@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 1, 2003 pp. 25–34 On complete objects in the category of T0 closure spaces D. Deses ∗ , E. Giuli and E. Lowen-Colebunders Abstract. In this paper we present an example in the setting of closure spaces that fits in the general theory on ‘complete objects’ as developed by G. C. L. Brümmer and E. Giuli. For V the class of epi- morphic embeddings in the construct Cl0 of T0 closure spaces we prove that the class of V-injective objects is the unique firmly V-reflective subconstruct of Cl0. We present an internal characterization of the V- injective objects as ‘complete’ ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. An external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. We prove that the construct of complete T0 closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins. 2000 AMS Classification: Primary: 54A05, 54B30, 18G05; Secondary: 54D10 Keywords: Complete object, firm, injective, complete lattice, T0 object, closure space. 1. Introduction A closure space (X,C) is a pair, where X is a set and C is a subset of the power set P(X) satisfying the conditions that X and ∅ belong to C and that C is closed for arbitrary unions. The sets in C are called open sets. A function f : (X,C) → (Y,D) between closure spaces (X,C) and (Y,D) is said to be continuous if f−1(D) ∈C whenever D ∈D. Cl is the construct of closure spaces as objects and continuous maps as morphisms. Some isomorphic descriptions of Cl are often used e.g. by giving the collection of all closed sets (the so called Moore family [4]) where, as usual, the closed sets are the complements of the open ones and continuity is defined accordingly. Another isomorphic ∗The first author is ‘aspirant’ of the F.W.O.-Vlaanderen. 26 D. Deses, E. Giuli and E. Lowen-Colebunders description is obtained by means of a closure operator [4]. The closure operation cl : P(X) →P(X) associated with a closure space (X,C) is defined in the usual way by x ∈ cl A ⇐⇒ (∀C ∈ C : x ∈ C ⇒ C ∩ A 6= ∅) where A ⊂ X and x ∈ X. This closure need not be finitely additive, but it does satisfy the conditions cl ∅ = ∅, (A ⊂ B ⇒ cl A ⊂ cl B), A ⊂ cl A and cl (cl A) = cl A whenever A and B are subsets of X. Continuity is then characterized in the usual way. Finally closure spaces can also be equivalently described by means of neighborhood collections of the points. These neighborhood collections satisfy the usual axioms, except for the fact that the collections need not be filters. So in a closure space (X,C) the neighborhood collection of a point x is a non empty stack (in the sense that with every V ∈N(x) also every W with V ⊂ W belongs to N(x)), where every V ∈ N(x) contains x and N(x) satisfies the open kernel condition. In the sequel we will just write X for a closure space and we’ll choose the most convenient form for its explicit structure. Motivations for considering closure spaces can be found in G. Birkhoff’s book [4] where he associates closures to binary relations in a natural way. Similar ideas appeared in G. Aumann’s work on contact relations with applications to social sciences [3] or in a more recent work of B. Ganter and R. Wille on formal contexts with applications in data analysis and knowledge representation [11]. In recent years closures have also been used in connection with quantum logic and in the representation theory of physical systems, see e.g. [2] or [16]. In these applications the T0 axiom we are dealing with plays a key role [20]. In 1940 G. Birkhoff’s motivation for considering closures also came from the observation that the collection of closed sets of a closure space forms a complete lattice. The interrelation between closures and complete lattices has been investigated by many authors and a general discussion of this subject can be found in M. Erné’s paper [10]. In the last section of our paper further investigation of the correspondence with complete lattices leads to an external characterization of the complete objects we are studying. For all categorical terminology we refer the reader to the books [1, 13] or [18]. 