() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 2, 2011 pp. 101-134 Dual attachment pairs in categorically-algebraic topology Anna Frascella, Cosimo Guido and Sergey A. Solovyov∗ Abstract The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category the- ory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation “∈” called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inher- ent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general set- ting for developing the attachment theory, motivated by the concept of (L, M)-fuzzy topological space of T. Kubiak and A. Šostak. 2010 MSC: 03E72, 54A40, 18B30, 18C99. Keywords: dual attachment pair, (lattice-valued) categorically-algebraic topology, (L, M)-fuzzy topology, (localic) algebra, (pre)image operator, quasi-coincidence relation, quasi-frame, spatialization, topological system, variety. ∗This research was partially supported by the ESF Project of the University of Latvia No. 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008. 102 A. Frascella, C. Guido and S. A. Solovyov 1. Introduction Motivated by the abundance of lattice-valued topological theories available in the literature and the lack of interaction means between them, this paper makes another step towards developing a new approach to (lattice-valued) topo- logical structures deemed to incorporate in itself the majority of the existing settings. Based in category theory and universal algebra, the new framework is called categorically-algebraic (catalg) topology [64] to underline its generat- ing theories. It originates from point-set lattice-theoretic (poslat) topology of S. E. Rodabaugh [61, 62], developed in the framework of lattice-valued powerset theories (motivated by algebraic theories (in clone form) of E. G. Manes [46], the basic example given by the theory of the powerset functor on the category Set of sets and maps, appended with its induced contravariant powerset func- tor), where the underlying algebraic structures for topology are semi-quantales. We replace semi-quantales with algebras (possibly having a class of non-finitary operations) from an arbitrary variety and consider an abstract category as the ground for topology. The framework obtained in this manner includes the most important approaches to (lattice-valued) topology, providing convenient means of intercommunication between them, and (that is more essential) ul- timately erasing the border between lattice-valued and crisp developments. Moreover, the amount of building blocks of the proposed theory is reduced to minimum, postulating the so-called “plug and play approach”, when additional requirements on the underlying setting are motivated by the need of additional properties. In particular, we never employ the framework of monadic topol- ogy, developed by several authors in the literature [18, 29, 31], as being too restrictive for our current purposes. Briefly speaking, we propagate the slogan: achieve more with less. On the other hand, it should be underlined imme- diately, that all essential properties of modern (lattice-valued) topology (e.g., compactness, separation axioms, connectedness, etc.) can be incorporated in the catalg setting. It is the theory of catalg spaces [67], which is currently undertaking the job. Moreover, the new framework is rapidly progressing in several other directions [65, 71, 72, 73, 74], influencing each other dramatically. It is the main purpose of this paper to show one of the important applications of the new theory, i.e., the development of a fruitful topological setting, based in a catalg generalization of the set-theoretic membership relation “∈”. The starting point for the proposed research topic lies in the concept of quasi- coincidence between a fuzzy point and a fuzzy set, introduced by P.-M. Pu and Y.-M. Liu [49, Definition 2.3′] with the aim to extend the standard approach to topology through neighborhood structures by fuzzifying the above relation “∈”. Given a set X, a fuzzy point ax (a map from X to the unit interval I = [0,1], taking value a at x and 0 elsewhere) is said to be quasi-coincident with a fuzzy set α (a map X α −→ I) provided that 1 − α(x) < a. Later on, Y.-M. Liu and M.-K. Luo [43, Definition 2.3.1] used a completely distributive lattice L, equipped with an order-reversing involution (−)′, to generalize the definition to a 66 (α(x))′. Moreover, Y.-M. Liu [42] showed that the quasi-coincidence Dual attachment pairs in categorically-algebraic topology 103 relation is the unique membership relation, which satisfies the four principles of a “reasonable” (generating a fruitful topological theory) membership relation. The next step was done by C. Guido (and V. Scarciglia) [25, 26, 27], who removed the requirement on the existence of an involution and introduced a lattice-valued analogue of the relation in question under the name of at- tachment. Given a complete lattice L, an attachment A in L is a family {Fa |a ∈ L\{⊥}} of completely prime ( ∨ S ∈ Fa implies S ⋂ Fa 6= ∅) fil- ters of L, indexed by its elements, with an additional requirement F⊥ = ∅. An L-point ax is said to be attached to an L-set α (denoted ax Aα) provided that α(x) ∈ Fa. The new notion not only generalizes quasi-coincidence relation (take L = I and let Fa = {b | 1 − a < b} for every a ∈ L), but also induces a functor from the category L-Top of L-topological spaces [33] to the category Top of topological spaces, which takes an L-topological space (X,τ) to the crisp space (SX,τ ⋆), where SX is the set of L-points of X and τ ⋆ = {α⋆ |α ∈ τ} with α⋆ = {ax |ax Aα}. The new functor is closely related to the well-known hypergraph functors [20, 32, 36, 53, 55, 58] used in lattice-valued mathemat- ics, bringing them under the common roof of attachment, and thereby remov- ing the difference in their definition by various authors, that prevents them from gaining in popularity in applications. The employed machinery is based in the concept of topological system of S. Vickers [75], introduced to merge point-set (topological spaces) and pointless (their underlying algebraic struc- tures - locales [35]) topology, which recently has raised an interest among the fuzzy researchers [12, 13, 14, 74] as a possible framework to incorporate both lattice-valued topology and its underlying algebraic structures (in most cases, particular kinds of the already mentioned semi-quantales). In [73], S. Solovyov generalized the approach even further, taking into con- sideration the fact that there exists a one-to-one correspondence between com- pletely prime filters of a complete lattice L and points of L (frame homomor- phisms from L to the two-element frame 2 = {⊥,⊤} [35]), which opens a possibility to define an attachment as a map L A −→ Frm(L,2) (omitting the re- quirement F⊥ = ∅ as never influencing the essential properties of the theory). The above-mentioned catalg approach to topology in hand, he introduced the notion of variable-basis attachment for an arbitrary variety of algebras. Definition 1.1. Let A be a variety of algebras and let A (−)∗ −−−→ Setop be a functor, which takes an A-algebra A to its underlying set. An (A-)attachment is a triple F = (ΩF,Σ F, ), where ΩF and ΣF are A-algebras, and ΩF −→ A(ΩF,ΣF) is a map. An attachment morphism F1 f −→ F2 is a pair of A- homomorphisms (ΩF1,ΣF1) (Ωf,Σ f) −−−−−−→ (ΩF2,ΣF2) such that for every a1 ∈ ΩF1 and every a2 ∈ ΩF2, ( 2(a2))(Ω f(a1)) = (Σf ◦ 1((Ωf) ∗op(a2)))(a1). AttA is the category of attachments and their homomorphisms, concrete over the product category A × A. The main achievement of [73] is a common framework (based in catalg at- tachment) for the majority of instances of hypergraph functor, providing a 104 A. Frascella, C. Guido and S. A. Solovyov convenient tool for exploring their features, and the explicit study of cate- gorical properties of attachment and its generated functors (in the sense of [25, 26]), which appear to have a right adjoint for a particular attachment type called spatial, generalizing the respective property of the hypergraph functor of U. Höhle [32]. The advantage of the last result is the extension of the achieve- ment of U. Höhle to all of the above instances of hypergraph functor, taking the appropriate underlying variety in each case. Moreover, this fact illustrates the main contribution of the catalg setting itself, whose essence is: prove once for many. It is the goal of this paper to continue the once started line of research by presenting a dual version of attachment. The meta-mathematical induce- ment for the new approach was given by the observation that both partially ordered sets and categories provide the means for dualization of their results. The real push, however, was taken up by the authors in their wish to change the setting of lattice-valued attachment of [26] from filters to ideals. After a brief discussion on the topic, the following crucial observations came to light. Observation 1. The case of a complete chain L provides a possibility to define an attachment A on L as a family F⊥ = ∅ and Fa = {b ∈ L |a < b} for every a ∈ L\{⊥}. More particularly, given an L-point ax and an L-set α, ax Aα iff α(x) ∈ Fa iff a < α(x). If α(x) 6= ⊤, then Gα(x) = {b ∈ L |b < α(x)} is a completely prime ideal of L, which is the notion dual to that of completely prime filter. This suggests the family G⊤ = ∅ and Ga = {b ∈ L |b < a} for every a ∈ L\{⊤} as a possible substitute for A, thereby turning the attachment condition ax A α from α(x) ∈ Fa into a ∈ Gα(x). Observation 2. In the wake of [73], the concluding section of [26] intro- duced the notion of generalized attachment, based in the category QFrm of quasi-frames (Definition 2.2 of this paper), which