Categorical foundations of topology with applications to quantaloid enriched topological spaces

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Categorical foundations of topology with applications to quantaloid enriched topological spaces夡 Ulrich Höhle∗ Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität, Gaußstraße 20, D-42097 Wuppertal, Germany

Abstract Based on ordered monads this paper uncovers the categorical basis of topology in terms of a categorical formulation of neighborhood axioms. Here dense subobjects, lower separation axioms and regularity receive a purely categorical representation. In the case of appropriate submonads of the double presheaf monad this theory is applied to quantaloid-enriched topological spaces which form a common framework for many valued topology as well as for non-commutative topology. As an illumination of this situation two examples are given: the first one is chosen from probability theory and has the following characteristics: Weak convergence of -smooth probability measures is topological. The Hausdorff separation axiom is valid. Dirac measures form a dense subset. The second example is related to operator theory and explains the topologization of spectra of non-commutative C ∗ -algebras. © 2013 Elsevier B.V. All rights reserved. Keywords: Ordered monad; Categorical neighborhood axiom; Lower separation axiom; Regularity; Quantaloid enriched category; Quantaloid-enriched topology; Many valued topology; Non-commutative topology; Smooth Borel probability measure; Spectrum of C ∗ -algebra

1. Introduction In the history of general topology the discovery of its monadic basis happens fairly late. It is Manes who proves in his PhD-thesis 1967 that compact Hausdorff spaces and algebras of the ultrafilter monad are equivalent concepts (cf. [30]). Three years later, as an extension of Manes’ Theorem, Barr establishes the equivalence between topological spaces in general and relational algebras of the ultrafilter monad (cf. [3]). Since that time this monadic basis is a solid part of general topology. In this paper we explain two things: First we recall ordered monads as a categorical framework for general topology (cf. [17]) and show that neighborhood systems are equivalent to principal relational algebras. As an illustration of this theory we develop basic topological notions, such as density, separation axioms, regularity, by purely categorical terms. In this context we emphasize that the formulation of the regularity axiom seems to require an enrichment of the underlying ordered monad and is based on an additional property which guarantees the existence of certain left adjoints. Second, for any small quantaloid Q we introduce the double presheaf monad on the category of Q-enriched categories. This monad is ordered and satisfies the additional property mentioned above. In this framework neighborhood systems on a Q-category X can be characterized by “topologies”— these are extremal subobjects of the free cocompletion of X 夡 In memory of Gisela Höhle. ∗ Tel.: +49 57314989762.

E-mail address: [email protected]. 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.03.010 Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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and satisfy the important property of being internally closed under arbitrary joins. Hence “topologies” on X are right Q-submodules of the right Q-module of contravariant presheaves on X and vice versa. In order to capture traditional axioms of topology it is necessary to study specific submonads of the double presheaf monad. The intersection axiom and the non-emptiness of neighborhood systems lead to the submonad of Q-enriched filters. Here, the indispensable generality of the intersection axiom demands to impose a premultiplication on the underlying quantaloid Q. Then neighborhood systems on Q-categories can be characterized by Q-enriched topologies — these are certain right Q-submodules of the right Q-module of contravariant presheaves which satisfy all axioms of “open” contravariant presheaves (see Definition 6.8 and Remark 6.9). Further, we give two examples of Q-enriched topological spaces: the first one is related to probability theory and deals with the topologization of the Q-category of -smooth Borel probability measures. We construct a Q-enriched topology which characterizes weak convergence, is Hausdorff separated and satisfies the important property that Dirac measures are dense. The second example is related to operator theory and explains the construction of a Q-enriched topological space for the spectrum of a non-commutative C ∗ -algebra. If we disregard the tensor condition which is responsible for the internalization of arbitrary joins (cf. Remark 5.6(d)), then this Q-enriched topology can be understood as the non-commutative topology characterizing the spectrum of non-commutative C ∗ -algebras. As illumination, we also give an explicit description of “openness” in the finite dimensional case. Various results in Sections 6 and 7 — e.g. the axioms of enriched topologies and the construction of the noncommutative topology determined by the spectrum of a non-commutative C*-algebra — have been obtained in a cooperation with Kubiak during summer 2011. Finally, we close this introduction with a comment on the presented material. Even though the reader should have some basic knowledge in category theory, we do not hesitate to include some well known facts from the theory of relations in abstract categories (see Section 2) and some fundamental properties of quantaloid-enriched categories (cf. Section 5). We hope that this approach makes this paper also accessible to an audience being not so familiar with enriched category theory. 2. Ordered monads We begin with a brief survey on ordered monads (see also [17]). For this purpose we fix a finitely complete (E, M)category C (cf. [14]) and assume that the class E of epimorphisms is stable under pullbacks. Because of the (E, M)factorization property the class M of monomorphisms is always stable under pullbacks and contains all equalizers (cf. [30, Chapter 3 4.12]). In order to fix notation we first recall the bicategory of relations in C. Let A and B be C-objects. A relation (, ) : A → B from A to B for M is a span      

R ? ??  ?? ?

A

B

, / A × B is contained in M. from A to B s.t. the universal arrow R If (, ) is a span from A to B, then the image arrow  ,   of ,  in the sense of the (E, M)-factorization property ,

R ? ?? ?? ??  ?? ?? 

R

/ A×B ?      ,   ? 

with  ∈ E,  ,   ∈ M

determines always a relation ( ,  ) : A → B for M. Relations (1 , 1 ) : A → B and (2 , 2 ) : B → C can be composed. The composition is defined in two steps. First we compose the relation as spans — this means that we compute the pullback of the first component 2 Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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of the second relation along the second component 1 of the first relation: 2

S   1    R1

1

/ R2     2  / B

2

/C

1

 A The resulting pullback square determines arrows 1 and 2 . Second, we compute the image arrow (1 · 1 ) , (2 · 2 )  of (1 · 1 ), (2 · 2 ) in the sense of the (E, M)-factorization property. Then the composition (2 , 2 ) ◦ (1 , 1 ) of (1 , 1 ) with (2 , 2 ) has the form: (2 , 2 ) ◦ (1 , 1 ) = ((1 · 1 ) , (2 · 2 ) ).

(2.1)

Obviously, the composition of relations is unique up to an isomorphism. Further, on the class R(A, B) of relations from A to B there exists a preorder defined as follows. If 1

 

A

R1

?? 1 ??  B

and

2

 

A

R2

?? 2 ??  B

 / are relations, then (1 , 1 ) (2 , 2 ) iff there exists an arrow R1 R2 s.t. 1 = 2 ·  and 1 = 2 · . It is easily seen that the composition of relations preserves in both variables. Hence we can summarize the previous results as follows.

Theorem 2.1. Let C be M-wellpowered. Then objects of C and relations between them together with their composition and preorder form a bicategory Rel(C). Before we proceed, we recall the meaning of adjointness in Rel(C). For this purpose let (, ) be a relation from A to B and (, ) be a relation from B to A. Then (, ) is left adjoint to (, ) (notation: (, )⵬(, )) iff (id A , id A ) (, ) ◦ (, ) and (, ) ◦ (, ) (id B , id B ). Lemma 2.2. Let A



/ B be a C-morphism. Then (id A , )⵫(, id A ).