2. The construct of T0 spaces 2.1. As is well known [9] Cl is a topological construct in the sense of [1]. Cl0 is the subconstruct consisting of its T0-objects. Applying Marny’s definition [15] we say that a closure space X is a T0-object in Cl if and only if every morphism from the indiscrete object I2 on the two point set {0, 1} to X is constant. This equivalently means that for every pair of different points in X there is a neighborhood of one of the points not containing the other one. Cl0 is an extremally epireflective subconstruct of Cl [15] and as such it is initially structured in the sense of [17, 18]. In particular Cl0 is complete and cocomplete and well-powered, it is an (epi - extremal mono) category and an (extremal epi - mono) category [13]. Also from the general setting it follows that monomorphisms in Cl0 are exactly the injective continuous maps and The category of T0 closure spaces 27 a morphism in Cl0 is an extremal epimorphism if and only if it is a regular epimorphism if and only if it is surjective and final. 2.2. In order to describe the epimorphisms and the extremal monomorphisms in Cl0 we need the regular closure operator determined by Cl0 as introduced in [8, 9]. Given a closure space X and a subset M ⊂ X one defines the regular closure of M in X as follows. A point x of X is in the closure of M if and only if (i) for every T0 closure space Z and every pair of morphisms f,g : X → Z, M ⊂{f = g} =⇒ f(x) = g(x). Using the fact that Cl0 is the epireflective hull in Cl of the two point Sierpinski space S2, we obtain the following equivalent description. (ii) For every pair of morphisms f,g : X → S2, M ⊂{f = g} =⇒ f(x) = g(x). Quite similar to the topological situation one can prove yet another equivalent formulation. (iii) For every neighborhood V of x: V ∩ cl{x}∩M 6= ∅ In each of the equivalent cases we’ll write x ∈ clXb M. It was shown in [9] that the regular closure clb = {clXb : P(X) →P(X)}X∈|Cl| defines a closure operator on Cl. By the equivalent description (ii) clb coincides with the Zariski closure operator as considered in [7] and [12]. The equivalent formulation (iii) is the formula for the b-closure (or front closure) in Top. For this reason we will also call clb the b-closure operator on Cl. It follows from theorem 2.8 in [9] that the epimorphisms in Cl0 are the b- dense continuous maps. So in fact the inclusion functor Top0 ↪→ Cl0 preserves epimorphisms. One observes that this is not so for the inclusion functor from Top0 to the construct PrTop0 of T0 pretopological spaces. Using arguments analogous to the ones used in the topological case, one proves that Cl0 is cowell-powered. The closure operator clb is idempotent and grounded and is easily seen to be hereditary in the sense that for a closure space Y , a subspace X and M ⊂ X ⊂ Y we have clXb M = cl Y b M ∩X Using this fact one can prove that a morphism in Cl0 is an extremal monomor- phism if and only if it is a regular monomorphism if and only if it is a b-closed embedding. Explicit proofs of the previous statements have been worked out in [19]. 28 D. Deses, E. Giuli and E. Lowen-Colebunders 3. Injective objects in Cl0 and firmness In this paragraph we consider a particular class of morphisms in Cl0. Let V be the class of epimorphic embeddings, i.e. the class of all b-dense embeddings. This class V satisfies the following conditions: (α) closedness under composition, (β) closedness under composition with isomorphisms on both sides. (α) and (β) are standing assumptions made in [5] and enable us to apply to Cl0 the theory developed in that paper. A T0 closure space B is V-injective if for each v : X → Y in V and f : X → B there exists f′ : Y → B such that f′◦v = f. In this case f′ is called an extension of f along v. Inj V denotes the full subcategory of all V-injective objects in Cl0. Proposition 3.1. The two point Sierpinski space S2 is V-injective in Cl0. Next consider RCl0 ({S2}), the epireflective hull of S2 in Cl0. In view of the properties of Cl0 listed in paragraph 2, this hull consists of all b-closed subspaces of powers of S2. Recall that a reflective subcategory is V-reflective if the reflection morphisms all belong to V. Proposition 3.2. A T0 closure space is V-injective if and only if it is a b-closed subspace of some power of S2. Proof. In view of theorem 37.1 in [13] Inj V is epireflective in Cl0 and since it contains S2 we immediately