are complete lattices, with the respective morphisms preserving arbitrary ∨ and binary ∧ (the empty meet is excluded). The new category gives rise to the notion of completely prime quasi-filter of a complete lattice L as the preimage of {⊤} under a quasi-frame map (quasi-point) L p −→ 2, which allows, apart from the standard filters, also the empty one. A generalized attachment in a quasi-frame L is then a map L −→ QFrm(L,2) (cf. Definition 1.1). The particular case of a complete chain L suggests the following definition of the map : ( (a))(b) = { ⊤, a < b, ⊥, otherwise, providing a generalized attachment F = (L,2, ). Brief consideration brings a new map L � −→ 2L defined by (�(a))(b) = ( (b))(a), which induces the triple G = (L,2,�). An important property of the map is its preservation of ∨ and binary ∧, i.e., (�( ∨ S))(b) = ⊤ iff b < ∨ S iff b < s for some s ∈ S iff (�(s))(b) = ⊤ for some s ∈ S iff ( ∨ s∈S �(s))(b) = ⊤, whereas (�(s∧t))(b) = ⊤ iff b < s ∧ t iff b < s, b < t iff (�(s))(b) ∧ (�(t))(b) = ⊤. On the other hand, the map �(a) is not ∨ -preserving for a 6= ⊥, since (�(a))(⊥) = ⊤ 6= ⊥. In Dual attachment pairs in categorically-algebraic topology 105 such a manner, the so-called dual attachment pair (F,G) arises, where duality means ax F α iff ( (a))(α(x)) = ⊤ iff �(α(x))(a) = ⊤ iff ax Gα. The above remarks provide an opening for a new definition of attachment, called in this paper dual attachment. Apart from concrete applications to the al- ready developed theory, the concept represents a catalg extension of the notion of “duality” in mathematics (should not be mixed with the theory of catalg du- alities [71, 72] dealing with topological representations of algebraic structures). The attentive reader will see that catalg “duality” is neither categorical duality (as, e.g., the dual of a category), nor algebraic duality (as, e.g., the dual of a partially ordered set), but truly categorically-algebraic “duality”. It will be the topic of our forthcoming papers to find the proper place for such kind of dual- ities in mathematics, whereas this manuscript is bound to consider categorical properties of dual attachment and the functors arising from it. It appears that the concept still retains a close relation to topological systems of S. Vickers. On the other hand, the results of this paper clearly show that the nature of the two notions is essentially different, the latter being equipped with an internal topology extracted by the procedure of spatialization of systems introduced by S. Vickers [75], whereas the former providing a way of interaction (natural transformation) between two topological theories, resulting in a functor be- tween the categories of the respective topological structures. The achievement finally resolves the question (posed in the fuzzy community) on relationships between the two concepts and a possible common framework for both of them (non-existent due to the principally different categorical perspectives of the no- tions). Moreover, the just mentioned crucial property of attachment gave rise to the study of one of the authors on general relationships between catalg topo- logical theories and their induced catalg topological structures (see Section 6 of this manuscript for the respective definitions), partly announced during the presentation of [65] and currently being developed as the subject of a forth- coming paper, similar by the approach (but not the results) to the widely used in categorical algebra algebraic theories of F. W. Lawvere [41]. This paper uses both category theory and universal algebra, relying more on the former. The necessary categorical background can be found in [1, 28, 45, 46]. For the notions of universal algebra we recommend [7, 9, 23, 46]. Although the authors tried to make the paper as much self-contained as possible, some details are still omitted and left to the reader. 2. Dual attachment with its induced categories and functors In this section, we introduce the notion of dual attachment and consider its related categories and functors. The cornerstone of the approach is the concept of algebra. The structure is to be thought of as a set with a family of operations defined on it, satisfying certain identities, e.g., semigroup, monoid, group and also (that is different from the standard theory of universal algebra) complete lattice, frame, quantale. The classes of finitary algebras (those induced by a set of finitary operations) are usually described in universal algebra as either varieties or equational classes [7, 9, 23], which coincide due to the well-known 106 A. Frascella, C. Guido and S. A. Solovyov HSP-theorem of G. Birkhoff [6]. To incorporate the algebraic structures used in lattice-valued topology (where set-theoretic unions are usually replaced by arbitrary joins), this paper extends the approach of varieties to cover its needs. Definition 2.1. (1) Let ΩΩΩ = (nλ)λ∈Λ be a (possibly proper) class of cardinal numbers. An ΩΩΩ-algebra is a pair (A,(ωAλ )λ∈Λ), comprising a set A and a fam- ily of maps Anλ ω A λ −−→ A (nλ-ary primitive operations on A). An ΩΩΩ- homomorphism (A,(ωAλ )λ∈Λ) ϕ −→ (B,(ωBλ )λ∈Λ) is a map A ϕ −→ B such that ϕ ◦ ωAλ = ω B λ ◦ ϕ nλ for every λ ∈ Λ. Alg(ΩΩΩ) is the construct of ΩΩΩ-algebras and ΩΩΩ-homomorphisms. (2) Let M (resp. E) be the class of ΩΩΩ-homomorphisms with injective (resp. surjective) underlying maps. A variety of ΩΩΩ-algebras is a full subcate- gory of Alg(ΩΩΩ) closed under the formation of products, M-subobjects and E-quotients. The objects (resp. morphisms) of a variety are called algebras (resp. homomorphisms). (3) Given a variety A, a reduct of A is a pair (‖ − ‖,B), where B is a variety such that ΩΩΩB ⊆ ΩΩΩA and A ‖−‖ −−→ B is a concrete functor. From now on, every concrete category is supposed to be equipped with the underlying functor | − | to its respective ground category (cf. Definition 1.1). For the sake of shortness, the fact will be never mentioned explicitly again. An experienced reader will probably be able to find numerous examples to back the new notion. Below, we extend the list with several more items, all of which (except the last one) come from the realm of lattice-valued topology [61, 62] and will be used throughout the paper. Definition 2.2. (1) Given Ξ ∈ { ∨ , ∧ }, a Ξ-semilattice is a partially ordered set having arbitrary Ξ. CSLat(Ξ) is the variety of Ξ-semilattices. (2) A semi-quantale (s-quantale) is a ∨ -semilattice equipped with a binary operation ⊗ (multiplication). SQuant is the variety of s-quantales. (3) An s-quantale is called DeMorgan provided that it is equipped with an order-reversing involution (−)′. DmSQuant is the variety of DeMor- gan s-quantales. (4) An s-quantale is called unital (us-quantale) provided that its multipli- cation has the unit . USQuant is the variety of us-quantales. (5) A quantale is an s-quantale whose multiplication is associative and distributes across ∨ from both sides. Quant is the variety of quantales. (6) A quasi-frame (q-frame) is an s-quantale whose multiplication is ∧. QFrm is the variety of q-frames. (7) A semi-frame (s-frame) is a unital q-frame. SFrm is the variety of s-frames. (8) A frame is an s-frame which is a quantale. Frm is the variety of frames. Dual attachment pairs in categorically-algebraic topology 107 (9) A closure semilattice (c-semilattice) is a ∧ -semilattice, with the singled out bottom element ⊥. CSL is the variety of c-semilattices. The reader should bear in mind that all varieties of Definition 2.2 have complete lattices as objects. Moreover, all of them except DmSQuant, Quant and Frm are reducts of the variety CLat of complete lattices. To continue the topic, we remark that CSLat( ∨ ) is a reduct of SQuant; SQuant is a reduct of USQuant and QFrm; USQuant is a reduct of SFrm; Set is a reduct of any variety. Also notice that the categories SFrm and QFrm, having essentially the same objects (complete lattices), differ significantly on morphisms. The last item of Definition 2.2 was motivated by the concept of strong (⊤-preserving) quantale homomorphism [37, 38], and would provide an additional example for the concept of catalg topology introduced later on in the paper. For the sake of convenience, from now on we use the following notations, which differ from the respective category-theoretic ones (see, e.g., [14, 57, 61] for the motivation). An arbitrary variety is denoted A, B, C, etc. The cate- gorical dual of a variety A is denoted LoA (the “Lo” comes from “localic”), whose objects (resp. morphisms) are called localic algebras (resp. homomor- phisms). Several other categories introduced in the paper (but always related to varieties) employ similar notation for their duals. Following the already accepted designation of [35], the dual of Frm is denoted Loc , whose objects are called locales. To distinguish maps (or, more generally, morphisms) and homomorphisms, the former are denoted f,g,h, reserving ϕ,ψ,φ for the latter. Given a homomorphism ϕ, the respective localic one is denoted ϕop and vice versa. Given an algebra A of a variety A (or an object of a related category), SA stands for the subcategory of LoA comprising the identity 1A on A as the only morphism. We will occasionally use the notation SAA, to underline the originating variety of the algebra A. Given a set X, an algebra A and an element a ∈ A, X aX −−→ A denotes the constant map with value a. A few words are due to the many-valued framework employed in the paper. Following [73], we extend the concept of lattice-valued set to that of algebraic one, which is defined as follows (recall the underlying functor of Alg(ΩΩΩ)). Definition 2.3. Let X be a set and let A be an algebra of a