Proof. Let us consider the following commutative diagram: A/   /// //  //   //    // id id A  // A S ?   ?    ?? //     ?? // ?? /      ?? // ?? /     ? / o / A A ? R ??   ?   ?? ??   ? ?? id   ??  id A  ?? A  ?   ??  ??    ? ??   ??   ?      A B A Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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where the inner rhombus is a pullback diagram. Then (id A , id A ) (, ) = (, id A ) ◦ (id A , ) follows. On the other hand, if we decompose ,  into an epimorphism  ∈ E followed by a monomorphism  1 , 2  ∈ M, then 1 and 2 necessarily coincide. Hence (id A , ) ◦ (, id A ) (id B , id B ) holds trivially.  For later purposes we need the following theorem. 

Theorem 2.3. Let A

  

A

≤A

A

??  A ??  A



// B

and

be a pair of parallel C-morphisms, and let   

B

≤B

B

??  B ??  B

be relations for M. Then the following assertions are equivalent: (i) There exists a C-morphism ≤A ≤A     A , A      A× A



×



/ ≤B

making the following diagram commutative:

/ ≤B     ,   B B   / B×B

(ii) The relation ( A ,  A ) (, id A ) ◦ ( B ,  B ) ◦ (id A , ) holds. Proof. (a) First we compute (, id A ) ◦ (( B ,  B ) ◦ (id A , )). By definition of the composition of relations for M the following diagrams are commutative: X  O /// //  //    //  1 //  // //   m1  //m 2  X ? ?? 2 ///  1  ??  // ??   //     / ≤ A B  ?? ///    ???? ?? /

B  ?? ?? /    ?  B ? /  id A  A B B

with 1 ∈ E

Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Z  O /// //  //   2 // //   //   // n 1  //n 2 Z ?  ?? 2 //  ??   1 ?? ///    //   X  A ? // ? ?? / ?? m 2     ?? // ??     m 1 ? id A ?? /  A B A

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with 2 ∈ E

where the respective inner rhombus is a pullback square.  / ≤B with  · A = (b) ((i) ⇒ (ii)) If we assume that assertion (i) holds — i.e. there exists an arrow ≤A

B ·  and  ·  A =  B · , then we conclude from the universal property of the pullback in the first diagram that there 1 / X with A = 1 · 1 and  = 2 · 1 . Hence the relation exists an arrow ≤A m 2 · 1 · 1 =  B · 2 · 1 =  B ·  =  ·  A follows. Because of the universal property of the pullback square in the second diagram there exists an arrow 2 / Z with the properties: ≤A 1 · 1 = 1 · 2 and 2 · 2 =  A . Finally, we obtain n 1 · (2 · 2 ) = m 1 · 1 · 2 = m 1 · 1 · 1 = 1 · 1 = A , n 2 · (2 · 2 ) = 2 · 2 =  A . Hence assertion (ii) holds. (c) ((ii) ⇒ (i)) Let us assume that there exists an arrow ≤A let 1 be the pullback of 1 along 1 — i.e. Z   1

 X

1



/ Z  with A = n 1 ·  and  A = n 2 · . Further,

// Z 1

1

 / / X

Then we consider the following commutative diagram: ≤A O



/ Z O

2

≤A O

2  

1

3

≤A

/ Z O

 

/ Z 

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where the inner squares are pullback squares. Since E is stable under pullbacks, the C-morphisms 1 , 2 and 3 are elements of E. Further, we obtain  · A · 2 · 3 =  · n 1 ·  · 2 · 3 =  · n 1 · 2 · 1 ·   =  · m 1 · 1 · 1 ·   =  · m 1 · 1 ·  1 ·   =  · 1 ·  1 ·   = B · ( 2 ·  1 ·  ),  ·  A · 2 · 3 =  · n 2 ·  · 2 · 3 =  · n 2 · 2 · 1 ·   =  · 2 · 1 ·   = m 2 · 1 · 1 ·   = m 2 · 1 ·  1 ·   =  B · ( 2 ·  1 ·  ), Hence the relation ( × ) ·  A ,  A  · ( 2 · 3 ) =  B ,  B  · ( 2 ·  1 ·  )

(2.2)

1 ·   and obtain the following holds. Further, we apply the (E, M)-factorization property to ( × ) ·  A ,  A  and 2 ·  commutative diagrams:

Since 2 · 3 ∈ E and  B ,  B  ∈ M, we conclude from (2.2) and the uniqueness of the (E, M)-factorization that there  / exists an isomorphism C1 C2 making the following diagram commutative:

Hence assertion (i) follows.  Based on the previous terminology and results we recall the concept of partially ordered objects in C. Definition 2.4. Let A be an object of C. Then a pair of parallel C-morphisms ≤ partial ordering on A in the sense of the (E, M)-category C iff ≤



//



//

A

with codomain A is a

A satisfies the following conditions:

(P0) ( , ) : A → A is a relation for M. (P1) The relation ( , ) : A → A is reflexive and transitive (cf. [26, p. 95]) — i.e. (1 A , 1 A ) ( , ), ( , ) ◦ ( , ) ( , ). (P2) The relation ( , ) : A → A is antisymmetric — i.e. the diagonal of the pullback square A / 

 ≤/

 id A , id A 

, 

 / A× A '

 ,

/≤ 

coincides with the diagonal of A. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

U. Höhle / Fuzzy Sets and Systems

It is well known that a pair of parallel C-arrows ≤ and for every C-object X the binary relation  

⇐⇒

∃ X



/≤



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A is a partial ordering on A iff  ,  is contained in M

s.t.  = ·  and  =  · 

(2.3)

is an ordinary partial ordering on hom(X, A) (cf. [17, Proposition 2.1.1]). Moreover, with regard to internal posets (cf. [15]) a simple application of Theorem 2.3 says that the ordinary binary relation specified in (2.3) is equivalent to (id A , id A ) (, id A ) ◦ ( B ,  B ) ◦ (id A , ). A triple (A, A ,  A ) is called an ordered object in C if A is a C-object and ( A ,  A ) is a partial ordering on A. Let  / B is isotone iff there exists an (A, A ,  A ) and (B, B ,  B ) be partially ordered objects. A C-morphism A  / ≤B making the following diagram commutative: arrow ≤A  / ≤B ≤A       ,    A , A    B B      / A × A × B × B

Because of Theorem 2.3 a C-morphism  is isotone iff the following relation holds: ( A ,  A ) (, id A ) ◦ ( B ,  B ) ◦ (id A , ). Hence isotonicity means order preservation. In this context one can make two simple observations: First partially ordered objects and isotone morphisms form a category denoted by Po(C). Second, the composition of isotone morphisms preserves the partial ordering defined in (2.3). Hence we have the following proposition. Proposition 2.5. Partially ordered objects, isotone morphisms and the partial orderings between morphisms form a 2-category.  Comment. Since regular categories (cf. [11]) are (extremal epi, mono)-categories, it is interesting to see that in the case of regular categories the assertion of Proposition 2.5 also appears in Section 2.2 in [15]. After these preliminaries we are ready to lay down the axioms of an ordered monad. First the forgetful functor from Po(C) to C is denoted by F. An endofunctor T of C factors through Po(C) iff there exists a functor S from C to Po(C) such that the composition of S with F coincides with T. Unfortunately, S is in general not uniquely determined by T. Therefore we use the following convention: If T factors through Po(C), then we assume that a functor S with F · S = T is fixed. For the sake of simplicity we will not distinguish between T (X ) and S(X ) where X is an object of C. Now we fix a monad T = (T, , ) on C. First we recall the Kleisli composition C(Y, T (Z )) × C(X, T (Y ))