have RCl0 ({S2}) ⊂ Inj V. Moreover RCl0 ({S2}) clearly is V-reflective, so if B is V-injective, the reflection morphism v : B → RB belongs to V. We have f′◦v = 1B where f′ is the extension of 1B : B → B along v. Then clearly v is an isomorphism and therefore B ∈ |RCl0 ({S2})|. � Remark 3.3. The notion of V-injectivity in Cl0 differs from injectivity related to the class of all embeddings. V-injectivity is a strictly weaker condition as is shown by the example B = {(0, 0, 1), (0, 1, 1), (0, 1, 0), (1, 1, 1)} which is a b-closed subspace of S32 and hence is V-injective by proposition 3.2. However B is not injective in Cl0 with respect to embeddings. We use the terminology of [5] (which slightly differs from [6]). A class U of morphisms in a category X (satisfying the standing assumptions (α) and (β)) is said to be (i) a subfirm class: if there exists a U-reflective subcategory with reflector R such that Rf is an isomorphism whenever f is in U. (ii) a firm class: if there exists a U-reflective subcategory with reflector R such that Rf is an isomorphism if and only if f is in U. In these cases the corresponding subcategory is said to be (sub-)firmly U- reflective and it coincides with Inj U [5]. Again we consider the particular class V of b-dense embeddings in Cl0. In view of the equivalent descriptions given in 2.2 and the fact that RCl0 ({S2}) = Inj V, the class Inj V is V-reflective. So we can apply theorems 1.4 and 1.14 in [5] to formulate the following result. The category of T0 closure spaces 29 Proposition 3.4. The class V of b-dense embeddings is a firm class of mor- phisms in Cl0 and Inj V is the unique firmly V-reflective subcategory of Cl0. In the context of an epireflective subcatgory X of a topological category, with S the class of embeddings in X and V the class of epimorphic embeddings, the notion of V-injective object can be linked to a few others, as discussed in [6]. An object X in X is said to be S-saturated if an X-morphism f : X → Y is an isomorphism whenever f is in V. X is said to be absolutely S-closed if an X-morphism f : X → Y is a regular monomorphism whenever f ∈S. In the particular situation where moreover in X extremal monomorphisms coincide with regular monomorphisms and where Inj V is (sub-)firmly V-re- flective, it was shown in [6] that for an object X in X one has X is V-injective ⇐⇒ X is S-saturated ⇐⇒ X is absolutely S-closed. From the results in paragraph 2 and from Proposition 3.4 we can conclude that the V-injective objects of Cl0 coincide with the S-saturated or equivalently with the absolutely S-closed T0 objects. These properties have also been considered by Diers [7] in the setting of T-sets and the objects fulfilling the equivalent conditions were called algebraic T-sets. Our example also fits in that context. 4. Internal characterizations via complete objects The results displayed so far in paragraph 3 are quite similar to the well known topological situation on V-injective objects in Top0. In that setting these objects can be internally characterized as T0 topological spaces for which every nonempty irreducible closed set is the closure of a point, i.e. as sober spaces [14, 6]. In this paragraph we give an internal characterization of the V-injective objects in Cl0. This description for Cl0, when applied to Top0 will turn out to deal with a notion much stronger than sobriety. Definition 4.1. Let X be a closure space. For A⊂P(X) we write stackA = {B ⊂ X | ∃A ∈A : A ⊂ B} and A is said to be a stack if A = stackA. A nonempty stack is said to be open based if A = stack{G ∈ A | G open}. A proper open based stack A is said to be fundamental if A contains a member of every open cover of every element of A, i.e. whenever A ∈A and G is an open cover of A then ∃G ∈G : G ∈A. More briefly a fundamental nonempty open based stack is called an O-stack. As an easy example we note that in every closure space the neighborhood collection N(x) of a point x is an O-stack. Proposition 4.2. On a closure space X and for A⊂P(X) we have: A is an O-stack if and only if there exists a (closed ) nonempty set F ⊂ X such that A = stack{G ⊂ X | G open, G∩F 6= ∅}. 