variety A. An (A-)algebraic set in X is a map X α −→ |A|. The underlying idea of the new setting is based in a direct algebraization of the classical frameworks of L. A. Zadeh [77] and J. A. Goguen [21], which can be easily restored by choosing an appropriate variety A. Despite the fact that the theory of algebraic sets provides a nice challenge for research, the current paper will not develop the topic off the bounds of its interests. To distinguish algebraic sets from other maps, from now on, they will be denoted α, β, γ. All preliminaries in their places, the new notion of attachment is ready to introduce (recall that Set stands for the category of sets and maps). Definition 2.4. Let B be a variety and let B (−)∗ −−−→ Setop be a functor such that B∗ = |B|. A dual (B-)attachment is a triple G = (ΩG,ΣG,�), where 108 A. Frascella, C. Guido and S. A. Solovyov ΩG, ΣG are B-algebras, and ΩG � −→ ΣG|Ω G| is a B-homomorphism. A dual attachment morphism G1 f −→ G2 is then a pair of B-homomorphisms (ΩG1,ΣG1) (Ωf,Σ f) −−−−−−→ (ΩG2,ΣG2) such that for every b1 ∈ ΩG1 and every b2 ∈ ΩG2, (�2(Ωf(b1)))(b2) = (Σf ◦ �1(b1))((Ω f) ∗op(b2)). ATTB is the category of dual attachments and their homomorphisms, concrete over the product category B × B. To convince the reader that Definition 2.4 gives a category, we check the closure under composition. Given two ATTB-morphisms G1 f −→ G2, G2 g −→ G3 and b1 ∈ ΩG1, b3 ∈ ΩG3, one easily gets that (�3(Ω(g ◦ f)(b1)))(b3) = Σg ◦ (�2(Ωf(b1)))((Ω g) ∗op(b3)) = Σg ◦ Σf ◦ (�1(b1))((Ω f) ∗op ◦ (Ωg)∗ op (b3)) = Σ(g◦f)◦(�1(b1))((Ω g ◦ Ωf) ∗op(b3))=(Σ(g◦f)◦(�1(b1)))(((Ω(g ◦ f)) ∗op)(b3)). An attentive reader will notice striking similarities between the categories AttA (Definition 1.1) and ATTB (to distinguish the new type of attachment, capital letters are used in the notation of the respective category). A somewhat deeper insight into their nature reveals not less striking differences in their behavior, one of which being ready for display on the spot. In [73], a full embedding A � � EA // AttA was provided, showing that AttA gave a proper extension of its underlying variety A. In the new framework, a similar procedure results in an (in general, non-full) embedding under certain requirements only. Proposition 2.5. Suppose there exists a nullary operation ωλ0 of B, satis- fying the identity ωλ(〈ωλ0 〉nλ) = ωλ0 for every B-operation ωλ (implying that ωλ0 is the unique nullary operation of B). Then there exists an (in general, non-full) embedding B � � EB // ATTB, EB(B1 ϕ −→ B2) = (B1,B1,�1) (ϕ,ϕ) −−−→ (B2,B2 �2), with Bi �i −−→ B |Bi| i given by �i(b) = ω Bi λ0 . Proof. To show that the functor is correct on objects, notice that given λ ∈ ΛB and bi ∈ B for i ∈ nλ, (�(ω B λ (〈bi〉nλ)))(b) = ω B λ0 (b) = ωBλ0 = ω B λ (〈ω B λ0 〉nλ) = ωBλ (〈(�(bi))(b)〉nλ) = (ω B |B| λ (〈�(bi)〉nλ))(b) for every b ∈ B. To check the correctness on morphisms, notice that given b1 ∈ B1 and b2 ∈ B2, one gets, (�2(ϕ(b1)))(b2) = ω B2 λ0 (b2) = ω B2 λ0 = ϕ(ωB1 λ0 ) = (ϕ ◦ ωB1 λ0 )(ϕ∗op(b2)) = (ϕ ◦ (�1(b1)))(ϕ ∗op(b2)). The embedding properties of EB follow directly from the definition of the functor. The claim on non-fullness requires an additional assumption that there exist two different B-homomorphisms B1 ϕ // ψ // B2. Then EB(B1) (ϕ,ψ) −−−→ EB(B2) is an ATTB-morphism, since given b1 ∈ B1 and b2 ∈ B2, (�2(ϕ(b1)))(b2) = ω B2 λ0 (b2) = ω B2 λ0 = ψ(ωB1 λ0 ) = (ψ ◦ ωB1 λ0 )(ϕ∗op(b2)) = (ψ ◦ (�1(b1)))(ϕ ∗op(b2)). By the fact that ϕ 6= ψ, we obtain that (ϕ,ψ) is not in the image of EB, thereby concluding the proof of the proposition. � An example for Proposition 2.5 is the variety CSLat( ∨ ) of ∨ -semilattices, which gives rise to the respective non-full embedding. The variety Frm of Dual attachment pairs in categorically-algebraic topology 109 frames, however, does not fit into the proposed framework, having more than one nullary operation, but its reduct QFrm suits well. In one word, in some cases the category ATTB provides a proper extension of its underlying variety. To continue, we need additional notions from the framework of categorically- algebraic (catalg) topology, introduced recently [64] as an extension of the point- set lattice-theoretic (poslat) topology of S. E. Rodabaugh [57, 61]. The full development of the theory will be given in Section 6 of this paper, whereas here, we just borrow some of its building blocks. By analogy with its predecessor, the new setting is based in a generalization of the backward powerset theory employed by the classical topological setting. The intuition for the new concept comes from the so-called (pre)image opera- tors [61], well-known for every working mathematician. Recall that given a set map X f −→ Y , there exist the maps P(X) f → −−→ P(Y ) (resp. P(Y ) f ← −−→ P(X)) such that f→(S) = {f(x) |x ∈ S} (resp. f←(T) = {x |f(x) ∈ T}). The operators have already been extended to powersets of lattice-valued sets (see [10, 21, 56, 77]) and the latter one can be lifted to a more general setting. Proposition 2.6. Given a variety A, every subcategory C of LoA induces a functor Set × C (−)← −−−→ LoA defined by ((X1,A1) (f,ϕ) −−−→ (X2,A2)) ← = A X1 1 ((f,ϕ)←)op −−−−−−−→ AX22 with (f,ϕ) ←(α) = ϕop ◦ α ◦ f. Proof. The proof consists of easy calculations and can be found in [69, 70]. � For the sake of convenience, the functor Set × SA (−)← −−−→ LoA (the so- called fixed-basis approach, whereas the full framework is referred to as the variable-basis approach) is denoted by (−)←A , omitting the notation for 1A in its definition. The functor of Proposition 2.6 has the merit of incorporating in itself the majority of the approaches to powersets of many-valued mathematics. The most crucial of its properties is the fact that it gives rise to a category of catalg (strictly speaking, its variety-based reduction [67]) topological spaces, providing a common framework for many approaches to (lattice-valued) topology. Definition 2.7. Let A be a variety and let C be a subcategory of LoA. A C-topological space (C-space) is a triple (X,A,τ), where (X,A) is a Set × C- object, and τ (C-topology on (X,A)) is a subalgebra of AX. A C-continuous map (X1,A1,τ1) (f,ϕ) −−−→ (X2,A2,τ2) is a Set × C-morphism (X1,A1) (f,ϕ) −−−→ (X2,A2) such that ((f,ϕ) ←)→(τ2) ⊆ τ1. C-Top is the category of C-spaces and C-continuous maps, which is concrete over the product category Set × C. The category SA-Top is denoted A-Top, whose objects (resp. morphisms) are shortened to (X,τ) (resp. f). It should be underlined that the category C-Top is a particular instance of a more general approach to catalg topology, developed in Section 6 of this paper. The main advantages of the new framework have already been described in an abstract way in Introduction and would be illustrated by concrete examples in Section 6. At the moment, the reader should notice that apart from serving 110 A. Frascella, C. Guido and S. A. Solovyov as a convenient tool for developing the attachment theory, the new setting provides the (much needed) means of interaction between hugely diversified (lattice-valued) topological theories available in the modern literature. It appears that the framework of attachment provides a more general cate- gory for topology than C-Top. Definition 2.8. Given a variety B and a subcategory D of LoATTB, a D-topological space (D-space) is a triple (X,G,τ), where (X,G) is a Set × D-object, and τ (D-topology on (X,G)) is a subalgebra of (ΩG)X. A D- continuous map (X1,G1,τ1) (f,g) −−−→ (X2,G2,τ2) is then a Set × D-morphism (X1,G1) (f,g) −−−→ (X2,G2) with ((f,(Ωg) op )←)→(τ2) ⊆ τ1. D-Top is the cate- gory of D-spaces and D-continuous maps, concrete over the category Set× D. For the sake of brevity, the category SG-Top is denoted G-Top, employ- ing the shortened notations of the category A-Top. Under the assumption used at the beginning of Proposition 2.5, there exists the embedding func- tor LoB-Top � � ETop // LoATTB-Top, ETop((X1,B1,τ1) (f,ϕ) −−−→ (X2,B2,τ2)) = (X1,EB(B1),τ1) (f,EB(ϕ)) −−−−−−→ (X2,EB(B2),τ2), which in general is not full (us- ing the machinery of the proof of Proposition 2.5, ETop(∅,B1,B ∅ 1 ) (!,(ϕ,ψ)) −−−−−→ ETop(∅,B2,B ∅ 2 ) is continuous, but never belongs to the image of ETop). The new functor makes the diagram LoB-Top |−| �� � � ETop // LoATTB-Top |−| �� Set × LoB � � 1Set×E op B // Set × LoATTB commute, showing that (in some cases) the category LoATTB-Top provides a proper extension of the category LoB-Top. Moreover, it appears that the former category induces another functor, which has more importance in the cur- rent developments. The new definition requires an additional (and very signifi- cant) notion related to catalg topology. This time, it is the concept of topological system introduced by S. Vickers [75] as a common framework for incorporating both topological spaces and their underlying algebraic structures – locales [35], thereby trying to merge point-set and pointless topology. Recently, the notion was successfully extended to include the case of lattice-valued topologies, the most significant results in the field achieved by J. T. Denniston, A. Melton, S. E. Rodabaugh [11, 12, 13, 14], C. Guido [25, 26] and S. Solovyov [66, 74]. Dual attachment pairs in categorically-algebraic topology 111 Definition 2.9. Let A be a variety and let C, D be subcategories of LoA. A (C,D)-topological system ((C,D)-system) is a tuple D = (ptD,ΣD,ΩD, |=), where (ptD,ΣD,ΩD) is a Set × C × D-object and ptD × ΩD |= −→ ΣD is a map (ΣD-satisfaction relation on (pt D,ΩD)) such that ΩD |=(x,−) −−−−−→ ΣD is an A-homomorphism for every x ∈ ptD. A (C,D)-continuous map D1 f −→ D2 is a Set×C×D-morphism (ptD1,ΣD1,ΩD1) (pt f,(Σf)op,(Ωf)op) −−−−−−−−−−−−−→(ptD2,ΣD2,ΩD2) such