◦

/ C(X, T (Z ))

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/ T (Y ) and Y / T (Z ) are arrows in C, then the Kleisli composition of  with  is an of T (cf. [30]). If X arrow  ◦  from X to T (Z ) defined as the composition of the following arrows: X



/ T (Y )

T ( )

◦

/ T 2 (Z )     Z  '  T (Z )

Definition 2.6. A monad T on C is said to be ordered iff T satisfies two properties: (1) The endofunctor T factors through Po(C). (2) The Kleisli composition of T is isotone in both variables separately. Obviously, the Kleisli category CT (cf. [30]) associated with an ordered monad T is always a 2-category. Hence for 1

every pair of C-arrows Z 

2

// T (X ) the implication



1 2 ⇒ 1 2 

holds where we have used the well known fact i = i ◦ 1T (Z ) (i = 1, 2). Before we proceed, we make a brief historical digression. Remark 2.7. (a) Ordered monads (more precisely preordered monads) on the category Set of sets appeared first in Werner Gähler’s paper on Monads and convergence (cf. [12,13]). (b) Ordered monads on finitely complete (epi, extremal mono)-categories appeared in [17]. (c) Monads on Set factoring coherently through the category of partially ordered sets (cf. [39]) and ordered monads on Set in the sense of Definition 2.6 are the same concepts. In the following considerations we will see that each ordered monad T forms an appropriate framework for the study of a principal relational T-algebra. First we recall the concept of relational T-algebras introduced by Barr 1970. Let T be a (not necessarily ordered) monad on C, and let X be an object of C. The triple (X, , ) is a relational T-prealgebra if (, ) is a relation from T (X ) to X for M. A relational T-prealgebra is a relational T-algebra (or lax T-algebra in a more modern terminology) iff the relation (, ) satisfies the following axioms (cf. [3]): (A1) (id X , id X ) (, ) ◦ (id X ,  X ), (A2) (, ) ◦ (T () , T () ) (, ) ◦ (1T 2 (X ) ,  X ) where ◦ denotes the composition of relations and T () , T ()  is the image arrow of T (), T () in the sense of the (E, M)-factorization property. With regard to later applications we need a characterization of the previous axioms (A1) and (A2). For this purpose we introduce the coextension of relations. Let (, ) be a relation from T (A) to B. The coextension (, ) : T (A) → T (B) of (, ) is determined by the image arrow of  , T () w.r.t. the (E, M)-factorization property. The next lemma explains that the coextension of relations can be written as a composition of two relations. Lemma 2.8. Let (, ) be a relation form T (A) to B and T () , T ()  be the image arrow of T (), T () in the sense of the (E, M)-factorization property. Then the following relation holds: (, ) = (T () , T () ) ◦ ( A , idT 2 (A) ). Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Proof. First we consider the (E, M)-factorization of T (), T () — i.e. T (R)

T (),T ()

JJ JJ J  JJ$ $

C

/ T 2 (A) × T (B) t: tt tt t :t T () ,T () 

with  ∈ E.

Then we put (, ) = (T () , T () ) ◦ ( A , idT 2 (A) ) and observe that the following diagram is commutative: D / O    ///   /    ///  //    //   //    //   //   //   //    // CO ?   //  T ()   ??? // ? id  C   ?   ??   //   ??  //    ?   /  //   T 2 (A) o T () T (R) C ? ///  ?? ?? /    ?? ?? //  ??     ? /   T () T () ???//    A idT 2 (A) ???  ? /     2 T (A) T (B) T (A) Hence  A · T (), T () = ,  ·  ·  is the (E, M)-decomposition of  , T ().  The following result is a categorical abstraction of Proposition 2.2 in [3]. Theorem 2.9. Let T = (T, , ) be a monad on C and (X, , ) be a relational T-prealgebra. Then the following equivalences hold: (i) (, ) satisfies (A1) iff ( X , id X ) (, ). (ii) (, ) satisfies (A2) iff (, ) ◦ (, ) (, ). Proof. The proof of both equivalences is based on the adjoint situation formulated in Lemma 2.2. The details are as follows: (a) If (A1) holds, then we obtain: ( X , id X ) = (id X , id X ) ◦ ( X , id X ) (, ) ◦ (id X ,  X ) ◦ ( X , id X ) (, ). On the other hand, if we assume ( X , id X ) (, ), then (A1) can be verified as follows: (id X , id X ) ( X , id X ) ◦ (id X ,  X ) (, ) ◦ (id X ,  X ). (b) In order to prove the equivalence (ii) we make additionally use of Lemma 2.8. In fact, if (A2) holds, we obtain: (, ) ◦ (, ) (, ) ◦ (idT 2 (X ) ,  X ) ◦ ( X , idT 2 (X ) ) (, ). On the other hand, if assume (, ) ◦ (, ) (, ), then (A2) can be verified as follows: (, ) ◦ (T () , T () ) (, ) ◦ (, ) ◦ (idT 2 (X ) ,  X ) (, ) ◦ (idT 2 (X ) ,  X ).



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Finally, in the framework of an ordered monad T we introduce some special properties of relational T-prealgebras. The next lemma is a preparation for Definition 2.11. Lemma 2.10. Let (X, , ) be a relational T-prealgebra and ( T (X ) , T (X ) ) be the fixed partial ordering on T (X ) in  / T (X ) making the following the sense of the ordered monad T. Then there exists at most one C-morphism X diagram into a pullback square: R  ,    X × T (X ) 1

Proof. Let X

2

×idT (X )

/ ≤T (X )     T (X ) ,T (X )    / T (X ) × T (X )

(2.4)

// T (X ) be a pair of parallel C-morphisms making respectively the diagram (2.4) into a pullback

1 // square. Then we conclude from the reflexivity of ( T (X ) , T (X ) ) that there exist universal arrows X R 2 with the following properties:

provided

id X =  · 1 =  · 2 , 1 =  · 1 , 2 =  · 2 . Because of i ·   (i = 1, 2) we conclude from the previous relation  1 =  1 ·  ·  2  · 2 =  2 and 2 = 2 ·  · 1  · 1 = 1 . Hence 1 = 2 follows from the anti-symmetry of ( T (X ) , T (X ) ).  Definition 2.11. Let T be an ordered monad and (X, , ) be a relational T-prealgebra. Then (X, , ) is 1 // T (X ) with 1 2 the following implication holds: if 1 factors • isotone if for any pair of morphism S 2 through , then also 2 factors through   / T (X ) s.t. the diagram (2.4) is a pullback square. • principal if there exists a morphism X