30 D. Deses, E. Giuli and E. Lowen-Colebunders Proof. For F nonempty it is clear that A = stack{G | G open, G∩F 6= ∅} = stack{G | G open, G∩ cl F 6= ∅} is an O-stack. Conversely let A be an O-stack. Let F = {x ∈ X | N(x) ⊂ A}. F clearly is nonempty since otherwise there would exist an open cover of X of which all members are not in A. If G is open and G∩F 6= ∅ then N(x) ⊂ A for some point x ∈ G. So G ∈A. On the other hand, if G is open and belongs to A then G has to intersect F. If not, there would exist an open cover of G of which all members are not in A. So finally we can conclude that A = stack{G | G open, G∩F 6= ∅}. Remark that the set F = {x ∈ X | N(x) ⊂A} is in fact closed. � Definition 4.3. A T0 closure space X is called complete if every O-stack is a neighborhood collection N(x) for some (unique) point x ∈ X. The uniqueness of the point follows from the T0 condition. In view of Propo- sition 4.2 we get the following equivalent description. Proposition 4.4. A T0 closure space X is complete if and only if every nonempty closed set is the closure of a (unique) point. Proof. If F is closed and nonempty then there is a point x ∈ X such that stack{G | G open, G∩F 6= ∅} = N(x). Then clearly F = cl{x}. Conversely if A is an O-stack, as in the proof of Proposition 4.2, let F be the nonempty closed set F = {x ∈ X | N(x) ⊂A}. Now F = cl{x} implies A = N(x). � Let CCl0 be the full subconstruct of Cl0 consisting of the complete objects. Proposition 4.5. Complete T0 objects are absolutely S-closed. Proof. Let X be a complete T0 space and let f : X → Y be an embedding in Cl0. We prove that f(X) is b-closed in Y . Let a ∈ Y \ f(X). Either clY {a}∩f(X) = ∅ or clY {a}∩f(X) = clf(X) {f(z)} for some z ∈ X. In the latter case, let U = Y \ clY {f(z)}, then by the T0 condition on Y we have a ∈ U. Moreover U ∩ clY {a}∩f(X) = ∅. So in both cases we can conclude that a 6∈ clYb f(X). � Proposition 4.6. CCl0 is V-reflective in Cl0. Proof. First we construct the reflector R : Cl0 → CCl0. Let X be a T0 closure space and let X̂ = {A |A is a O-stack on X}. On X̂ we define a closure space as follows. For G ⊂ X open let Ĝ = {A O-stack | G ∈A} The category of T0 closure spaces 31 then {Ĝ | G ⊂ X open} defines a T0 closure structure on X̂. If A is an O-stack on X then  = stack{Ĝ | G ⊂ X, G open, G ∈A} is its neighborhood collection in X̂. If Ψ is an O-stack on X̂ then Ψ̌ = stack{G | G ⊂ X, G open, Ĝ ∈ Ψ} is an O-stack on X. It follows that X̂ is a complete T0 closure space. Let rX : X → X̂ be the natural injective map sending x ∈ X to N(x) ∈ X̂. Clearly for G ⊂ X open, we have r−1X (Ĝ) = G and hence rX is an embedding (in Cl0). This embedding is b-dense since for an O-stack A and G ∈ A there exists x ∈ G such that N(x) ⊂A and therefore Ĝ∩ cl X̂ {A} ∩rX(X) 6= ∅. Now let B be a complete T0 closure space and f : X → B a continuous map. By Proposition 3.4 Inj V is V-reflective, so that the reflection map, say sB, belongs to V. Hence sB is a b-dense embedding and by Proposition 4.5 sB is also b-closed, therefore it is an isomorphism. Thus B is V-injective. So there is an extension f′ of f along rX : X → X̂. Since rX is an epimorphism in Cl0 this extension moreover is unique. � Corollary 4.7. CCl0 is V firmly reflective in Cl0 and therefore coincides with the class of all V-injective T0 objects. Remark 4.8. The previous conclusion combined with the characterization of CCl0 given in Proposition 4.4 and the remarks at the end of paragraph 3, imply the result stated in example 9.7 in [7] that algebraic T0 closure spaces are those for which every nonempty closed set is the closure of a point. 5. An external characterization via the natural correspondence with complete lattices In the topological counterpart on complete T0 objects the duality between sober topological spaces and spatial frames leads to an external characterization of ‘completeness’. In this paragraph we base our external characterization on the correspondence between closure spaces and complete lattices. Let CLat∨,1 be the category whose objects are complete lattices and whose morphisms are maps preserving arbitrary joins and the top element. The dual category will be denoted CLatop∨,1. To every closure space X we associate the lattice O(X) of its open subsets. With f : X → Y we associate the map O(Y ) → O(X) : G 7→ f−1(G). This correspondence defines a functor Ωc : Cl → CLatop∨,1. 