that for every x ∈ ptD1 and every b ∈ ΩD2, it follows that |=1(x,Ωf(b)) = Σf(|=2(pt f(x),b)). (C,D)-TopSys is the category of (C,D)-systems and (C,D)-continuous maps, concrete over the product category Set × C × D. For the sake of shortness, the category (LoA,LoA)-TopSys is denoted LoA-TopSys, whereas the category (SA,LoA)-TopSys is denoted A-TopSys. To provide the intuition for the concept, we list two important examples. Example 2.10. Loc-TopSys is precisely the category of lattice-valued topo- logical systems introduced by J. T. Denniston, A. Melton and S. E. Rodabaugh in [12]. Its subcategory 2-TopSys ( 2 is the two-element frame {⊥,⊤}) is iso- morphic to the category TopSys of S. Vickers [75]. Example 2.11. Given a set K, the subcategory K-TopSys of LoSet-TopSys is isomorphic to the category Chu(Set,K) (or just ChuK) comprising Chu spaces over a given set K [5, 48]. In particular, Chu2 is the category Cont of contexts of formal concept analysis [19, 76], and also the category IntSys of interchange systems introduced recently by J. T. Denniston, A. Melton and S. E. Rodabaugh [13] in connection with certain aspects of program semantics (the so-called predicate transformers) initiated by E. W. Dijkstra [16]. Sharing the same definition, the categories Chu2, Cont and IntSys have quite different motivating theories. The framework of Definition 2.9 is closely related to the category LoA-Top, allowing the extension of the system spatialization procedure, introduced by S. Vickers [75] to extract their inherent topology. Theorem 2.12. (1) There exists a full embedding LoA-Top � � E // LoA-TopSys defined by E((X1,A1,τ1) (f,ϕ) −−−→ (X2,A2,τ2)) = (X1,A1,τ1, |=1) (f,ϕ,((f,ϕ)←)op) −−−−−−−−−−−→ (X2,A2,τ2, |=2), where |=i(x,α) = α(x). (2) There exists a functor LoA-TopSys Spat −−−→ LoA-Top, which is de- fined by the formula Spat(D1 f −→ D2) = (pt D1,ΣD1,τ1) (pt f,(Σf)op) −−−−−−−−→ (pt D2,ΣD2,τ2), where τj = {|=j(−,b) |b ∈ ΩDj}. (3) Spat is a right-adjoint-left-inverse to E. (4) The category LoA-Top is isomorphic to a full (regular mono)-coreflec- tive subcategory of the category LoA-TopSys. 112 A. Frascella, C. Guido and S. A. Solovyov The attentive reader has probably already guessed that the name “Spat” in the second item of Theorem 2.12 comes from “spatialization”. The new category of Definition 2.8 in hand, we can proceed to the definition of a new functor. Proposition 2.13. There is a functor LoATTB-Top EATT −−−−→ LoB-TopSys, which is given through the formula EATT((X1,G1,τ1) (f,g) −−−→ (X2,G2,τ2)) = (X1×| ΩG1|,ΣG1,τ1, |=1) (f×(Ω g)∗op,(Σg)op,((f,(Ωg)op)←)op) −−−−−−−−−−−−−−−−−−−−−−−−→(X2×| ΩG2|,ΣG2, τ2, |=2), where |=i((x,b),α) = (�i(α(x)))(b). Proof. To show that the functor in question is correct on objects, notice that given λ ∈ ΛB and αi ∈ τ for i ∈ nλ, it follows that |=((x,b),ωτλ(〈αi〉nλ)) = (�((ω τ λ(〈αi〉nλ))(x)))(b) = (�(ω ΩG λ (〈αi(x)〉nλ)))(b) = (ω (Σ G)|Ω G| λ (〈�(αi(x))〉nλ))(b) = ω ΣG λ (〈(�(αi(x)))(b)〉nλ) = ωΣGλ (〈|=((x,b),αi)〉nλ). To check the preservation of continuity, use the fact that for (x,b) ∈ X1×| ΩG1| and α ∈ τ2, |=1((x,b),(f,(Ω g) op )←(α)) = (�1(((f,(Ω g) op )←(α))(x)))(b) = (�1(Ωg ◦ α ◦ f(x)))(b) = (Σg ◦ �2(α ◦ f(x)))((Ω g) ∗op(b)) = Σg ◦ |=2((f(x),(Ω g) ∗op(b)),α) = Σg ◦ |=2(f × (Ωg) ∗op(x,b),α). � It should be noticed at once that despite the notation, the functor of Propo- sition 2.13 never needs to be an embedding. In fact, the merits of the functor in question are highly dependant on the properties of the employed functor B (−)∗ −−−→ Setop. On the other hand, it is possible to restrict the domain of EATT and obtain an embedding. Below we suggest two possible approaches, the first of which being rather straightforward. Proposition 2.14. Given a dual B-attachment G such that ΩG is non-empty, the restriction G-Top EGATT=EATT | ΣG-TopSys G-Top −−−−−−−−−−−−−−−−→ ΣG-TopSys is an embedding. Proof. Given a G-continuous map (X1,τ1) f −→ (X2,τ2), E G ATT((X1,τ1) f −→ (X2,τ2)) = (X1 × | ΩG|,τ1, |=1) (f×1|Ω G|,(f ← Ω G) op) −−−−−−−−−−−−−→ (X2 × | ΩG|,τ2, |=2) (recall our shortened notation for fixed-basis topological spaces) that implies the de- sired property, the condition on ΩG excluding the case of the constant functor mapping everything to the empty system. � The second approach is more sophisticated. The restriction in question is provided by the concept of stratified topological space (the idea of stratification is due to R. Lowen [44], the term itself coined by P.-M. Pu and Y.-M. Liu [50]). Dual attachment pairs in categorically-algebraic topology 113 Definition 2.15. LoATTB-Top ∅k is the full subcategory of LoATTB-Top of non-empty stratified spaces, i.e., spaces (X,G,τ) such that both X and ΩG are non-empty, and for every a ∈ ΩG, the constant map aX is in τ. The notation “(−)k” for stratified spaces comes from [51, 52] and is already widely accepted among the researchers, motivating us to follow their steps. Proposition 2.16. The restriction LoATTB-Top ∅k E ∅k ATT =EATT|LoATTB-Top ∅k −−−−−−−−−−−−−−−−−−→ LoB-TopSys provides an embedding. Proof. Let (X1,G1,τ1) (f1,g1)// (f2,g2) // (X2,G2,τ2) be a pair of LoATTB-Top∅k-mor- phisms. To show that E∅k ATT embeds objects, notice that E∅k ATT (X1,G1,τ1) = E∅k ATT (X2,G2,τ2) implies X1 × | ΩG1| = X2 × | ΩG2|, ΣG1 = ΣG = ΣG2, τ1 = τ = τ2 and (X1 × | ΩG1|) × τ |= 1 −−→ ΣG = (X2 × | ΩG2|) × τ |= 2 −−→ ΣG. The assumption on non-emptiness (which can not be avoided) pro- vides X1 = X = X2 and | ΩG1| = Y = | ΩG2|. To show that ΩG1 = ΩG2, take some x0 ∈ X and then, given λ ∈ ΛB and bi ∈ Y for i ∈ nλ, ω ΩG1 λ (〈bi〉nλ) = (ω τ1 λ (〈bi〉nλ))(x0) = (ω τ2 λ (〈bi〉nλ))(x0) = ω Ω G2 λ (〈bi〉nλ), im- plying ΩG1 = ΩG = ΩG2. To show that �1 = �2, employ the existing x0 to get that for every b1,b2 ∈ ΩG, (�1(b1))(b2) = (�1(b1(x0)))(b2) = |=1((x0,b2),b1) = |=2((x0,b2),b1) = (�2(b1))(b2). To show faithfulness of E∅k ATT , use the fact that E∅k ATT (f1,g1) = E ∅k ATT (f2,g2) implies f1×(Ωg1) ∗op = f2 ×(Ωg2) ∗op, Σg1 = Σg = Σg2 and (f1,(Ωg1) op )← = (f2,(Ωg2) op )←. The non-emptiness requirement provides f1 = f = f2 and also (Ωg1) ∗op = (Ωg2) ∗op. To verify that Ωg1 = Ωg2, use the fact that given b ∈ ΩG2, (Ωg1)(b) = (Ωg1 ◦ b ◦ f)(x0) = ((f,(Ωg1) op )←(b))(x0) = ((f,(Ωg2) op )←(b))(x0) = (Ω g2)(b). � At the end of this section, we finally define the main object of our inter- est, namely, a particular functor. It provides an analogue of the functor H, introduced in [73] as a generalization of the functor L-Top (−)⋆ −−−→ Top of [26] (already mentioned in Introduction), with the aim to produce a convenient framework for studying categorical properties of the hypergraph functors. It is one of the main goals of this paper to explore the nature of the new functor and its relationships to its predecessors. Definition 2.17. There exists a functor LoATTB-Top HATT −−−−→ LoB-Top = LoATTB-Top EATT −−−−→ LoB-TopSys Spat −−−→ LoB-Top, HATT((X1,G1,τ1) (f,g) −−−→ (X2,G2,τ2))=(X1×| ΩG1|,ΣG1, τ̃1) (f×(Ω g)∗op,(Σg)op) −−−−−−−−−−−−−→(X2×| ΩG2|,ΣG2, τ̃2), where τ̃i = {α̃ = (�i(α(−)))(−) |α ∈ τi}. Given a LoATTB-object G, the respective fixed-basis functor G-Top HGATT −−−−→ ΣG-Top is defined by the formula HGATT((X1,τ1) f −→ (X2,τ2)) = (X1 × | ΩG|, τ̃1) f×1|Ω G| −−−−−−→ (X2 × | ΩG|, τ̃2). 114 A. Frascella, C. Guido and S. A. Solovyov Unlike [73], we are not going to touch the topic of hypergraph functors in this paper, restricting our attention to the functor HATT itself. We begin with the remark that certain properties of the dual attachment G can help to provide an embedding property for the resulting functor HGATT. Definition 2.18. A dual attachment G is called (1) Ω-spatial provided that for every b1,b2 ∈ ΩG such that b1 6= b2, �(b1) 6= �(b2) (� is injective). (2) Σ-spatial provided that for every b1,b2 ∈ ΩG such that b1 6= b2, there exists some b ∈ ΩG such that (�(b))(b1) 6= (�(b))(b2). After brief consideration, the reader will easily see that Ω-spatiality and Σ- spatiality are quite different notions. A nice example on the topic is provided by the category ATTSet. The map I � −→ II, taking every a ∈ I to the identity 1I, gives a dual attachment G = (I,I,�), which is Σ-spatial but not Ω-spatial. On the other hand, changing the definition to take every a ∈ I to the constant map aI, provides an attachment which is Ω-spatial but not Σ-spatial. Examples for more complicated varieties can be found in [73]. Proposition 2.19. Given an Ω-spatial attachment G with the property that ΩG is non-empty, the functor G-Top HGATT −−−−→ ΣG-Top is an embedding. Proof. It will be enough to show the injectivity on objects. Given two spaces (X1,τ1) and (X2,τ2) such that H G ATT(X1,τ1) = H G ATT(X2,τ2), the non-empti- ness of ΩG implies X1 = X = X2. To show that τ1 = τ2, notice that given α1 ∈ τ1, α̃1 ∈ τ̃1 = τ̃2 and, therefore, α̃1 = α̃2 for some α2 ∈ τ2. Given x ∈ X, (�1(α1(x)))(b) = α̃1(x,b) = α̃2(x,b) = (�2(α2(x)))(b) for every b ∈ ΩG and that implies α1(x) = α2(x) by Ω-spatiality of G. As a result, α1 = α2, providing τ1 ⊆ τ2. The converse inclusion is similar. � 3. Dual attachment pairs This section clarifies the word “dual” in the term “dual attachment” used in this paper. The motivation for the choice comes from a particular property of attachment found out in [26], namely, the existence of a functor L-Top (−)⋆ −−−→ Top (already mentioned in Introduction), which takes an L-topological space (X,τ) to the crisp space (SX,τ ⋆), where SX is the set of L-points of X, and τ⋆ consists of the sets α⋆ for every α ∈ τ, comprising precisely those L-points, which are attached to the particular α in question. A catalg analogue of the above-mentioned functor has been already considered in [73], whose counter- part for the current setting is given