Because of Lemma 2.10 the morphism  in Definition 2.11 is uniquely determined by (, ). Further, we conclude from the universal property of (2.4) and the transitivity of partial orderings that every principal relational T-prealgebra is isotone. Therefore the question arises under which condition does the converse hold? In this context we point out a result established by Hofmann and Tholen [16]: Let T F be the ordinary filter monad on Set. Then every isotone relational T F -algebra is principal. The proof of the previous statement depends heavily on the axiom of choice! 3. Topological T-spaces On the basis of ordered monads we specify the categorical axioms of neighborhood systems. Definition 3.1. Let T be an ordered monad on C and ◦ be the Kleisli composition in the sense of T. A neighborhood  / T (X ) satisfying the following axioms: system on a C-object X is a C-morphism X (N1)   X , (N2)   ◦ . Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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If  is a neighborhood system on X, then the pair (X, ) is called a topological T-space. It is interesting to see that topological T-spaces are structured objects in the sense of the Kleisli category CT associated with T. More precisely, if we view CT as 2-category, then topological T-spaces are comonads in the sense of CT (cf. [5]). In particular, the antisymmetry of the partial ordering on hom(X, T (X )) implies that every neighborhood system is idempotent w.r.t. the Kleisli composition.  / Y is continuous w.r.t. 1 and 2 if the Let (X, 1 ) and (Y, 2 ) be topological T-spaces. A C-morphism X following condition holds: (C) 2 ·  T () · 1 .

(Continuity)

If we make use of the Kleisli composition, then continuity can be characterized as follows: 2 ◦ (Y · ) (Y · ) ◦ 1 . Hence continuity is again a property which can be expressed by the data of CT . In particular, it follows immediately from the previous relation that topological T-spaces and continuous morphisms form a category denoted by Top(C). In the next theorem we explain the relationship between topological T-spaces and principal relational T-algebras. Theorem 3.2. Let (X, , ) be a principal relational T-prealgebra and X mined by the pullback (2.4). Then the following assertions hold:



/ T (X ) be the C-morphism deter-

(a)  satisfies (N1) iff (X, , ) satisfies (A1). (b)  satisfies (N2) iff (X, , ) satisfies (A2). Proof. Because of  ·   the equivalence of (N1) and (A1) follows immediately from Theorem 2.9(i) and the universal property of the pullback square in (2.4). In order to verify the equivalence between (N2) and (A2) we proceed as follows. Because of Theorem 2.9(ii) it is sufficient to prove the equivalence between (N2) and the relation (, ) ◦ (, ) (, ).

(3.1) ( ) , T () 

For this purpose we put (, ) = (, ) ◦ (, ) and recall that is the image arrow of sense of the (E, M)-factorization property. Then we consider the following commutative diagram: S  O /// //  //   //  //  //   //  //  S ? //   ?? //  ?   ??  //  ? 2  ?  1 //    R ? //  T (R) ?? ?? //    ??T ()  ?? //     ??   ?? /   ( )    T (X ) T (X ) X

 , T ()

in the

(3.2)

where ∈ E and the inner rhombus is a pullback diagram. Because of the (E, M)-factorization property it is sufficient  / to show that (3.1) is equivalent to the existence of a C-morphism S R provided with the properties  · 2 =  ·  and ( ) · 1 =  · .

(3.3)

(a) Let us assume that (N2) holds. Since the Kleisli composition is isotone,  ·   implies  · T ()  —  / ≤T (X ) with  · T (),   =  T (X ) , T (X )  · . Because of this means that there exists a morphism T (R) Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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T (),   = T () , ( )  · ,  ∈ E we conclude again from the (E, M)-factorization property that the relation  · T () ( ) holds where we have used the fact that  T (X ) , T (X )  is contained in M. Now we apply (N2)  ·  · 2  ·  ·  · 2  ·  · 2 =  · T () · 1 ( ) · 1 .  / and obtain  ·  · 2 ( ) · 1 . Hence the existence of a morphism S R satisfying (3.3) follows immediately from the universal property of diagram (2.4). (b) Let us assume the existence a morphism  satisfying (3.3). Since the partial ordering on T (X ) is reflexive, we  / R with 1 X = · and  = ·. conclude from the universal property of (2.4) that there exists a C-morphism X  / S provided Now we apply the universal property of the rhombus in the diagram (3.2) and obtain a morphism X

with the following properties:  = 2 ·  and  · T () ·  = 1 · . Finally, if we compose the right side of the equations in (3.3) with , then we get  ·  ·  =  · 2 ·  =  ·  = 1 X ,

(3.4)

 ·  ·  = ( ) · 1 ·  = ( ) ·  · T () ·  =  X · T () · T () ·  =  X · T () ·  =  ◦ .

(3.5)

Because of  ·   the neighborhood axiom (N2) follows from (3.4) and (3.5).  Since C is finitely complete, every C-morphism  from X to T (X ) determines a relational T-prealgebra (X, , ) s.t. the diagram R ,

 X × T (X )

×idT (X )

/ ≤T (X )    ,  T (X ) T (X )   / T (X ) × T (X )

(3.6)

is a pullback square. Hence the previous theorem says that topological T-spaces and principal relational T-algebras are equivalent concepts. Because of this situation principal relational T-algebras receive a special name and are called limit relations. In the next section we show that the categorical neighborhood axioms work as one expects. Hence the following theory can be viewed as a confirmation of the chosen axiomatization.

4. Lower separation axioms and regularity Let C be a finitely complete (E, M)-category s.t. E is stable under pullbacks. Further, let T = (T, , ) be an ordered monad in C. Then every neighborhood system  on a C-object X induces a preorder

s t

//

X on X

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(so-called specialization order) by means of the following pullback square: / ≤T (X ) 



 T (X ) ,T (X ) 

s,t

 X×X

× X

s / The reflexivity of t / X follows immediately from (N1). In order to verify the transitivity of first form the pullback square:

2

S 1



t

(4.1)

 / T (X ) × T (X ) s t

//

X we

/  / X

s

Then we make use of the triple product law (cf. [30]) and obtain the following relation from (N2) and the isotonicity of the Kleisli composition:  · s · 1 ( ◦ ) · s · 1 =  ◦ ( · s · 1 )  ◦ ( X · t · 1 ) = ( ◦  X ) · t · 1 =  · s · 2  X · t · 2 .  / with Hence the universal property of the pullback square in (4.1) implies the existence of a C-morphism S s ·  = s · 1 and t ·  = t · 2 — i.e. (s, t) ◦ (s, t) (s, t) where ◦ denotes here the composition of relations.

