@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 1, 2003 pp. 15–24 The quasitopos hull of the construct of closure spaces V. Claes and G. Sonck Abstract. In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most inter- esting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull. 2000 AMS Classification: 18A35, 18B25, 18D15, 54A05, 54C35 Keywords: Topological construct, closure space, extensional topological construct, quasitopos, cartesian closed category, cartesian closed topological hull. 1. Introduction Cartesian closedness is an interesting property for topological constructs. It guarantees the existence of nice function spaces in the construct. This property has been studied extensively in the literature. A closure space is a set X endowed with a closure operator, i.e. a map cl : P(X) → P(X) satisfying the following conditions: cl ∅ = ∅, A ⊂ cl A, A ⊂ B ⇒ cl A ⊂ cl B and cl (cl A) = cl A. One notes that the closure is allowed to be non-additive. A map f : X → Y is said to be continuous if f(cl A) ⊂ cl (f(A)) for all subsets A ⊂ X. The construct of closure spaces and continuous maps is denoted by Cls. Cls is known to be a well-fibred topological construct. It is an interesting construct since non-additive closures arise in different fields of mathematics, in particular in algebra, geometry and analysis. Perhaps the best known example is the convex hull in vector spaces. Other examples are listed in the introductory chapter of [6]. There is also a strong relation between closures and complete lattices [4]. In recent years 16 V. Claes and G. Sonck closures have even been used in connection with quantum logic and in the representation theory of physical systems [13, 14, 19]. The construct Cls is not cartesian closed. In [3] the cartesian closed topo- logical hull of Cls was described. In this paper we will describe an even more convenient hull: the quasitopos hull. A topological construct is a quasitopos if it is both extensional and cartesian closed. The property of extensionality ensures the existence of one-point extensions in the topological construct, or, equivalently, that quotients and coproducts are preserved by pullbacks along embeddings. This property is extensively treated in [20]. The stronger concept topos is not interesting for topological categories, since the only topological topoi are isomorphic to the category Set [1, 22]. Quasitopoi (also called topo- logical universes for topological categories) were introduced by J. Penon [18] as a generalization of topoi. This generalization is broad enough to allow topo- logical examples, but not too broad for losing most useful properties of topoi. Topological universes are used for functional analysis and differential calculus (see e.g. [15, 16, 17]) and for a theory of holomorphic maps (see e.g. [5]). Also for a topological construct being a quasitopos is equivalent to being lo- cally cartesian closed i.e. all comma-categories being cartesian closed. It is well known that the construct Top of topological spaces and continuous maps is not a quasitopos: it is neither extensional nor cartesian closed. Schwarz [21] showed that the topological quasitopos hull of a construct –if it exists– can be described as the cartesian closed topological hull of the extensional topological hull. Therefore we first construct the extensional topological hull of Cls and then the cartesian closed topological hull of this extensional hull. Categorical terminology follows [1]. We will only consider constructs that are well-fibred. 