in Definition 2.17. It is the main purpose of this section to show that both functors coincide in case the respective at- tachments form a dual attachment pair. We begin with the definition of a generalized version of the above-mentioned functors, which require some addi- tional preliminaries contained in the following definition and proposition. Dual attachment pairs in categorically-algebraic topology 115 Definition 3.1. Let A be a variety and let (‖−‖,B) be a reduct of A. A dual B-attachment G is called A-derived provided that there exist A-algebras AΩ, AΣ such that ΩG = ‖AΩ‖, ΣG = ‖AΣ‖. Proposition 3.2. Let A be a variety, let (‖ − ‖,B) be a reduct of A and let A be an A-algebra. There exist two functors: (1) ‖A‖-Top Xt‖A‖A −−−−−→ A-Top defined by Xt‖A‖A((X1,τ1) f −→ (X2,τ2)) = (X1,〈τ1〉) f −→(X2,〈τ2〉) with 〈τi〉 the A-subalgebra of A generated by τi; (2) A-Top RdA‖A‖ −−−−−→ ‖A‖-Top defined by RdA‖A‖((X1,τ1) f −→ (X2,τ2)) = (X1,‖τ1‖) f −→ (X2,‖τ2‖). If G is an A-derived dual B-attachment, then there is a functor AΩ-Top H G ATT −−−−→ AΣ-Top defined by commutativity of the following diagram: AΩ-Top RdAΩ Ω G �� H G ATT // AΣ-Top G-Top HGATT // ΣG-Top. XtΣ GAΣ OO Proof. It is enough to verify that the functor Xt‖A‖A preserves continuity. We use the simple fact that given a homomorphism A1 ϕ −→ A2 and a subset S ⊆ A1, ϕ→(〈S〉) = 〈ϕ→(S)〉 [70]. As a result, (f←A ) →(〈τ2〉) = 〈(f ← A ) →(τ2)〉 ⊆ 〈τ1〉. � The reader should notice that “Xt” (resp. “Rd”) is the abbreviation for “extension” (resp. “reduction”), used to underline the action of the functor in question, i.e., to extend (resp. reduce) the algebraic structure. Both functors will play an important role in the subsequent developments. To compare the new functor with the already existing setting of [73], one should recall some results from its approach to the concept of attachment. Proposition 3.3. There exists a functor LoAttA-Top EAtt −−−→ LoA-TopSys, which is given by the formula EAtt((X1,F1,τ1) (f,g) −−−→ (X2,F2,τ2)) = (X1 × | ΩF1|,ΣF1,τ1, |=1) (f×(Ωg)∗op,(Σ g)op,((f, (Ωg)op)←)op) −−−−−−−−−−−−−−−−−−−−−−−−→(X2×| ΩF2|,ΣF2,τ2, |=2), |= i ((x,a),α) = ( i(a))(α(x)). The reader is advised to pay attention to the important fact that the only difference in the definition of the functors of Propositions 2.13, 3.3 concerns the respective satisfaction relation. Definition 3.4. There exists a functor LoAttA-Top HAtt −−−→ LoA-Top = LoAttA-Top EAtt −−−→ LoA-TopSys Spat −−−→ LoA-Top, HAtt((X1,F1,τ1) (f,g) −−−→ (X2,F2,τ2)) = (X1×| ΩF1|,ΣF1, τ̂1) (f×(Ωg)∗op,(Σ g)op) −−−−−−−−−−−−−→ (X2×| ΩF2|,ΣF2, τ̂2), 116 A. Frascella, C. Guido and S. A. Solovyov where τ̂i = {α̂ = ( i(−))(α(−)) |α ∈ τi}. Given a LoAttA-object F , the re- spective fixed-basis functor F-Top HFAtt −−−→ ΣF-Top is given by HFAtt((X1,τ1) f −→ (X2,τ2)) = (X1 × | ΩF |, τ̂1) f×1|Ω F| −−−−−−→ (X2 × | ΩF |, τ̂2). By analogy with Proposition 3.2(3), one obtains the functor AΩ-Top H F Att −−−→ AΣ-Top (notice the difference in the notations). The crucial question arises on when the two functors HFAtt and H G ATT coincide, and that is precisely the point for dual attachment pairs to come in play. Definition 3.5. Let A be a variety, let (‖ − ‖,B) and (‖ − ‖,C) be reducts of A, and let F and G be a B-attachment and a dual C-attachment respectively. (1) Both F and G are called reduct attachments. (2) The pair (F,G) is an attachment pair w.r.t. (A,B,C). (3) (F,G) is a related attachment pair provided that both F and G are A-derived, and AFΩ = AΩ = A G Ω, A F Σ = AΣ = A G Σ. (4) (F,G) is a dual attachment pair provided that (F,G) is a related at- tachment pair, and for every a1,a2 ∈ AΩ, ( (a1))(a2) = (�(a2))(a1) w.r.t. the maps |AΩ| // � // Set(|AΩ|, |AΣ|). Every related attachment pair gives two functors AΩ-Top H F Att // H G ATT // AΣ-Top. With a dual attachment pair in hand, these two functors coincide. Proposition 3.6. Given a dual attachment pair (F,G), HFAtt = H G ATT. Proof. Since the case of morphisms is clear, it will be enough to show equality of the functors on objects. Given an AΩ-space (X,τ), it is sufficient to verify that 〈τ̂〉 = 〈τ̃〉. Given α ∈ τ, α̂(x,a) = (( (−))(α(−)))(x,a) = ( (a))(α(x)) = (�(α(x)))(a) = ((�(α(−)))(−))(x,a) = α̃(x,a) for every (x,a) ∈ X × |AΩ|. Thus τ̂ = τ̃, implying 〈τ̂〉 = 〈τ̃〉. � A good illustration of Proposition 3.6 is provided by Observation 2 from Introduction, which gives a dual attachment pair (F,G) w.r.t. the varieties (CLat,QFrm,QFrm), where F = (L,2, ) is based on a complete chain L and the map ( (a1))(a2) = { ⊤, a1 < a2, ⊥, otherwise. To get more intuition for the attachment pair (F,G), one can represent every map (a) as a particular subset of L (the preimage of {⊤} under the map in question), i.e., (a) = {b ∈ L |a < b} =↾ a (notice that (⊤) = ∅). The dual attachment � is then the collection of sets �(a) = {b ∈ L |b < a} =⇂ a (�(⊥) = ∅ now) for every a ∈ L. It is easy to see that in this particular case, the attachment duality reduces to the usual, well-known in mathematics, order-theoretic duality. The main point of Proposition 3.6 is that the functors Dual attachment pairs in categorically-algebraic topology 117 L-Top H F Att // H G ATT // (2-Top ∼= Top) generated by F and G coincide. On the other hand, straightforward computations backed by [73] show that HFAtt is precisely the functor L-Top (−)⋆ −−−→ Top of [26]. It follows that a two-fold representation of the already well-known notion is obtained. The reader should pay attention to the fact that the case L = 2 does not result in the identity functor on Top, since a topological space (X,τ) is taken to the space (X × |2|,〈 ˆ‖τ‖〉 = 〈 ˜‖τ‖〉), which has a different carrier set. The reader will easily find other examples on the topic. It is important to underline, however, that the case of an unrelated attachment pair can provide completely different (incomparable) functors HFAtt, H G ATT. 4. Existence of dual attachment pairs The previous section showed that the case of a dual attachment pair has the crucial property of equality of the derived functors. On the other hand, the functors of a just related attachment pair need not coincide which gives dual attachment pairs even more importance. A natural question on the existence of dual attachment pairs arises. This short section clarifies the situation. Proposition 4.1. Let F be a B-attachment which is a reduct attachment w.r.t. A. There exists a dual C-attachment G such that (F,G) is a dual attachment pair iff the following conditions are fulfilled: (1) F is A-derived (ΩF = ‖AΩ‖ and ΣF = ‖AΣ‖); (2) there exists a reduct (‖−‖,C) of A such that ‖AΩ‖ −→ C(‖AΩ‖,‖AΣ‖). Proof. For the necessity, notice that, firstly, AFΩ = AΩ = A G Ω, A F Σ = AΣ = A G Σ, and, secondly, given λ ∈ ΛC and ai ∈ ‖AΩ‖ for i ∈ nλ, ( (a))(ω ‖AΩ‖ λ (〈ai〉nλ)) = (�(ω ‖AΩ‖ λ (〈ai〉nλ)))(a) = (ω ‖AΣ‖ |AΩ| λ (〈�(ai)〉nλ))(a) = ω ‖AΣ‖ λ (〈(�(ai))(a)〉nλ) = ω ‖AΣ‖ λ (〈( (a))(ai)〉nλ) for every a ∈ ‖AΩ‖. The sufficiency is slightly more sophisticated. Define the required dual attachment G by ΩG = ‖AΩ‖, ΣG = ‖AΣ‖ (the respec- tive reducts are taken in the variety C) together with ΩG � −→ (ΣG)|Ω G| given by (�(a1))(a2) = ( (a2))(a1). The only challenge now is to show that � is a C-homomorphism. Given λ ∈ ΛC and ai ∈ ΩG for i ∈ nλ, (�(ω ‖AΩ‖ λ (〈ai〉nλ)))(a) = ( (a))(ω ‖AΩ‖ λ (〈ai〉nλ)) = ω ‖AΣ‖ λ (〈( (a))(ai)〉nλ) = ω ‖AΣ‖ λ (〈(�(ai))(a)〉nλ) = (ω ‖AΣ‖ |AΩ| λ (〈ai〉nλ))(a) for every a ∈ ΩG. � An example for the proposition is provided by the dual attachment pair (F,G) with F = (L,2, ), mentioned at the end of the previous section, the second of the requirements (the first being obvious) verified in Introduction as follows: ( (a))( ∨ S) = ⊤ iff a < ∨ S iff a < s for some s ∈ S iff ( (a))(s) = ⊤ for some s ∈ S iff ( ∨ s∈S (a))(s) = ⊤, whereas ( (a))(s ∧ t) = ⊤ iff a < s ∧ t 118 A. Frascella, C. Guido and S. A. Solovyov iff a < s and a < t iff ( (a))(s) ∧ ( (a))(t) = ⊤. The converse way (from dual attachment to attachment) is equally easy and can be run through as follows. Proposition 4.2. Let G be a dual C-attachment which is a reduct attachment w.r.t. A. There exists a B-attachment F such that (F,G) is a dual attachment pair iff the following conditions are fulfilled: (1) G is A-derived (ΩG = ‖AΩ‖ and ΣG = ‖AΣ‖); (2) there exists a reduct (‖ − ‖,B) of A such that ‖AΩ‖ � −→ ‖AΣ‖ |AΩ| is a B-homomorphism. Proof. For the necessity notice that, firstly, AGΩ = AΩ = A F Ω, A G Σ = AΣ = A F Σ, and, secondly, given λ ∈ ΛB and ai ∈ ‖AΩ‖ for i ∈ nλ, ((�(ω ‖AΩ‖ λ (〈ai〉nλ)))(a) = ( (a))(ω ‖AΩ‖ λ (〈ai〉nλ)) = ω ‖AΣ‖ λ (〈( (a))(ai)〉nλ) = ω ‖AΣ‖ λ (〈(�(ai))(a)〉nλ) = (ω ‖AΣ‖ |AΩ| λ (〈�(ai)〉nλ))(a) for every a ∈ ‖AΩ‖. To show the sufficiency, define an attachment F by ΩF = ‖AΩ‖, ΣF = ‖AΣ‖ together with ΩF −→ B(Ω F,ΣF) given by ( (a1))(a2) = (�(a2))(a1). It should be verified that (a) is a B-homomorphism for every a ∈ ΩF . Given λ ∈ ΛB and ai ∈ ΩF for i ∈ nλ, ( (a))(ω ‖AΩ‖ λ (〈ai〉nλ)) = (�(ω ‖AΩ‖ λ (〈ai〉nλ)))(a) = (ω ‖AΣ‖ |AΩ| λ (〈�(ai)〉nλ))(a) = ω ‖AΣ‖ λ (〈(�(ai))(a)〉nλ) = ω ‖AΣ‖ λ (〈( (a))(ai)〉nλ). � We close this section with the remark (already mentioned in Introduction) that the concept of duality for attachment used in this paper is developed in the framework of arbitrary algebras (possibly) void of any kind of order relation and, therefore, our current setting is not the duality induced by a partial order. On the other hand, the approach is neither a duality of category theory, since the underlying category of the respective attachment is not dualized. Based on the employed framework of catalg topology, the type of duality presented in the paper could be called categorically-algebraic. It will be the topic of our forthcoming papers to find the proper place of the new notion in mathematics. The current manuscript will continue exploring another aspects of attachment. 