Definition 4.1. Let (X, ) be a topological T-space and (X, , ) be the corresponding limit relation. Then (X, ) is called • Kolmogoroff separated (or satisfies the T0 -axiom) if the specialization order on X is anti-symmetric (cf. (P2)) and thus a partial ordering on X, • Fréchet separated (or satisfies the T1 -axiom) if the diagram / ≤T (X )

X

 X , X 

1 X ,1 X 

 X×X

× X

 / T (X ) × T (X )

is a pullback square — i.e. the specialization order is discrete, • Hausdorff separated (or satisfies the T2 -axiom) if the first component  of the limit relation is a C-monomorphism. It is evident that the separations axioms introduced in Definition 4.1 depend on the structure of the underlying ordered monad. As an illustration of this fact we consider the following situation: let T F be the ordinary filter monad and T S be the semifilter monad — i.e. the submonad of the double power set monad which consists of non-empty families of non-empty subsets closed under the formation of super-sets. Then in the case of T F we have the traditional Hausdorff separation axiom, while in the case of T S the Hausdorff separation axiom is empty — this means that Hausdorff separated topological T S -spaces do not exist. After this digression we return to the general case. We begin with the following hierarchy of separation axioms. Lemma 4.2. T2 ⇒ T1 ⇒ T0 . Proof. The implication T1 ⇒ T0 is trivial. In order to verify T2 ⇒ T1 we proceed as follows. Because of (N1)  / ≤T (X ) making the diagram there exists a morphism X Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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X

(

)

–

/ ≤T (X )

1 X ,1 X 

 ,

 X×X

× X

(4.2)

 / T (X ) × T (X )

commutative. We show that the previous diagram is a pullback square. It is sufficient to verify the following implication:  · k1  X · k2 ⇒ k1 = k2 . k1

/ X and Z

k2

/ X with  · k1  X · k2 . The universal  / R s.t. k1 =  ·  and property of the pullback square in (2.4) guarantees the existence of a morphism Z  / R with 1 X =  ·  and  X =  · . Hence  X · k2 =  · . Again because of (N1) there exists a morphism X we obtain

Therefore we choose a pair of C-morphisms Z

 ·  =  X · k2 =  ·  · k2 .

(4.3)

Since  is monic, we infer from (4.3) that  =  · k2 holds. Thus the relation k1 =  ·  =  ·  · k2 = k2 follows.



The previous lemma motivates the question: Under which condition does T0 imply T2 ? The standard condition is here the regularity axiom. But, unfortunately for a categorical formulation of the regularity axiom we need an enrichment of the underlying ordered monad. Thus as an additional standing assumption we assume the following condition for the rest of the paper:  h / T (X ) has a right inverse, then the left adjoint of h  exists — If the extension h  of a C-morphism Z (R) i.e. the (unique) existence of an isotone C-morphism with idT (X ) h  · (h  )∗ and (h  )∗ · h  idT (Z ) .

Under the assumption of (R) we are now in the position to construct the closure operator associated with a neighborhood system. Because of (N1) we first observe that the extension  of the first component  of the limit relation has a right inverse. Hence  has a left adjoint, and the closure operator cl is defined by cl = T () · ( )∗ . Then the regularity axiom attains the following categorical formulation. Definition 4.3. A topological T-space is regular iff its neighborhood system  is invariant under the closure operator — i.e.  = cl · . The next remark shows that regularity in the previous sense coincides with the traditional concept of regularity in the case of the filter monad. Remark 4.4. Let T F be the filter monad and (X, ) be a topological T F -space — i.e. a topological space in the traditional sense. Further, let TF (X ) be the set of all (proper) filters on X and      TF (X ) 

R ? ??  ?? ? X

be the corresponding limit relation. Then TF (R)



/ TF (X ) has the form

 (G) = {A ⊆ X, |{r ∈ R | A ∈ (r )} ∈ G}, G ∈ TF (R). Since the neighborhood axiom (N1) implies that for every non-empty subset A of X the set {r ∈ R | A ∈ (r )} is also non-empty, the left adjoint of  is given by ( )∗ (F) = {B ⊆ R|∃A ∈ F, {r ∈ R | A ∈ (r )} ⊆ B}, F ∈ TF (X ). Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Further, we infer from the intersection axiom of filters that for any subset A of X the topological closure A of A coincides with set ({r ∈ R | A ∈ (r )}). Hence for any filter F on X we obtain cl (F) = {C ⊆ X | −1 (C) ∈ ( )∗ (F)} = {C ⊆ X | ∃A ∈ F, A ⊆ C}. Theorem 4.5. Let (X, ) be a regular topological T-space object. If (X, ) is Kolmogoroff separated, then (X, ) is also Hausdorff separated. Proof. Let (X, ) be a topological T-space object and (X, , ) be the corresponding limit relation. In order to verify 1 / R and U 2 / R, and show that 1 = 2 the T2 -axiom we consider the situation  · 1 =  · 2 with U holds. Since ,  is monic, it is sufficient to establish  · 1 =  · 2 .

(4.4)

First we use the fact  ·   and conclude from the regularity axiom:  ·  · 1 = cl ·  ·  · 1 cl ·  · 1 = cl ·  · 2 = T () · ( )∗ ·  ·  R · 2 T () ·  R · 2 =  X ·  · 2 . Hence the relation ··1  X ··2 holds. Interchanging the role of 1 and 2 we also obtain ··2  X ··1 . Because of the universal property of the pullback square in (2.4) there exists a pair of C-morphisms U provided with the following properties:

12 21

// X

 · 1 =  · 12 ,  X ·  · 2 =  · 12 ,  · 2 =  · 21 ,  X ·  · 1 =  · 21 . Because of the universal property of the pullback square     s,t    

X×X

1 X × X

/ R     ,   / X × T (X )

there exists a further pair of C-morphisms U

1 2

//

satisfying the following properties:

 · 1 = s ·  1 ,  · 2 = t ·  1 ,  · 2 = s ·  2 ,  ·  1 = t ·  2 . Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Finally, we invoke the T0 -axiom (i.e. the anti-symmetry of the specialization order) and conclude from the universal property of the pullback square U / ?OO // ??OOOOO // ??? OOOO OO // ??? OO2O // ?? OOO ?? OOO // ? OOO  // /' / X 1 // //  id X , id X  //  //  //  /   / X s,t

   t,s   ×X

that the relation  · 1 = s · 1 = t · 1 =  · 2 holds. Thus (4.4) is verified.  We finish this section with an application of the T2 -axiom. For this purpose we need the concept of density. Definition 4.6. Let (X, ) be a topological T-space. An M-subobject of X represented by U /  / T (U ) with  T (m) · . if there exists a C-morphism X

m

/ X is dense,

The next proposition shows that continuous morphisms are uniquely determined on dense M-subobjects provided their codomains are Hausdorff separated. Proposition 4.7. Let (X,  X ) be a topological T-space and U /

m

/ X a dense M-subobject, and (Y, Y ) a Haus1

dorff separated topological T-space. Then for continuous morphisms X

2

//

Y

the following implication holds:

1 · m = 2 · m ⇒ 1 = 2 . m /  / T (U ) Proof. Since the M-subobject of X represented by U / X is dense, there exists a morphism X with  X T (m) · . Then we infer from the continuity of i (i = 1, 2) that the following relation holds:

Y · i T (i · m) · . Now let (, ) be the limit relation corresponding to Y . Because of the universal property of the pullback (3.6) for each i ∈ {1, 2} there exists a unique arrow i making the following diagram commutative: X 7TTTT 77 TTTTTT TTTT 77 TTTT 77 TTTT 77  TTTT 77 i TTTT TTTT 77 * $ 77 / ≤T (Y ) R i ,T (i ·m)· 77 77 77 77  , 77 , 77    / T (Y ) × T (Y ) Y × T (Y ) Y ×idT (Y )