2. The extensional topological hull of Cls In this section, we construct the extensional topological hull of Cls. We do this in the same way as has been done for Top: we weaken the axioms of the closure operator. Definition 2.1. [10, 11, 20] A topological construct A is called extensional if it has representable partial morphisms to all A-objects, where • A partial morphism from X to Y is a morphism f : Z → Y whose domain Z is a subspace of X. • Partial morphisms to Y are representable provided Y can be embed- ded via the addition of a single point ∞Y into an object Y ] with the property that for every partial morphism f : Z → Y from X to Y , the map fX : X → Y ], defined by fX(x) = f(x) if x ∈ Z, fX(x) = ∞Y if x ∈ X \ Z, is a morphism. The object Y ] is called the one-point extension of Y . Extensional was called hereditary in [10] and [20]. In [20] Schwarz proved that the one-point extension of an object Y (if it exists) carries the smallest The quasitopos hull of Cls 17 (i.e. coarsest) structure that makes Y a subspace. Thus the one-point exten- sion is unique (up to isomorphism). There is no smallest closure on {0, 1,∞} which makes the Sierpinski space 2 a subspace, so the construct Cls can not be extensional. We shall now obtain an extensional supercategory of Cls by dropping the idempotency of the closure operator. Definition 2.2. A preclosure space (X, cl) is a set X structured by a preclosure operator cl : P(X) →P(X) satisfying the following conditions. For A,B ⊂ X : (C1) cl ∅ = ∅, (C2) A ⊂ cl A, (C3) A ⊂ B ⇒ cl A ⊂ cl B. A function f : (X, clX) → (Y, clY ) between preclosure spaces is continuous iff f(clXA) ⊂ clY f(A) for all A ⊂ X. The construct of preclosure spaces and continuous maps is denoted by PrCls. A preclosure space can also be described in an isomorphic way using neigh- borhoods. For a preclosure space (X, cl) and x ∈ X, the collection of neighbor- hoods V(x) = {V ⊂ X | x /∈ cl (X\V )} satisfies the following three conditions: (V1) X ∈V(x), (V2) ∀V ∈V(x) : x ∈ V , (V3) ∀V ∈V(x) : V ⊂ W ⇒ W ∈V(x). Conversely, if the family (V(x))x∈X satisfies the conditions (V1), (V2) and (V3) for each x ∈ X, then a unique preclosure operator cl on X exists, such that for each x ∈ X, V(x) are the neighborhoods of x. This closure operator is defined by: cl A = {x ∈ X | ∀V ∈ V(x) : V ∩ A 6= ∅} for all A ⊂ X. A function f : (X,µ) → (Y,η) is continuous iff f−1(η(f(x))) ⊂ µ(x) for every x ∈ X. From the definition of PrCls follows immediately that Cls is a full subcat- egory of PrCls. Proposition 2.3. PrCls is a topological construct. Proof. For a structured source (fi : X → (Yi, cli))i∈I in PrCls, the initial preclosure operator on X is defined by: clA = ⋂ i∈I f−1i (cli(fi(A))) for each A ⊂ X or with neighborhoods: V(x) = {V ⊂ X | ∃i ∈ I, ∃W ∈ Vi(fi(x)) : f−1i (W) ⊂ V}. � It is clear that Cls is closed under formation of initial structures in PrCls, and thus we have: Proposition 2.4. Cls is a bireflective subconstruct of PrCls. Theorem 2.5. PrCls is extensional. Proof. Analogously as for PrTop [11]. � Definition 2.6. [9, 11] An extensional topological construct B is called an extensional topological hull of a construct A if B is a finally dense