5. Natural transformations induced by attachment It was shown in [26] that every attachment A in a complete lattice L provides a frame homomorphism LX (−)⋆ −−−→ P(SX) for every set X (simply taking α ∈ L X to the set α⋆ of all L-points attached to α), which gives rise to the already mentioned functor L-Top (−)⋆ −−−→ Top. This section extends the map to our catalg setting, resulting in consequences important for the whole development. Proposition 5.1. Let F be a B-attachment and let G be a dual C-attachment. For every set X, there exist a B-homomorphism (ΩF)X (−) −−−→ (ΣF)X×|ΩF| Dual attachment pairs in categorically-algebraic topology 119 defined by α (x,b) = ( (b))(α(x)), and a C-homomorphism (ΩG)X (−)� −−−→ (ΣG)X×|Ω G| defined by α�(x,b) = (�(α(x)))(b). If (F,G) is a related attach- ment pair, then the maps have the same (co)domain |AΩ| X (−)� // (−) // |AΣ| X×|AΩ| . If (F,G) is a dual attachment pair, then the maps coincide. Proof. To prove the first claim, we show that the map (−)� provides a C- homomorphism. Given λ ∈ ΛC and 〈αi〉nλ ∈ (ΩG) X, (ω (Ω G)X λ (〈αi〉nλ)) �(x,b) = (�((ω (Ω G)X λ (〈αi〉nλ))(x)))(b) = (�(ωΩGλ (〈αi(x)〉nλ)))(b) = (ω (Σ G)|Ω G| λ (〈�(αi(x))〉nλ))(b) = ωΣGλ (〈(�(αi(x)))(b)〉nλ) = ω ΣG λ (〈α � i (x,b)〉nλ) = (ω (ΣG)X×|Ω G| λ (〈α�i 〉nλ))(x,b) for every (x,b) ∈ X × | ΩG|. The second claim is obvious. For the last state- ment, notice that given α ∈ |AΩ| X, α�(x,a) = (�(α(x)))(a) = ( (a))(α(x)) = α (x,a) for every (x,a) ∈ X × |AΩ|. � After a closer scrutiny, it appears that the homomorphisms of Proposition 5.1 are actually components of natural transformations (the fact, never mentioned in [26]). To prove the claim, start with the preliminary remark that given a dual C-attachment G, there exists a functor Set (−×|ΩG|)←Σ G −−−−−−−−−→ LoC defined by commutativity of the following triangle: Set (−×|Ω G|)←Σ G '' −×|Ω G| // Set (−)←Σ G �� LoC, where − × | ΩG| is the standard product functor [1] defined by the formula (− × | ΩG|)(X1 f −→ X2) = X1 × | ΩG| f×1|Ω G| −−−−−−→ X2 × | ΩG|. Proposition 5.2. Every dual C-attachment G provides a natural transforma- tion (− × | ΩG|)←Σ G ((−)�) op −−−−−−→ (−)←Ω G. Proof. Given a map X1 f −→ X2, one has to verify commutativity of the diagram (ΩG)X2 f ← Ω G �� (−)�X2 // (ΣG)X2×|Ω G| (f×1|Ω G|) ← Σ G �� (ΩG)X1 (−)�X1 // (ΣG)X1×|Ω G|, and that follows from the fact that (((−)�X1 ◦f ← ΩG)(α))(x,b) = (α◦f) �(x,b) = (�(α ◦f(x)))(b) = α�(f(x),b) = (α� ◦ (f × 1|Ω G|))(x,b) = (((f × 1|Ω G|) ← Σ G ◦ (−)�X2)(α))(x,b) for every α ∈ (ΩG) X2 and every (x,b) ∈ X1 × | ΩG|. � 120 A. Frascella, C. Guido and S. A. Solovyov Similarly, one gets a natural transformation (−×| ΩF |)←Σ F ((−) ) op −−−−−−→ (−)←Ω F for a given B-attachment F , which reduces to the setting of [26] for B being the variety Frm of frames. In case of a dual attachment pair, (as one might expect) both natural transformations coincide. To generalize the passage of [26] from a natural transformation to a functor, we need some additional notions from the realm of catalg topology, presented briefly in the subsequent section. 6. Categorically-algebraic topology and attachment In this section we recall from [71] basic concepts of categorically-algebraic (catalg) topology (see also [64, 66, 72]), which bring to light the crucial property of attachment, i.e., generation of a functor between two topological settings. The approach is motivated by the currently dominating in the fuzzy community point-set lattice-theoretic (poslat) topology introduced by S. E. Rodabaugh [55] and developed by P. Eklund, C. Guido, U. Höhle, T. Kubiak, A. Šostak and the initiator himself [17, 24, 33, 39, 40, 57]. The main advantage of the new setting is the fact that apart from incorporating as special subcases the most important approaches to (lattice-valued) topology and providing convenient means of interaction between them, the catalg framework ultimately erases the border between crisp and many-valued developments, producing a theory which underlines the algebraic essence of the whole (not only lattice-valued) mathematics, thereby propagating algebra as the main driving force of modern exact sciences. It should be noticed immediately that some parts of the theory have already been used throughout the paper (Definition 2.7). The current section provides a more rigid foundation for the approach and backs it by several motivating examples, to give the flavor of fruitfulness of the new theory. The setting is based in a mixture of powerset theories of [61, Definition 3.5] (see also [60, 62]) and topological theories of [1, Exercise 22B]. Definition 6.1. A variety-based backward powerset theory (vbp-theory) in a category X (the ground category of the theory) is a functor X P −→ LoA to the dual of a variety A. To get the intuition for the concept, the reader is advised to recall the functor of Proposition 2.6, providing the main example for the notion and incorporating many approaches to powerset operators popular in lattice-valued mathematics. Example 6.2. (1) Set×S2 P=(−)← −−−−−−→ LoCBool, where CBool is the variety of complete Boolean algebras (complete, complemented, distributive lattices) and 2 = {⊥,⊤}, provides the standard preimage operator, mentioned before Proposition 2.6. (2) Set × SI Z=(−)← I −−−−−−→ DmLoc (cf. Definition 2.2), where I = [0,1] is the unit interval, gives the fixed-basis fuzzy approach of L. A. Zadeh [77]. Dual attachment pairs in categorically-algebraic topology 121 (3) Set × SL G1=(−) ← L −−−−−−→ Loc provides the fixed-basis L-fuzzy approach of J. A. Goguen [21]. The setting was changed to Set × SL G2=(−) ← L −−−−−−→ LoUQuant in [22]. The machinery can be generalized to an arbitrary variety A and the theory Set × SA SAA=(−) ← A −−−−−−−→ LoA, which unites the previous items in one common fixed-basis framework. (4) Set × C RC1 =(−) ← −−−−−−−→ DmLoc, where C is a subcategory of DmLoc, gives the variable-basis poslat approach of S. E. Rodabaugh [54]. The setting has been extended to Set × C RC2 =(−) ← −−−−−−−→ LoUSQuant in [61] and then reduced to Set × Loc R3=(−) ← −−−−−−→ Loc in [11, 14]. (5) Set×FuzLat E=(−)← −−−−−→ FuzLat provides the variable-basis approach of P. Eklund [17], motivated by those of S. E. Rodabaugh [54] and B. Hut- ton [34]. Notice that FuzLat is the dual of the variety HUT of completely distributive DeMorgan frames called Hutton algebras [57]. The machinery can be generalized to an arbitrary variety A and the theory Set×C SCA=(−) ← −−−−−−−→ LoA, which unites the previous items in one common variable-basis framework. On the next step, we provide another level of abstraction, which has never been used in the above-mentioned theories of S. E. Rodabaugh. Definition 6.3. Let X be a category and let T = (P,(‖ − ‖,B)) contain a vbp-theory X P −→ LoA in the category X and a reduct (‖ − ‖,B) of A. The variety-based topological theory (vt-theory) in X induced by T is the functor X T=‖−‖op◦P −−−−−−−−→ LoB. Since a vt-theory T is completely determined by the respective pair T , we use occasionally the notation (P,B) instead of T . It is important to underline that the aim of an additional level of abstraction is to remove the unused topological structure provided by powerset theories, the move, motivated by the observation that the standard backward powerset theory is based in Boolean algebras, whereas the respective topological theory is reduced to frames (the case of closure spaces mentioned below provides another good example). On the other hand, the case of coincidence between powerset and topological theories is not excluded in our framework. The reader will see that the subsequent developments will often provide a topological theory only, without any explicit reference to its generating powerset theory. Definition 6.4. Let T be a vt-theory in a category X. Top(T) is the cat- egory, concrete over X, whose objects (T-topological spaces) are pairs (X,τ), comprising an X-object X and a subalgebra τ of T(X) (T-topology on X), and whose morphisms (X,τ) f −→ (Y,σ) are those X-morphisms X f −→ Y , which satisfy ((T(f)) op )→(σ) ⊆ τ (T-continuity). 122 A. Frascella, C. Guido and S. A. Solovyov The significance of the category Top(T) is the fact that it unites many of the existing topological frameworks in mathematics. To give the reader the flavor of their abundance and the fruitfulness of the new unifying framework, we provide a short list of examples illustrating the notion of catalg topology. Example 6.5. (1) Top((P,Frm)) is isomorphic to the category Top of topological spaces and continuous maps. (2) Top((P,CSL)) is isomorphic to the category Cls of closure spaces and continuous maps, studied by D. Aerts et al. [2, 3]. (3) Top((Z,Frm)) is isomorphic to the category I-Top of fixed-basis fuzzy topological spaces, introduced by C. L. Chang [8]. (4) Top((G2,UQuant)) is isomorphic to the category L-Top of fixed-basis L-fuzzy topological spaces of J. A. Goguen [22]. (5) Top((RCi ,USQuant)) is isomorphic to the category C-Topi, i ∈ {1,2} for variable-basis poslat topology of S. E. Rodabaugh [54, 61]. (6) Top((E,Frm)) is isomorphic to the category FUZZ for variable-basis poslat topology of P. Eklund [17], motivated by those of S. E. Rod- abaugh [54] and B. Hutton [34]. (7) Top((SAA,A)) (resp. Top((S LoA A ,A))) is isomorphic to the fixed- (resp. variable-) basis category A-Top (resp. LoA-Top) used in the former approach to catalg topologies of [70] (resp. [69]) as well as in the pre- vious sections of this paper (Definition 2.7). The reader should notice the fact that the second item of Example 6.5 is never included in the setting of topological