Finally, we assume 1 · m = 2 · m. Since  is a monomorphism, 1 = 2 follows. Hence 1 and 2 coincide.  Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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5. The double presheaf monad In the following consideration we will construct the double presheaf monad on the category of quantaloid-enriched categories. We show that this monad is ordered in a natural way and satisfies condition (R). Hence topological T-spaces in the sense of the double presheaf monad exist. In order to fix notation we begin with some basic definitions and properties from the theory of quantaloids (cf. [36]). Because of the relationship between small quantaloids and many-valued logics we think of the composition as a binary algebraic operation and use a notation for right- and left-implication which has its historical roots in Lambeck’s work in linguistics 1958. Let Q be a small quantaloid; this means that Q is a small category provided with the following additional properties: • each hom-set is a complete lattice, • the composition · of morphisms preserves arbitrary joins in both variables. Objects (resp. morphisms) of Q are denoted by small Roman (resp. Greek) letters. Further, we write Q(a, b) for the hom-set hom(a, b). By the special adjoint functor theorem, for every morphism a o b both maps · : Q(c, b) → Q(c, a) and

· : Q(a, c) → Q(b, c)

have right adjoints  and  which are determined by    = { ∈ Q(c, b)| ·  ≤ },   = { ∈ Q(a, c)| · ≤ }. It is easily seen that the following relations hold: (  ) · (  ) ≤   and (  ) · (  ) ≤   . Further, Q(a, a) is always a unital quantale. In this sense quantaloids can be viewed as “varying unital quantales”. The unit of Q(a, a) is always denoted by 1a . For concepts not defined herein we refer to [34,36]. e / d / obj(Q) and X × X A Q-category is a triple X = (X, e, d) where X is a set, X mor(Q) are maps subjected to the following axioms for all x, y, z ∈ X (cf. [7,45]): (Q1) d(x, y) ∈ Q(e(y), e(x)), (Q2) 1e(x) ≤ d(x, x), (Q3) d(x, y) · d(y, z) ≤ d(x, z). A Q-category X is skeletal if for all x, y ∈ X the following implication holds: e(x) = e(y) and 1e(x) ≤ d(x, y) ∧ d(y, x) imply x = y. Every Q-category X has an underlying preorder determined by x y

⇐⇒

e(x) = e(y) and 1e(x) ≤ d(x, y).

(5.1)

Since a kind of basic functor (cf. [9]) is involved in the previous construction, we attach a special name to the preorder defined in (5.1) and call the intrinsic preorder of the Q-category X. In this context, the map d can be understood as a Q-enrichment of its underlying intrinsic preorder. A Q-category is skeletal iff its intrinsic preorder is antisymmetric. Hence from an order-theoretic point of view the property of being skeletal can be interpreted as an antisymmetry axiom. Before we turn to morphisms between Q-categories, we recall two strategic examples of Q-categories which can be found in [42]. Example 5.1. With every object a of Q one can associate two Q-categories Sa = (Sa , ea , da ) and Sa = (S a , ea , d a ) as follows: • Sa is the set of all morphisms of Q with dom( ) = a, ea ( ) = codom( ) and da ( 1 , 2 ) = 1  2 . • S a is the set of all morphisms of Q with codom( ) = a, ea ( ) = dom( ) and d a ( 1 , 2 ) = 1  2 . Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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It is easily seen that Sa and Sa are skeletal. Moreover, the intrinsic partial ordering of Sa restricted to the fiber over b coincides with the partial ordering on the hom-set Q(b, a) of the underlying quantaloid. In contrast to Sa the intrinsic partial ordering of Sa restricted to the fiber over b coincides with the opposite partial ordering of the hom-set Q(a, b). This observation will be of some importance, when the double presheaf monad will be provided with an appropriate partial ordering. If X = (X, e, d) is a Q-category, then X is called the carrier of X. In order to avoid confusion we sometimes add an index to e and d referring to the carrier X.  / Y is a map X  / Y satisfying the following axioms for all x, y ∈ X : A Q-functor X (M1) e X (x) = eY ((x)), (M2) d X (x, y) ≤ dY ((x), (y)). In an obvious way Q-categories and Q-functors form a category denoted by Cat(Q). Then the following properties hold: • Q-functors are always isotone maps w.r.t. the intrinsic preorder. • Cat(Q) is a complete and cocomplete, wellpowered category. Hence Cat(Q) is an (epi, extremal mono)category. • A Q-functor  is an epimorphism if  viewed as a map is surjective. Hence epimorphisms are stable under pullbacks. • A Q-functor  is an extremal monomorphism iff  viewed as a map is injective and satisfies the following condition: (EM) d X (x1 , x2 ) = dY ((x1 ), (x2 )), x1 , x2 ∈ X . • Cat(Q) is a 2-category.

(Fully Faithfulness)

In particular, the terminal object 1 = (X 1 , e1 , d1 ) in Cat(Q) has the form: X 1 = obj(Q), e1 = idobj(Q) , d1 (a, b) =



Q(b, a).

The next remark recalls the concept of Q-enriched presheaves. Remark 5.2. Presheaves of type a are Q-functors taking their values either in Sa or in Sa . A presheaf of type a is contravariant iff its codomain is Sa or it is covariant iff its codomain is Sa . Since in the absence of involutive quantaloids the concept of dual Q-categories does not exist, it is necessary to make some comments on the chosen terminology. The first observation is that in the enriched setting we have hom-objects or hom-arrows, but not hom-sets. Hence we need something like a basic functor which removes the enrichment from hom-objects respectively homarrows. In the case of Q-categories this procedure leads to the underlying intrinsic preorder. Then in the presence of Sa the functor related to the enriched presheaf reverses the direction of arrows, while in the case of Sa the concerning functor preserves the direction of arrows. The last observation can be understood as a motivation to accept the previous use of adjectives like contravariant and covariant. The next proposition explains a characterization of presheaves. Proposition 5.3. (a) A map X conditions:

f

/ mor(Q) is contravariant presheaf on X of type a iff f satisfies the following

(PS1) f (x) ∈ Q(a, e(x)) for all x ∈ X . (PS2) d(y, x) · f (x) ≤ f (y) for all x, y ∈ X . (b) A map X

f

(Left Extensionality)

/ mor(Q) is covariant presheaf on X of type a iff f satisfies the following conditions:

(PS3) f (x) ∈ Q(e(x), a) for all x ∈ X . (PS4) f (x) · d(x, y) ≤ f (y) for all x, y ∈ X .

(Right Extensionality)

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Notation. In order to make the type of contravariant (resp. covariant) presheaves f explicit we write the pair (a, f ) (resp. ( f, a)) for f where the first (resp. second) component of this pair labels the type of f. After these preparations we recall monads of Q-enriched presheaves. Remark 5.4 (Monad of contravariant presheaves). We begin with the construction of the Q-category of contravariant presheaves (cf. [42]). For this purpose let X be a Q-category. On the set P (X) of all contravariant presheaves on X we introduce the structure (e , d ) of a Q-category as follows: e (a, f ) = a, d ((a, f ), (b, g)) =



f (x)  g(x).

x∈X

Then P (X) := (P (X), e , d ) is called the Q-category of contravariant presheaves on X. Obviously, P (X) is skeletal, and the intrinsic partial ordering on the carrier of P (X) can be characterized as follows: (a, f ) (b, g) ⇐⇒ a = b and f (x) ≤ g(x) for all x ∈ X

(5.2)

where ≤ denotes the partial ordering on the respective hom-sets of Q. It is well known that P (X) is the free cocompletion of X. This statement is a motivation to recall the monad of contravariant presheaves.  / Y, then First the object function X  P (X) can be completed to an endofunctor P as follows: if X P ()(a, f ) = (a,  ( f )) where the contravariant presheaf  ( f ) on Y is given by  ( f )(y) =



dY (y, (x)) · f (x),

y ∈ Y.