extension of A with the property that any finally dense embedding of A into an extensional topological construct can be uniquely extended to B. 18 V. Claes and G. Sonck The extensional topological hull of a construct – if it exists – is unique up to isomorphism. Theorem 2.7. [10, 11] The extensional topological hull B of a construct A is characterized by the following properties: (1) B is an extensional topological construct. (2) A is finally dense in B. (3) {Y ] | Y ∈ |A|} is initially dense in B. The proofs of the following propositions are similar to those in [11]. Proposition 2.8. Cls is a finally dense subcategory of PrCls. Definition 2.9. The one-point extension 2] of the Sierpinski space is the preclosure space 3 with underlying set {0, 1, 2} (we denote ∞ by 2) and neigh- borhoods: V(0) = V(2) = 0̇ ∩ 1̇ ∩ 2̇ and V(1) = 1̇ ∩ 2̇. Proposition 2.10. {3} is initially dense in PrCls. Now from Theorem 2.5, Proposition 2.8, Proposition 2.10 and Theorem 2.7 we have the following result: Theorem 2.11. PrCls is the extensional topological hull of Cls. 3. The cartesian closed hull of PrCls An object X in a category with finite products is exponential if X ×− has a right adjoint. In a well-fibred topological construct D, this notion can be characterized as follows: X is exponential in D iff for each D-object Y the set HomD(X,Y ) can be supplied with the structure of a D-object – a function space or a power object Y X – such that (1) the evaluation map ev : X ×Y X → Y is a D-morphism, and (2) for each D-object Z and each D-morphism f : X × Z → Y the map f∗ : Z → Y X defined by f∗(z)(x) = f(x,z) is a D-morphism. It is well known that in the setting of a topological construct D, an object X is exponential in D iff X × − preserves final episinks [7, 8]. Moreover, small fibredness of D ensures that this is equivalent to the condition that X ×− preserves quotients and coproducts. A well-fibred topological construct D is said to be cartesian closed (or to have function spaces) if every object is exponential. It was shown in [3] that Cls is not cartesian closed. In fact, the class of exponential objects consists precisely of all indiscrete closure spaces. We show that exponential objects are unchanged if we replace Cls by PrCls. Proposition 3.1. A preclosure space X is an exponential object in PrCls if and only if X is indiscrete. Proof. Suppose µ is an admissible PrCls-structure on PrCls(X, 3). Take x ∈ X and V ∈VX(x). Then f : X → 3 defined by f−1(1) = {x},f−1({1, 2}) = V is a PrCls-morphism. By continuity of the evaluation map ev in (x,f) there exists B ∈Vµ(f) such that ev(X ×B) ⊂{1, 2}. This implies X = f−1({1, 2}) The quasitopos hull of Cls 19 and so V = X. If X is indiscrete, then the PrCls-structure µ on PrCls(X, 3) given by Vµ(f) = {PrCls(X, 3)} if f−1(1) = ∅ and Vµ(f) = stack{{g ∈ PrCls(X, 3); g(X) ⊂{1, 2}}} if f−1(1) 6= ∅ is admissible and proper. � We first recall the definitions of CCT hull, multimorphism, strictly dense subcategory, power-closed collection and the construction of the CCT hull pre- sented by J. Adámek, J. Reiterman and G. E. Strecker [2]. Then we use this method to construct the CCT hull of PrCls. Definition 3.2. [12] A cartesian closed topological construct B is called a cartesian closed topological hull (CCT hull) of a construct A if B is a finally dense extension of A with the property that