theories of S. E. Rodabaugh [61], which are based explicitly on s-quantales (and, therefore, on ∨ -semilattices), whereas closure spaces rely on c-semilattices (and, therefore, on ∧ -semilattices). It appears (as an experienced reader might guess) that in order to deal successfully with the categories of the form Top(T), it is enough to consider their generating topological theories T . Proposition 6.6. Let X T1 −→ LoA, Y T2 −→ LoA be vt-theories, let X F −→ Y be a functor, and let T2 ◦ F η −→ T1 be a natural transformation. There exists a functor Top(T1) Hη −−→ Top(T2) given by Hη((X1,τ1) f −→ (X2,τ2)) = (F(X1),(η op X1 )→(τ1)) F(f) −−−→ (F(X2),(η op X2 )→(τ2)). Proof. It will be enough to show that the functor Hη preserves continuity and that follows immediately from commutativity of the diagram T1(X2) (T1(f)) op �� η op X2 // T2 ◦ F(X2) (T2◦F(f)) op �� T1(X1) η op X1 // T2 ◦ F(X1), Dual attachment pairs in categorically-algebraic topology 123 since ((T2 ◦ F(f)) op )→((η op X2 )→(τ2)) = ((T2 ◦ F(f)) op ◦ η op X2 )→(τ2) = (η op X1 ◦ (T1(f)) op )→(τ2) ⊆ (η op X1 )→(τ1). � As an example of the obtained result, one can look at Proposition 5.2 and the remark just afterward, providing two functors Top((−)←Ω G) H�op −−−→ Top((−)←Σ G) and Top((−) ← Ω F ) H op −−−→ Top((−)←Σ F ), which essentially are the fixed-basis functors G-Top HGATT −−−−→ ΣG-Top and F-Top HFAtt −−−→ ΣF-Top of Definitions 2.17 and 3.4 respectively. Moreover, it appears that a more gen- eral framework is available. Start by defining the required topological theories, together with an additional functor and a natural transformation. Proposition 6.7. There exist topological theories (1) Set×LoATTC T ATT Ω −−−−→ LoC, where T ATTΩ ((X1,G1) (f,g) −−−→ (X2,G2)) = (ΩG1) X1 ((f,(Ω g)op)←)op −−−−−−−−−−−→ (ΩG2) X2 ; (2) Set × LoC T ATT Σ −−−−→ LoC = Set × LoC (−)← −−−→ LoC; together with a functor Set×LoATTC KATT×Ω −−−−→Set×LoC, KATT×Ω ((X1,G1) (f,g) −−−→ (X2,G2)) = (X1 × | ΩG1|,ΣG1) (f×(Ωg)∗op,(Σ g)op) −−−−−−−−−−−−−→ (X2 × | ΩG2|,ΣG2) and a natural transformation T ATTΣ ◦ K ATT ×Ω ((−)�) op −−−−−−→ T ATTΩ given by the maps of Proposition 5.1. Proof. Since the definitions of the functors are straightforward, the only thing to verify is correctness of the definition of the natural transformation. Consider a Set×LoATTC-morphism (X1,G1) (f,g) −−−→ (X2,G2) and check commutativity of the diagram (ΩG2) X2 (f,(Ωg)op)← �� (−)�(X2,G2) // (ΣG2) X2×|Ω G2| (f×(Ω g)∗op,(Σg)op)← �� (ΩG1) X1 (−)�(X1,G1) // (ΣG1) X1×|Σ G1|. Given α ∈ (ΩG2) X2 and (x,b) ∈ X1 × | ΩG1|, ((−)�(X1,G1) ◦ (f,(Ω g) op )←(α))(x,b) = (Ωg ◦ α ◦ f)�(x,b) = (�1(Ωg ◦ α ◦ f(x)))(b) = (Σg ◦ �2(α ◦ f(x))((Ω g) ∗op))(b) = Σg ◦ α�(f(x),(Ω g)∗ op (b)) = (Σg ◦ α� ◦ (f × (Ωg)∗ op ))(x,b) = ((f × (Ωg)∗ op ,(Σg) op )← ◦ (−)�(X2,G2)(α))(x,b). � 124 A. Frascella, C. Guido and S. A. Solovyov Propositions 6.6, 6.7 give a functor Top(T ATTΩ ) H�op −−−→ Top(T ATTΣ ), which is essentially (up to the change of the notation for the underlying variety) the functor LoATTC-Top HATT −−−−→ LoC-Top of Definition 2.17. In a similar way, one obtains the functor Top(T AttΩ ) H op −−−→ Top(T AttΣ ), which essentially provides the functor LoATTB-Top HAtt −−−→ LoB-Top of Definition 3.4. The reader should notice the significant difference between the ways of obtaining the functors HATT, HAtt by Propositions 6.6, 6.7 and in Definitions 2.17, 3.4, the latter relying on the framework of topological systems, whereas the former being based explicitly on catalg topology. It is up to the reader to decide, which way is more applicable in his/her framework. We would just like to notice that the final results of this section clearly show one of the main advantages of the notion of attachment, i.e., the fact that it provides a way of moving (natural transformation) between two topological theories, resulting in a functor between the categories of the respective topological structures. 7. Categorically-algebraic attachment The attentive reader will easily notice that although the definition of ob- jects of the category ATTB of Definition 2.4 provides a straightforward gen- eralization of the attachment notion of [26], the definition of morphism comes essentially “out of the blue”, the only its justification being the existence of the functor LoATTB-Top EATT −−−−→ LoB-TopSys of Proposition 2.13. The simple reason for the occurrence is the fact that the case of attachment morphisms has never been treated in [25, 26] and, therefore, there is actually nothing to compare with. It is the main goal of this section to provide a more trustworthy justification for the definition of morphisms of the category ATTB. We start with some new functors, which will be used in the subsequent pro- cedures. For the sake of convenience, in what follows, we change the notation (kept until now) for the underlying variety of ATTB from B to A. Proposition 7.1. Given a variety A, there exists a functor Setop×A (−)↼ −−−→ A defined by the formula ((X1,A1) (f,ϕ) −−−→ (X2,A2)) ↼ = AX11 (f,ϕ)↼ −−−−→ AX22 with (f,ϕ)↼(α) = ϕ◦α◦fop. The new functor satisfies the equality Setop×A (−)↼ −−−→ A = (Set × LoA (−)← −−−→ LoA)op. Proof. Correctness of the definition of the functor follows from the last claim (backed by Proposition 2.6), which is a consequence of the fact that given α ∈ AX11 , (f,ϕ) ↼(α) = ϕ ◦ α ◦ fop = (fop,ϕop)←(α). � To underline the motivating setting of the functor (−)←, the new one uses a similar notation (−)↼. The other functors are collected in the next definition. Dual attachment pairs in categorically-algebraic topology 125 Definition 7.2. Every variety A equipped with a functor A (−)∗ −−−→ Setop such that A∗ = |A|, gives rise to the following three functors: (1) A × A Π1 −−→ A, Π1((A1,A ′ 1) (ϕ,ψ) −−−→ (A2,A ′ 2)) = A1 ϕ −→ A2 (the first projection functor); (2) A × A K −→ Setop × A = A × A (−)∗×1A −−−−−−→ Setop × A; (3) Setop × A P −→ A = Setop × A (−)↼ −−−→ A. The next definition shows a more general approach to attachment, with the notion of morphism coming from the existing framework of comma categories. Definition 7.3. ATT ∗ A is the full subcategory of the comma category (Π1 ↓ P ◦ K), whose objects are precisely those (Π1 ↓ P ◦ K)-objects Π1(A1,A2) ϕ −→ P ◦ K(A′1,A ′ 2) for which (A1,A2) = (A ′ 1,A ′ 2). A natural question on the equivalence of Definition 7.3 and Definition 2.4 arises. It is the purpose of the next result to answer it positively. Proposition 7.4. The categories ATTA and ATT∗A are isomorphic. Proof. An ATT∗A-object G is a triple (ΩG,ΣG,�), where (ΩG,ΣG) is the object of the product category A × A and Π1(ΩG,ΣG) � −→ P ◦ K(ΩG,ΣG) = ΩG � −→ (ΣG)|Ω G| is an A-homomorphism. An ATT∗A-morphism G1 f −→ G2 is a pair of A-homomorphisms (ΩG1,ΣG2) (Ω f,Σ f) −−−−−−→ (ΩG2,ΣG2) making the following diagram commute: Π1(ΩG1,ΣG1) Π1(f) �� �1 // P ◦ K(ΩG1,ΣG1) P◦K(f) �� Π1(ΩG2,ΣG2) �2 // P ◦ K(ΩG2,ΣG2) that in its turn provides commutativity of the next diagram: ΩG1 Ωf �� �1 // (ΣG1) |Ω G1| ((Ω f)∗,Σ f)↼ �� ΩG2 �2 // (ΣG2) |Ω G2|, which is equivalent to the fact that given b1 ∈ ΩG1 and b2 ∈ ΩG2, it fol- lows that (�2 ◦ Ωf(b1))(b2) = ((((Ω f) ∗,Σf)↼ ◦ �1)(b1))(b2) = Σf ◦ �1(b1) ◦ (Ωf)∗ op (b2) = (Σf ◦ �1(b1))((Ω f) ∗op(b2)). Taking together, the remarks pro- vide an isomorphism ATTA K −→ ATT∗A, K(G1 f −→ G2) = G1 f −→ G2. � Having introduced an equivalent definition of attachment, we are going to generalize the results from the end of the last section to the new framework, 126 A. Frascella, C. Guido and S. A. Solovyov namely, derive a respective natural transformation between topological theo- ries. Now, however, we would like to provide a more explicit description of the machinery employed. To make the things easier, we begin with several additional properties of powerset operators. Proposition 7.5. Given a variety A and a set X, there exists a functor A (−)X→ −−−→ A, (A1 ϕ −→ A2) X → = A X 1 ϕ X → −−→ AX2 with ϕ X →(α) = ϕ ◦ α. Proof. To show that the functor is correct on morphisms (provides an A- homomorphism), notice that given λ ∈ ΛA and αi ∈ A X 1 for i ∈ nλ, it fol- lows that (ϕX→(ω A X 1 λ (〈αi〉nλ)))(x) = ϕ◦ω A1 λ (〈αi(x)〉nλ) = ω A2 λ (〈ϕ ◦ αi(x)〉nλ) = (ω A X 2 λ (〈ϕ ◦ αi〉nλ))(x) = (ω A X 2 λ (〈ϕX→(αi)〉nλ))(x) for every x ∈ X. � Notice that the morphism action of the functor of Proposition 7.5 has already been considered in [69, 70], where it has been observed (following, e.g., [56, 59]) that the variable-basis functor of Proposition 2.6 splits up as follows: A X2 2 f ← A2 �� (ϕop)X2→ // (f,ϕ)← '' A X2 1 f ← A1 �� A X1 2 (ϕop)X1→ // AX11 . In view of Proposition 7.5 and the above-mentioned remark, every commutative diagram in a variety A A1 ψ1 �� ϕ1 // A′1 ψ2 �� A2 ϕ2 // A′2 and every map X1 f −→ X2, provide the following commutative diagram: (7.1) AX21 (f,ψ op 1 ) ← '' (ψ1) X2 → �� (ϕ1) X2 → // A′1 X2 (ψ2) X2 → �� (f,ψ op 2 ) ← xx A X2 2 f ← A2 �� (ϕ2) X2 → // (f,ϕ op 2 ) ← N N N N N ''NN N N N A′2 X2 f ← A′ 2 �� A X1 2 (ϕ2) X1 → // A′2 X1. The last needed property of powerset operators is in the next proposition. Dual attachment pairs in categorically-algebraic topology 127 Proposition 7.6. For a Set×LoA×LoA-morphism (X1,A1,B1) (f,ϕop,ψop) −−−−−−−→ (X2,A2,B2), there exist A-homomorphisms (B |Ai| i ) Xi Θi −−→ B Xi×|Ai| i defined by (Θi(α))(x,a) = (α(x))(a), making the following diagram commute: (B |A2| 2 ) X2 (f,((ϕ∗,ψ)↼)op)← �� Θ2 // B X2×|A2| 2 (f×ϕ∗op,ψop)← �� (B |A1| 1 ) X1 Θ1 // B X1×|A1| 1 . Proof. There are two simple challenges to deal with. To show that Θi is an A-homomorphism, notice that given λ ∈ ΛA and αj ∈ (B |Ai| i ) Xi for j ∈ nλ, it follows that (Θi(ω (B |Ai| i )Xi λ (〈αj〉nλ)))(x,a) = ((ω (B |Ai| i )Xi λ (〈αj〉nλ))(x))(a) = (ω B |Ai| i λ (〈αj(x)〉nλ))(a) = ω Bi