(5.3)

x∈X

Further, the unit  : idCat(Q) → P and the multiplication  : P · P → P are determined by X

X

/ P (X),

X (x) = (e X (x), x ) where x (y) = d X (y, x),

y, x ∈ X,

and P (P (X)) 

X

F(x) =

/ P (X),  (a, f )∈P (X)

X (b, F) = (b,



F) where

f (x) · F(a, f ), x ∈ X.

Then the triple TP = (P ,  ,  ) is a monad on Cat(Q) and is called the monad of contravariant presheaves. In this context we make two observations. First the left-extensionality of contravariant presheaves f of type a is equivalent to f (x) = d ((e X (x),  x ), (a, f )), x ∈ X. Hence, if X is skeletal, then there exists at most one Q-functor P (X) mutative:

X

X

/ X s.t. the following diagram is com-



/X ?      idX 

P (X) O



(5.4)

Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Secondly, if there exists a Q-functor P (X) commutativity of the subsequent diagram P (P (X))    X     P (X)

P (  )





(

)

–

/ X, then the commutativity of the diagram (5.4) implies the

/ P (X)        / P (X)

(5.5)

Hence TP -algebras can be characterized as follows: a Q-category is a carrier of a TP -algebra iff there exists a  / X (so-called structure morphism) making the diagram (5.4) commutative. Obviously, the Q-functor P (X) carrier of TP -algebra is always skeletal. A further characterization of TP -algebras is the following one. A skeletal Q-category X is a carrier of a TP -algebra iff X

X

/ P (X) has a left adjoint  — i.e.

d X ((a, f ), x) = d ((a, f ), X (x)), (a, f ) ∈ P (X), x ∈ X. A skeletal Q-category X is cocomplete iff X is a carrier of a TP -algebra. In this context the structure morphism  can be understood as the formation of arbitrary internal joins in X. A Q-functor between skeletal and cocomplete Q-categories is cocontinuous iff  is a TP -homomorphism — this means the commutativity of the following diagram: P (X)    X     X

P ()



/ P (Y)     Y   /Y

In the case of skeletal and cocomplete Q-categories X and Y a characterization of cocontinuity can be given as follows: ∗  / Y is cocontinuous iff  has a right adjoint Q-functor Y / X — i.e. a Q-functor X d X (x, ∗ (y)) = dY ((x), y), x ∈ X, y ∈ Y. For further details the reader is refer to [42]. Before we proceed, we give a characterization of skeletal and cocomplete Q-categories by right Q-modules. We begin with a definition. 

/ obj(Q) over obj(Q) is a right Q-module Definition 5.5 (cf. Rosenthal [35]). (a) A bundle X iff every fiber of  is a complete lattice and for every pair (a, b) of objects of Q there exists a right action ∗ / −1 −1 ({a}) × Qop (a, b)  ({b}) satisfying the following conditions: (AC1) (x ∗ ) ∗  = x ∗ ( · ), (x) = a, ∈ Qop (a, b),  ∈ Qop (b, c), (AC2) if (x) = a, then x ∗ 1a = x, (AC3) ∗ preserves arbitrary joins in both variables separately. 



/ obj(Q) iff U is a subset / obj(Q) is a right Q-submodule of a right Q-module X (b) A right Q-module U of X and the inclusion map  from U to X is a Q-module homomorphism — i.e. the commutativity of the subsequent Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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diagram XO 

? U



/ obj(Q) ?     

and the property that the restrictions of  to the respective fibers are arbitrary join preserving and satisfy the condition (u ∗ ) = (u) ∗ for all u ∈ U , b ∈ obj(Q) and ∈ Qop ((u), b). Comment. Let s be the category of sup-lattices (cf. [23]). Then every right Q-module  can be viewed as an T / s-enriched functor Qop s where T (a) = −1 ({a}), a ∈ obj(Q), T (a)

T ( )

/ T (b),

T ( )(x) = x ∗ , ∈ Qop (a, b).

The next remark summarizes some results obtained by Stubbe in a something more general context (cf. [43]). Remark 5.6. (a) Let X = (X, e X , d X ) be a skeletal and cocomplete Q-category. We show that e X carries the structure of a right Q-module. First we notice that the restriction of the intrinsic partial ordering of X to each fiber of e X leads to a complete lattice. In fact, if  is the formation of internal joins, then for every object a of Q the join of a subset Z of e−1 X ({a}) is given by

  z  z∈Z

where  z  (y) = d X (y, z), z ∈ Z , y ∈ X (see the definition of  ). Second, since (X, ) is a TP -algebra, the right action ∗ can be introduced as follows: x ∗ = ( x · ) where ∈ Qop (e X (x), b).

(5.6)

In particular, the axiom (AC1) follows from the commutativity of the diagram (5.5). It is not difficult to verify the remaining axioms (AC2) and (AC3). Hence e X is a right Q-module w.r.t. the restriction of the intrinsic partial ordering to its fibers and the right action defined by (5.6). In this context, we mention that the Q-enriched preorder d X and the structure morphism  can be rediscovered as follows:  (5.7) d X (x, y) = { ∈ Qop (e X (x), e X (y)) | x ∗ y},  x ∗ f (x). (5.8) (a, f ) = x∈X



/ obj(Q) is a right Q-module, then the Q-enriched preorder d X and the structure morphism  are (b) If X defined by (5.7) and (5.8). It is not difficult to show that (X, , d X ) is a skeletal and cocomplete category. Further, (X, , d X ) has the following properties: • The given partial ordering on the fibers of  coincides with the restriction of the intrinsic partial ordering of (X, , d X ). • The given right action on  coincides with the right action induced by (X, , d X ) according to (5.6). (c) It follows from (a) and (b) that skeletal and cocomplete Q-categories and right Q-modules are equivalent concepts. For more details the reader is referred to [43] (see e.g. [43, Theorem 4.12 and Corollary 4.13]). (d) The significance of right Q-modules consists in the way how they express the formation of internal joins. First, the partial ordering on their respective fibers is given by the restriction of the intrinsic partial ordering of the associated skleletal and cocomplete Q-categories. On this basis the formation of internal joins is divided into two parts: the first Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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part is the formation of external joins expressed by the structure of complete lattices; the second part is the tensor condition expressed by the right action (cf. (5.6)). Remark 5.7 (Monad of covariant presheaves). The construction of the data of the monad of covariant presheaves is dual to the construction of the monad of contravariant presheaves. We begin with the Q-category Pr (X) = (Pr (X), er , dr ) of covariant presheaves on X. • Pr (X) is the set of all covariant presheaves on X, • the maps er and dr are defined by  er ( f, a) = a, dr (( f, a), (g, b)) = f (x)  g(x),

(5.9)

x∈X

• Pr (X) is skeletal and its intrinsic partial ordering has the form: ( f, a) (g, b) ⇐⇒ a = b and g(x) ≤ f (x) for all x ∈ X

(5.10)

where ≤ denotes the partial ordering on the respective hom-sets of Q. The object function X  Pr (X) can be completed to an endofunctor Pr acting on morphisms X Pr ()(g, b) = (r (g), b) where  (g, b) ∈ Pr (X), r (g)(y) = g(x) · dY ((x), y), y ∈ Y.