any finally dense embedding of A into a cartesian closed topological construct can be uniquely extended to B. Definition 3.3. [2] Let K be a construct and let H,K be K-objects and X a set. A function h : X ×H → K is called a multimorphism if for each x ∈ X, h(x,−) : H → K defined by h(x,−)(y) = h(x,y) is a morphism. Definition 3.4. [2] Let K be a construct with quotients and finite products. A full subcategory H of K is said to be strictly dense in K provided that : (1) for each object K ∈ |K| there exists a productively final sink (Hi hi→ K)i∈I with Hi ∈ |H|, i.e., a final sink such that for each L ∈ |K| the sink (Hi ×L hi×1L−→ K ×L)i∈I is final as well. (2) H is well-fibred, closed under quotients, and has productive quotients (i.e., for each quotient e : A → B with A ∈ |H|, we have B ∈ |H| and e× 1H : A×H → B ×H is a quotient for each H ∈ |H|). Definition 3.5. [2] Let K be a construct with quotients and finite products and let H be strictly dense in K. A collection A of H-objects (A,α) with A ⊂ X is said to be power-closed in X provided that A contains each H-object (A0,α0), A0 ⊂ X, with the following property: Given a multimorphism h : X ×H → K with H ∈ |H| and K ∈ |K| such that for each (A,α) ∈ A the restriction h|A : (A,α) ×H → K is a morphism, then the restriction h|A0 : (A0,α0) ×H → K is also a morphism. We denote by PCH(K) the category of power-closed collections in H. Ob- jects are pairs (X, A), where X is a set and A is a power-closed collection of H-objects in X. Morphisms f : (X, A) → (Y, B) are functions from X to Y such that for each (A,α) ∈ A the final object of the restriction fA : (A,α) → f(A) is in B. If H = K then we simply write PC(K). Theorem 3.6. [2] Any construct K which has quotients and finite products that are preserved by the forgetful functor, and which has a strictly dense subcategory H, has a CCT hull. Moreover, this hull is precisely the category of power-closed collections in H. Proposition 3.7. In PrCls arbitrary products of quotients are quotients. 20 V. Claes and G. Sonck Proof. Let Xi fi→ Yi be a quotient in PrCls for any i ∈ I, which means that for every Ai ⊂ Yi : clYiAi = fi(clXi(f −1 i (Ai))). Let f = ∏ i∈I fi : ∏ i∈I Xi → ∏ i∈I Yi and A ⊂ ∏ i∈I Yi, (yi)i∈I ∈ clA. Since all fi : Xi → Yi are quotients, we have: yi ∈ clYi(prYiA) = fi(clXif −1 i (prYiA)) = fi(clXiprXi(f −1(A)) for all i ∈ I. This implies: (yi)i∈I ∈ f(clf−1(A)). � From the previous proposition follows that the construct PrCls is strictly dense in itself. Our aim is to give an explicit description of the objects of the construct PC(PrCls) in terms of preclosures. Proposition 3.8. If X is a set and C is a power-closed collection in X, then there exists a unique collection A⊂P(X) and for all A ∈A a unique PrCls- structure αA on A such that C = {(A,β) ∈ |PrCls| : A ∈A, αA 6 β}. Proof. For A we take the set of underlying sets of objects in C and for A ∈A we set αA the final PrCls-structure on A for the sink (1A : (A,β) → A)(A,β)∈C. It remains to prove that (A,αA) ∈ C for all A ∈ A. Take A ∈ A and take a multimorphism h : X × (H,γ) → 3, with (H,γ) ∈ |PrCls|, such that for all (C,δ) ∈ C the restriction h|C : (C,δ) × (H,γ) → 3 is a PrCls-morphism. Then h|A : (A,β) × (H,γ) → 3 is a PrCls-morphism for all (A,β) ∈ C with underlying set A. So if (a,y) ∈ A × H satisfies h(a,y) = 1, (h|A)−1({1, 2}) is a neighborhood of (a,y) in (A,β) × (H,γ) for all (A,β) ∈ C, and one of the following cases arises: (1) ∃(A,β) ∈ C, ∃W ∈Vγ(y) with A×W ⊂ (h|A)−1({1, 2}). (2) ∀(A,β) ∈ C, ∃Vβ ∈Vγ(a) with Vβ ×H ⊂ (h|A)−1({1, 2}). In case (1), A×W is a neighborhood of (a,y) in (A,αA) × (H,γ). In case (2), V = ⋃ {Vβ | (A,β) ∈ C} is a neighborhood of a in αA and V ×H ⊂ (h|A)−1({1, 2}). We conclude that h|A : (A,αA) × (H,γ) → 3 is a PrCls-morphism. � Proposition 3.9. With the notation of Proposition 3.8 the collection A has the properties (A1) ∀x ∈ X, {x}∈A, (A2) A′ ⊂ A ∈A =⇒ A′ ∈A, and the PrCls-structures αA satisfy (B1) A′ ⊂ A ∈A =⇒ αA′ 6 αA|A′, (B2) ∀A ∈A, ∀x ∈ A, ∀V ∈VαA(x), ∃T ∈VαA(x) with T ⊂ V and {x}∈Vα(A\T)∪{x}(x). Proof. For the difficult part take x ∈ A ∈ A and V ∈ VαA(x) and suppose V 6= A (otherwise we can take T = V = A). A PrCls-structure µ on A is given by Vµ(x) = VαA(x) \{W ∈P(A) | W ⊂ V}, Vµ(y) = VαA(y) if y 6= x. The quasitopos hull of Cls 21 Clearly (A,µ) does not belong to C and so there exists a multimorphism h : X × (H,γ) → 3, with (H,γ) ∈ |PrCls|, such that for all (C,δ) ∈ C, h|C : (C,δ) × (H,γ) → 3 is a PrCls-morphism and h|A : (A,µ) × (H,γ) → 3 is not a PrCls-morphism. There exists y ∈ H such that h(x,y) = 1 and h|A : (A,µ)×(H,γ) → 3 is not continuous at (x,y). Since h|A : (A,αA)×(H,γ) → 3 is continuous one of the following cases arises: (1) ∃W ∈Vγ(y) with A×W ⊂ (h|A)−1({1, 2}). (2) ∃T ′ ∈VαA(x) with T ′ ×H ⊂ (h|A)−1({1, 2}). Since h|A : (A,µ) × (H,γ) → 3 is not continuous at (x,y) we can suppose (2). Then T = {t ∈ A | {t}×H ⊂ (h|A)−1({1, 2})} belongs to VαA(x) and satisfies T ×H ⊂ (h|A)−1({1, 2}) and T ⊂ V . Now h|(A\T)∪{x} : ((A\T) ∪{x},α(A\T)∪{x}) × (H,γ) → 3 is continuous at (x,y) and so one of the following cases arises: (1) ∃W ′ ∈Vγ(y) with ((A\T) ∪{x}) ×W ′ ⊂ (h|(A\T)∪{x})−1({1, 2}). (2) ∃T ′′ ∈Vα(A\T)∪{x}(x) with T ′′ ×H ⊂ (h|(A\T)∪{x})−1({1, 2}). In the first case A×W ′ would be a neighborhood of (x,y) in (A,µ) × (H,γ) with A×W ′ ⊂ (h|A)−1({1, 2}), so we can suppose (2). For T ′′ we have T ′′ ⊂ (A\T) ∪{x} as well as T ′′ ⊂ T , so T ′′ = {x}. � Proposition 3.10. If X is a set, A⊂P(X) a collection of subsets satisfying the conditions (A1 ) and (A2 ); and if for each A ∈ A a PrCls-structure αA is given such that (B1 ) and (B2 ) are satisfied, then the set C = {(A,β) ∈ |PrCls| : A ∈A, αA 6 β} is a power-closed collection in X. Proof. Take a PrCls-object (A0,α0) with A0 ⊂ X that does not belong to C. Then either A0 /∈ A or both A0 ∈ A and αA0 66 α0. In both cases we give a multimorphism h : X × (H,γ) → 3, with (H,γ) ∈ |PrCls|, such that for all (A,β) ∈ C, h|A : (A,β) × (H,γ) → 3 is a morphism and h|A0 : (A0,α0) × (H,γ) → 3 is not a morphism. First suppose A0 /∈ A and take x0 ∈ A0. The PrCls-object (H,γ) and multi- morphism h : X × (H,γ) → 3 are defined as follows: H = A∪{∞} (∞ /∈A), Vγ(y) = {H} if y ∈A, Vγ(∞) = stackH{{∞,A} | x0 ∈ A ∈A}, h : X ×H → 3 : (x,y) 7→   1 if (x,y) = (x0,∞), 2 if (x,y) ∈ ⋃ x0∈A∈A A×{A,∞}\{(x0,∞)}, 0 in all other cases. For all (A,β) ∈ C, h|A : (A,β) × (H,γ) → 3 is continuous since if x0 ∈ A the neighborhood A×{A,∞} of (x0,∞) in (A,β) × (H,γ) satisfies A×{A,∞}⊂ (h|A)−1({1, 2}). However h|A0 : (A0,α0) × (H,γ) → 3 is not continuous at (x0,∞) since for V ∈ Vα0 (x0) we have V × H 6⊂ (h|A0 )−1({1, 2}) (because (x0,φ) ∈ V ×H but h(x0,φ) = 0) and for x0 ∈ A ∈A we have A0 ×{A,∞} 6⊂ (h|A0 )−1({1, 2}) (for