λ (〈(αj(x))(a)〉nλ) = ω Bi λ (〈(Θi(αj))(x,a)〉nλ) = (ω B Xi×|Ai| i λ (〈Θi(αj)〉nλ))(x,a) for every (x,a) ∈ Xi×|Ai|. Commutativity of the above-mentioned diagram follows from the fact that given α ∈ (B |A2| 2 ) X2 and (x1,a1) ∈ X1 × |A1|, ((Θ1 ◦ (f,((ϕ ∗,ψ)↼) op )←)(α))(x1,a1) = (Θ1((ϕ ∗,ψ)↼ ◦ α ◦ f))(x1,a1) = ((ϕ ∗,ψ)↼ ◦ α ◦ f(x1))(a1) = ψ ◦ (α ◦ f(x1)) ◦ ϕ ∗op(a1) = ψ ◦ (Θ2(α))(f(x1),ϕ ∗op(a1)) = ψ◦Θ2(α)◦(f ×ϕ ∗op)(x1,a1) = ((f ×ϕ ∗op,ψop)←◦ Θ2)(x1,a1). � Everything in its place, we consider the analogues for our current setting of the functors of Proposition 6.7, which are Set × LoATT∗A T ATT Ω −−−−→ LoA, Set × LoATT∗A KATT×Ω −−−−→ Set × LoA and Set × LoA T ATT Σ −−−−→ LoA. Our aim is to provide an equivalent description of the natural transformation between them given in Proposition 6.7. The new approach is bound to clarify its nature, vaguely touched in Proposition 5.1. Proposition 7.7. There is a natural transformation T ATTΣ ◦ K ATT ×Ω ((−)�) op −−−−−−→ T ATTΩ defined by T ATT Ω (X,G) (−)�(X,G) −−−−−−→ T ATTΣ ◦ K ATT ×Ω (X,G) = (ΩG) X � X −−→ (ΣG|Ω G|)X Θ −→ ΣGX×|ΩG|, where the map Θ comes from Proposition 7.6. Proof. The claim is a consequence of the fact that every Set × LoATT∗A- morphism (X1,G1) (f,g) −−−→ (X2,G2) makes the following diagram commute: (ΩG2) X2 (f,(Ωg)op)← �� � X2 2 // ((ΣG2) |Ω G2|)X2 (f,(((Ωg)∗,Σg)↼)op)← �� Θ2 // (ΣG2) X2×|ΩG2| (f×(Ωg)∗op,(Σ g)op)← �� (ΩG1) X1 � X1 1 // ((ΣG1) |Ω G1|)X1 Θ1 // (Σ G1) X1×|Ω G1|, 128 A. Frascella, C. Guido and S. A. Solovyov where the left rectangle uses commutativity of the second diagram of Propo- sition 7.4 in the form of resulting Diagram (7.1), and the right rectangle is a direct consequence of Proposition 7.6. � The reader should pay attention to the fact that the new description clari- fies the categorical background of the natural transformation in question, which was defined in an algebraic way in Propositions 5.1, 6.7. Categorical approach brings more universality in play, reducing dramatically the algebraic depen- dence on points, thereby opening a possibility to define attachment in a more general category than Set. It will be the subject of our forthcoming papers to provide such an extended definition of attachment. 8. Conclusion: lattice-valued categorically-algebraic topology The notion of dual attachment introduced in this paper clarified completely the categorical nature of the concept of attachment considered in [25, 26, 73]. As the main achievement, we showed that it provides a way of interaction (nat- ural transformation) between two topological theories (Propositions 6.7, 7.7), which results in a functor between the respective categories of topological struc- tures (see remarks after Proposition 6.7). It was the framework of categorically- algebraic (catalg) topology, which helped to discover this important property. Moreover, we have already remarked (see, e.g., Example 6.5) that the catalg approach incorporates the majority of the existing (lattice-valued) topological settings, erasing the border between crisp and many-valued frameworks. A significant drawback of the concept, however, is its inability to include the the- ory of (L,M)-fuzzy topological spaces of T. Kubiak and A. Šostak [40]. The striking difference of their approach from those used in the paper is the fact that a topological space is defined as a pair (X,T), where T is a lattice-valued subalgebra of T(X) for an appropriate topological theory T of Definition 6.3. This observation in hand, the concept of lattice-valued catalg topology has been introduced in [65], based in a suitably defined notion of lattice-valued alge- bra. The latter notion has already appeared in [68], motivated by the concept of fuzzy group of A. Rosenfeld [63] and its generalization of J. M. Anthony and H. Sherwood [4]. The employed machinery goes in line with the general procedure of, e.g., J. N. Mordeson and D. S. Malik [47] as follows. Definition 8.1. Let A, L be varieties, the latter having the variety CSLat( ∨ ) as a reduct, and let C be a subcategory of L. An (A,C)-algebra is a triple (A,µ,L), where A is an A-algebra, L is a C-algebra and |A| µ −→ |L| is a map such that for every λ ∈ Λ and every 〈ai〉nλ ∈ A nλ, ∧ i∈nλ µ(ai) ≤ µ(ω A λ (〈ai〉nλ)). An (A,C)-homomorphism (A1,µ1,L1) (ϕ,ψ) −−−→ (A2,µ2,L2) is an A × C-morphism (A1,L1) (ϕ,ψ) −−−→ (A2,L2) such that ψ ◦ µ1(a) ≤ µ2 ◦ ϕ(a) for every a ∈ A1. C-A is the category of (A,C)-algebras and (A,C)-homomorphisms, which is concrete over the product category A × C. Dual attachment pairs in categorically-algebraic topology 129 An important moment arising from the definition should be underlined at once. In [15], A. Di Nola and G. Gerla introduced the category C(τ) as a “general approach to the theory of fuzzy algebras”. It is easy to see that C(τ) is isomorphic to the subcategory CLat-Alg(ΩΩΩ) = of CLat-Alg(ΩΩΩ), with the same objects and with morphisms (A1,µ1,L1) (ϕ,ψ) −−−→ (A2,µ2,L2) satisfying the identity ψ ◦ µ1 = µ2 ◦ ϕ (reflected in the notation “(−) =”). Moreover, [15] started to develop the theory of fuzzy universal algebra, some results of which can be easily extended to our approach. Bound to the topological nature of this paper, we will only notice that the category C-A provides a more appropriate fuzzification of universal algebra, which fuzzifies not only algebras, but also (and that is more important) their respective homomorphisms. The related topological stuff is an easy modification of Definitions 6.3, 6.4. Definition 8.2. Let T be a vt-theory in a category X, let L be a variety having CSLat( ∨ ) as a reduct, and let L be a subcategory of LoL. An L-valued vt- theory in X induced by T and L is the pair (T,L). Definition 8.3. Let (T,L) be an L-valued vt-theory in a category X. LTop(T) is the category, concrete over X × L, whose objects (L-valued T-topological spaces) are triples (X,T,L), comprising an X-object X, an L-object L and a (B,LoL)-algebra (T(X),T,L) (L-valued T-topology on X), and whose mor- phisms (X,T,L) (f,ψ) −−−→ (Y,S,M) are those X × L-morphisms (X,L) (f,ψ) −−−→ (Y,M), which satisfy the property of (T(X),T,L) (T(f),ψ) −−−−−→ (T(Y ),S,M) being a Lo(LoL-B)-morphism (L-valued T-continuity). It appears that lattice-valued catalg topology is truly a universal one, incor- porating all (up to the knowledge of the authors) existing topological settings (including the catalg one). The following examples justify the fruitfulness of the new notion (the reader should notice that an L-valued vt-theory (T,L) is occasionally denoted by (P,B,LoL), to underline its building blocks). Example 8.4. (1) LTop((SSL CLat ,SFrm,S CDCLat M )), where CLat is the variety of com- plete lattices and CDCLat is its subcategory of completely distributive lattices, provides a categorical accommodation of the theory of (L,M)- fuzzy topological spaces of T. Kubiak and A. Šostak [40]. (2) LTop((R3,Frm,Frm)) is isomorphic to the category Loc-F 2 Top of (L,M)-fuzzy topological spaces of J. T. Denniston, A. Melton and S. E. Rodabaugh [11], which was introduced as a variable-basis counter- part of the above-mentioned approach of T. Kubiak and A. Šostak. (3) LTop((P,Frm,SDMLocL )) provides the approach of U. Höhle [30]. (4) LTop(T) for L = S CSLat( ∨ ) 2 is isomorphic to the category Top(T) introduced in Definition 6.4. In view of the above-mentioned remarks, it seems natural to consider the notion of attachment in the more general lattice-valued framework, and, there- fore, one can postulate the following open problem. 130 A. Frascella, C. Guido and S. A. Solovyov Problem 8.5. What will be the concept of lattice-valued catalg attachment? It will be the topic of our further research to extend the already developed framework to the new setting. Table of categories A, B, C: varieties of algebras. 107 Alg(ΩΩΩ): ΩΩΩ-algebras. 106 AttA: variety-based attachments. 103 ATTB: dual variety-based attachments. 108 ATT ∗ A: (Π1 ↓ P ◦ K)-objects Π1(A1,A2) ϕ −→ P ◦ K(A′1,A ′ 2) for which (A1,A2) = (A ′ 1,A ′ 2). 125 C-A: lattice-valued algebras. 128 CBool: complete Boolean algebras. 120 (C,D)-TopSys: variable-basis variety-based topological systems. 111 Chu(Set,K): Chu spaces. 111 CLat: complete lattices. 107 Cont: contexts. 111 CSL: closure semilattices. 107 CSLat(Ξ): Ξ-semilattices. 106 C-Top: variable-basis variety-based topological spaces. 109 DmSQuant: DeMorgan semi-quantales. 106 Frm: frames. 106 FuzLat: dual of HUT. 121 HUT: Hutton algebras. 121 IntSys: interchange systems. 111 LoA: the dual category of a variety A. 107 LoATTB-Top ∅k: non-empty stratified variable-basis variety-based topo- logical spaces. 113 Loc: locales. 107 L-Top: fixed-basis lattice-valued topological spaces. 103 LTop(T): lattice-valued topological spaces induced by a topological the- ory T . 129 QFrm: quasi-frames. 106 Quant: quantales. 106 SA: subcategory of LoA whose only morphisms is the identity 1A. 107 Set: sets. 102 SFrm: semi-frames. 106 SQuant: semi-quantales. 106 Top: topological spaces. 103 Top(T): topological spaces induced by a topological theory T . 121 USQuant: unital semi-quantales. 106 131 References [1] J. 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De Giorgi”, University of Salento, P. O. Box 193, 73100 Lecce, Italy. Cosimo Guido (cosimo.guido@unisalento.it) Department of Mathematics “E. De Giorgi”, University of Salento, P. O. Box 193, 73100 Lecce, Italy. Sergey A. Solovyov (sergejs.solovjovs@lu.lv, sergejs.solovjovs@lumii.lv) Department of Mathematics, University of Latvia, Zellu iela 8, LV-1002 Riga, Latvia. Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris 29, LV-1459 Riga, Latvia. Dual attachment pairs in categorically-algebraic[2pt] topology. By A. Frascella, C. Guido and S. A. Solovyov