therefore, A-Sob is a reflective subcategory of A B -Top. Remark 79. The functor A B -Top P t AO A A-Sob is called the A-soberification functor. Given an A-space (X, τ), the A-spectrum of τ is called the A-soberification of (X, τ). Lemma 80. The composite O A P t A is a right adjoint functor to the embedding A-Spat E B op, and therefore, A-Spat is a coreflective subcategory of B op. Example 81. (1) For A = CBAlg, B = Frm, and A = 2, one gets the classical sobriety-spatiality equivalence of, e.g., [9]. (2) For A = B = SFrm, one gets the lattice-valued sobriety-spatiality equivalence of S. E. Rodabaugh [16]. References [1] J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Dover Publications (Mineola, New York), [2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), [3] J. T. Denniston and S. E. Rodabaugh, Functorial relationships between lattice-valued topology and topological systems, Quaest. Math. 32 (2009), no. 2, [4] P. Eklund, Categorical Fuzzy Topology, Ph.D. thesis, Åbo Akademi, [5] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, [6] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967), [7] J. A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973), [8] U. Höhle and A. P. Šostak, Axiomatic Foundations of Fixed-Basis Fuzzy Topology, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. Höhle and S. E. Rodabaugh, eds.), Kluwer Academic Publishers, 1999, pp [9] P. T. Johnstone, Stone Spaces, Cambridge University Press,

10 [10] E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic Publishers, [11] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976), [12] E. G. Manes, Algebraic Theories, Springer-Verlag, [13] H.-E. Porst and W. Tholen, Concrete Dualities, Category Theory at Work (H. Herrlich and H.-E. Porst, eds.), Heldermann Verlag, 1990, pp [14] S. E. Rodabaugh, A categorical accommodation of various notions of fuzzy topology, Fuzzy Sets Syst. 9 (1983), [15] S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets Syst. 40 (1991), no. 2, [16] S. E. Rodabaugh, Categorical Frameworks for Stone Representation Theories, Applications of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Academic Publishers, 1992, pp [17] S. E. Rodabaugh, Categorical Foundations of Variable-Basis Fuzzy Topology, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. Höhle and S. E. Rodabaugh, eds.), Kluwer Academic Publishers, 1999, pp [18] S. E. Rodabaugh, Powerset Operator Foundations for Poslat Fuzzy Set Theories and Topologies, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. Höhle and S. E. Rodabaugh, eds.), Kluwer Academic Publishers, 1999, pp [19] S. E. Rodabaugh, Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice- Valued Mathematics, Int. J. Math. Math. Sci (2007), [20] L. A. Zadeh, Fuzzy sets, Inf. Control 8 (1965),