x ∈ A0 \A the pair (x,A) satisfies (x,A) ∈ A0 ×{A,∞} and h(x,A) = 0). Now suppose A0 ∈ A and αA0 66 α0. Then there exist 22 V. Claes and G. Sonck x0 ∈ A0 and V ∈ VαA0 (x0) \Vα0 (x0). Take T ∈ VαA0 (x0), T ⊂ V such that {x0} ∈ VαA0\T∪{x0}(x0). If (A0 \T) ∪{x0} ⊂ A ∈ A, then using (B1) we can choose VA ∈VαA(x0) with VA∩((A0 \T)∪{x0}) = {x0}. So we can define the map h : X ×H → 3 (with (H,γ) as in the case A0 /∈A) as follows: h(x,y) =   1 if (x,y) = (x0,∞), 2 if (x,y) ∈ ⋃ {VA ×H | (A0 \T) ∪{x0}⊂ A ∈A}∪⋃ {A×{A,∞}\{(x0,∞)} | x0 ∈ A ∈A, (A0 \T) ∪{x0} 6⊂ A}, 0 in all other cases. Then h : X × (H,γ) → 3 is a multimorphism. Now for all (A,β) ∈ C, h|A : (A,β)×(H,γ) → 3 is a PrCls-morphism: if (A0\T)∪{x0}⊂ A then VA×H is a neighborhood of (x0,∞) in (A,β)×(H,γ) and VA×H ⊂ (h|A)−1({1, 2}) and if (A0\T)∪{x0} 6⊂A and x0 ∈ A then A×{A,∞} is a neighborhood of (x0,∞) in (A,β)×(H,γ) which is contained in (h|A)−1({1, 2}). Now we only have to prove that h|A0 : (A0,α0) × (H,γ) → 3 is not continuous at (x0,∞). Therefore we show that for x0 ∈ A ∈A, A0×{A,∞} 6⊂ (h|A0 )−1({1, 2}) and for W ∈Vα0 (x0) we have W ×H 6⊂ (h|A0 )−1({1, 2}). Take x0 ∈ A ∈A. If (A0 \T) ∪{x0}⊂ A then for x ∈ A0\T we have h(x,A) = 0. If (A0\T)∪{x0} 6⊂ A then h(x,A) = 0 for x ∈ ((A0 \T) ∪{x0}) \A. Finally take W ∈Vα0 (x0). Then h(x,{x0}) = 0 for x ∈ W \T . � Definition 3.11. [21] A quasitopos B is called a quasitopos hull of a construct A if B is a finally dense extension of A with the property that any finally dense embedding of A into a quasitopos can be uniquely extended to B. F. Schwarz [21] proved that the quasitopos hull of a construct (if it exists) can be described as the cartesian closed topological hull of the extensional topological hull. Using this we have the following proposition. Proposition 3.12. The quasitopos hull of Cls is the construct which has as objects the pairs (X,{(A,β) ∈ |PrCls| : A ∈ A,αA 6 β}), where X is a set, A ⊂ P(X) is a collection of subsets of X satisfying (A1 ) and (A2 ) (see Proposition 3.9 ), and for each A ∈A, αA is a PrCls-structure on A such that the properties (B1 ) and (B2 ) (see Proposition 3.9 ) are fulfilled. Morphisms f : (X, A) → (Y, B) are functions f : X → Y such that for each (A,β) ∈ A the final PrCls-object of the restriction f|A : (A,β) → f(A) is in B. In [3] we constructed the cartesian closed topological hull of Cls. This construct was denoted by L∗. The following diagram shows the bireflective inclusions into the three hulls of Cls under discussion. 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Schwarz, Hereditary topological categories and topological universes, Quaest. Math. 10 (1986), 197–216. 24 V. Claes and G. Sonck [21] F. Schwarz, Description of the topological universe hull, in: H. Ehrig et al. (eds.), Categorical Methods in Computer Science with Aspects from Topology (Proc. Berlin 1988), Lecture Notes Computer Science 393 (Springer, Berlin, 1989), 325–332. [22] O. Wyler, Are there topoi in topology?, in: E. Binz and H. Herrlich (eds.), Categorical Topology (Proc. Mannheim 1975), Lecture Notes Math. 540 (Springer, Berlin, 1976), 699–719. Received December 2001 Revised December 2002 V. Claes and G. Sonck Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium. E-mail address : vclaes@vub.ac.be ggsonck@vub.ac.be The quasitopos hull of the construct of closure spaces. By V. Claes and G. Sonck