32 D. Deses, E. Giuli and E. Lowen-Colebunders In order to define an adjoint for Ωc, let L be a complete lattice. A point of L is a surjective CLat∨,1 morphism L → 2 where 2 = {0, 1} is the two point complete lattice. In the sequel we’ll use pts(L) to denote the set of points of L and for u ∈ L we’ll write Σu = {ξ ∈ pts(L) | ξ(u) = 1}. Observe that in contrast to the topological and frame counterpart, for objects u and v in L, we always have u 6= v ⇒ Σu 6= Σv. With this notation we can describe the functor Σc : CLatop∨,1 → Cl sending a lattice L to the set pts(L), endowed with the closure structure {Σu | u ∈ L}, and f : M → L to ΣcM → ΣcL : ξ 7→ ξ ◦f. Proposition 5.1. For a complete lattice L, ΣcL is a complete T0 closure space. Proof. Consider two distinct ξ1,ξ2 ∈ pts(L) of ΣcL. There exist a u ∈ L such that ξ1(u) 6= ξ2(u). Hence either ξ1 ∈ Σu and ξ2 6∈ Σu or ξ1 6∈ Σu and ξ2 ∈ Σu. So Σc is T0. To prove the completeness we choose an O-stack A in ΣcL and consider v =∨ {u ∈ L | Σu 6∈ A}. Next we define the point ξ : L → 2 : u 7→ { 1 u 6≤ v 0 u ≤ v . We have that ξ(v) = 0, hence (Σu 6∈ A ⇒ ξ(u) = 0). Conversely, if Σu ∈ A then u 6≤ v since A is an O-stack. Thus (Σu ∈ A ⇒ ξ(u) = 1). Finally Σu ∈A ⇐⇒ ξ(u) = 1. Therefore A = N(ξ) in ΣcL. � Proposition 5.2. The restrictions Ωc : CCl0 → CLat op ∨,1 Σc : CLatop∨,1 → CCl0 define an equivalence of categories. Proof. The proof consists of three parts. (1) Let L be a complete lattice then L ' ΩcΣcL. Choose the isomorphism as follows: �L : Ω cΣcL → L : Σu 7→ u This is a well defined CLatop∨,1-isomorphism. (2) Let X be a complete T0 closure space then X ' ΣcΩcX. Define the following map ηX : X → pts(ΩcX) : x 7→ ξx where ξx(A) = { 1 x ∈ A 0 x 6∈ A , for all open sets A. ηX is injective since by the T0 property we have for x 6= y an open subset A such that ξx(A) 6= ξy(A). Therefore ξx 6= ξy. To show that ηX is surjective we choose a point ξ, and consider stack ξ−1(1). This is obviously a stack with an open basis, so that if The category of T0 closure spaces 33⋃ i∈I Ai ∈ stack ξ −1(1) then there exists B ∈ ξ−1(1) : B ⊂ ⋃ i∈I Ai. So we get 1 = ξ(B) ≤ ξ( ⋃ i∈I Ai) = ∨ i∈I ξ(Ai). Hence there exists an i ∈ I with Ai ∈ ξ−1(1) and so stack ξ−1(1) is an O-stack. Therefore there is an x ∈ X such that N(x) = stack ξ−1(1). We now have ξx(A) = 1 ⇔ A ∈N(x) ⇔∃B ∈ ξ−1(1) : B ⊂ A ⇔ ξ(A) = 1. Hence ξ = ξx. Moreover ηX and η −1 X are both continuous. This follows from η−1X (ΣA) = {x ∈ X | ξx(A) = 1} = A, ηX(A) = {ξx | x ∈ A} = {ξx | ξx(A) = 1} = ΣA, where A is open. (3) To see the naturality of η, consider continuous f : X → Y where X and Y are complete T0 closure spaces. We have the following compositions: (Σc(Ωc(f))◦ηX)(x) = (Σc(f−1))(ξx) = ξx◦f−1 = ξf(x) and ηY ◦f(x) = ξf(x). Hence η = (ηX)X∈|CCl0| is a natural isomorphism η : 1CCl0 ' ΣcΩc. The naturality of � follows since if h : L → M is a CLat∨,1- morphism, we have the compositions (�M ◦ Ωc(Σc(h)))(Σu) = �M ((Σc(h))−1(Σu)) = �M ({ξ ∈ pts(M) | ξ ◦h ∈ Σu}) = �M ({ξ ∈ pts(M) | ξ ◦h(u) = 1}) = �M (Σh(u)) = h(u) and h ◦ �L(Σu) = h(u). Therefore � = (�L)L∈|CLat∨,1| is a natural isomorphism � : ΩcΣc ' 1CLat∨,1 . 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Moore, Categories of representations of physical systems, Helv. Phys. Acta 68 (1995), 658–678. [17] L. D. Nel, Initially structured categories and Cartesian closedness, Canad. J. Math. 27 (6) (1975), 1361–1377. [18] G. Preuss, Theory of Topological Structures (D. Reidel Publishing Company, Dordrecht, 1988). [19] A. Van der Voorde, Separation axioms in extensiontheory for closure spaces and their relevance to state property systems, PhD Thesis Vrije Universiteit Brussel, July 2001. [20] B. Van Steirteghem, T0 separation in axiomatic quantum mechanics, Internat. J. The- oret. Phys. 39 (3) (2000), 955–962. Received December 2001 Revised November 2002 D. Deses and E. Lowen-Colebunders Department of Mathematics, Vrije Universiteit Brussel, 1050 Brussels, Belgium. E-mail address : diddesen@vub.ac.be evacoleb@vub.ac.be E. Giuli Dipartimento di Matematica Pura ed Applicata, Università di L’Aquila, 67100 L’Aquila, Italy. E-mail address : giuli@univaq.it On complete objects in the category of T0 closure spaces. By D. Deses, E. Giuli and E. Lowen-Colebunders