/ Y as follows:

(5.11)

x∈X

Further, the unit r and the multiplication r are given by •X

rX

/ Pr (X) rX

• Pr (Pr (X))

with rX (x) = ( xr , e X (x)) and  xr (y) = d X (x, y), x, y ∈ X , / Pr (X) 

(G)(x) =

with rX (G, b) = (G, b) and

G(g, c) · g(x), x ∈ X.

(g,c)∈Pr (X)

Then the triple TPr = (Pr , r , r ) is a monad on Cat(Q) which we call the monad of covariant presheaves. In this context we mention three facts: • Pr (X) is the free completion of X. • A skeletal Q-category X is complete iff X is a carrier of a TPr -algebra — i.e. there exists a Q-functor Pr (X) making the diagram

rX

X

/X



/X ?      idX  

Pr (X) O



(5.12)

commutative. • Skeletal complete Q-categories and left Q-modules are equivalent concepts. Finally, we note the important fact that cocompleteness and completeness are equivalent properties (cf. [42]). This result follows immediately from the existence of the following Q-functors: 

/ Pr (X),  (a, f ) = (  ( f ), a) with 

 ( f )(x) = f (z)  d X (z, x), (a, f ) ∈ P (X) P (X)

z∈X

Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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and r

/ P (X), r ( f, a) = (a, r ( f )) with Pr (X) 

r ( f )(x) = d X (x, z)  f (z), ( f, a) ∈ Pr (X). z∈X

In order to give a characterization of Q-functors taking their values in the Q-category of presheaves we need the important concept of distributors which goes back to the work of Jean Bénabou 1973 (cf. [6]). Let X and Y be Q-categories. From the point of view of fuzzy sets Q-enriched distributors are simply binary fuzzy relations being leftand right-extensional where the multiplication comes from the composition of morphisms in Q. The precise formulation is given in the next definition. Definition 5.8. Let X and Y be Q-categories. A distributor R : XY is map Y × X the following axioms (cf. [42, Definition 3.2]):

R

/ mor(Q) subjected to

(D1) R(y, x) ∈ Q(e X (x), eY (y)), (D2) dY (y2 , y1 ) · R(y1 , x) ≤ R(y2 , x), (D3) R(y, x1 ) · d X (x1 , x2 ) ≤ R(y, x2 ). Because of (Q1) and (Q3) the map d X is always a distributor from X to X. Distributors R1 : XY and R2 : YZ can be composed  R2  R1 (z, x) = R2 (z, y) · R1 (y, x), (x, z) ∈ X × Z . y∈Y

Because of (Q2) and (Q3) the identity of X w.r.t.  is given by d X . Hence distributors form a category Dist(Q), or more precisely a quantaloid. The relationship between relations and set-valued maps has an extension to distributors and Q-functors taking their values in Q-categories of enriched presheaves. The precise formulation of this result is given in the next proposition which goes back to Stubbe [42]. Proposition 5.9 (Stubbe [42]). Let X and Y be Q-categories. (a) There exists a unique bijective map  from Dist(Q)(X, Y) to Cat(Q)(X, P (Y)) satisfying the following property:  (R)(x) = (e(x), f x ),

f x (y) = R(y, x), (x, y) ∈ X × Y.

(b) There exists a unique bijective map r from Dist(Q)(X, Y) to Cat(Q)(Y, Pr (X)) satisfying the following property: r (R)(y) = ( f y , e(y)),

f y (x) = R(y, x), (x, y) ∈ X × Y.

Proof. Here we only prove (a) and leave the proof of (b) to the reader. Since for every distributor R : XY the axiom (D3) is equivalent to  d(x1 , x2 ) ≤ R(y, x1 )  R(y, x2 ), y∈Y

we conclude from the construction of the Q-category of contravariant presheaves that every distributor R : XY can R / P (Y) and vice versa. In particular, this bijective map  is determined by be identified with a Q-functor X  (R)(x) =  R (x) = (e(x), f x ),

f x (y) = R(y, x), x ∈ X, y ∈ Y.

After these preparations we are ready for the construction of the double presheaf monad which is the composition of the monad of contravariant presheaves with the monad of covariant presheaves. We will return to this aspect later (cf. Theorem 5.10 infra). Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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First we introduce directly the double presheaf monad as an algebraic theory in Cat(Q) where we use the terminology from [30]. We begin with some notation: P2 (X) is the Q-category of all covariant presheaves on the Q-category of all contravariant presheaves on X — i.e. P2 (X) = Pr (P (X)). Because of Proposition 5.9(b) every Q-functor  / P2 (Y) can be identified with its corresponding distributor R : P (Y)X. In particular, if we put (x) = X (x , e X (x)) for all x ∈ X , then the distributor R is given by R (x, (a, g)) = x (a, g), x ∈ X, (a, g) ∈ P (Y).

(5.13)

The data of the intended algebraic theory are defined as follows: (I) The object function X  T (X) is given by: T (X) = P2 (X). X / T (X) of the insertion − o f − the − variables  (II) For every Q-category X the X-component X is determined by the evaluation map — i.e. X (x) = ( x , e(x)), x (a, f ) = f (x), (a, f ) ∈ P (X). (III) The Kleisli composition-function ◦ : Cat(Q)(Y, T (Z)) × Cat(Q)(X, T (Y)) → Cat(Q)(X, T (Z)) is given by X



/ P2 (Y), Y



/ P2 (Z),

R◦ (x, (b, g)) = R (x, (b, g)),

where g(y) = R (y, (b, g)), (b, g) ∈ P (Z), x ∈ X,

y ∈ Y.

It is not difficult to show that T=(T, , ◦) is in fact an algebraic theory (in clone form) in Cat(Q). It is well known that in any algebraic theory the object function can be completed to an endofunctor and the multiplication is induced by the Kleisli composition. In the case of (T, , ◦) we can concretely describe the endofunctor T and the multiplication  as follows.  / Y be a Q-functor. Because of T () = (Y · ) ◦ id 2 Let X P (X) we first compute the distributors RidP2 (X) : P (X)P2 (X) and RY · : P (Y)X corresponding to the Q-functors idP2 (X) and Y ·  according to (5.13): RidP2 (X) ((, b), (a, f )) = (a, f ), (, b) ∈ Pr (P (X)), (a, f ) ∈ P (X),

(5.14)

RY · (x, (a, g)) = g((x)), x ∈ X, (a, g) ∈ P (Y).

(5.15)

Next we apply the Kleisli composition and obtain that for (, b) ∈ Pr (P (X) the expression T ()(, b) = ( , b) has the following form:  (a, g) = (a, gQ), (a, g) ∈ P (Y).

(5.16)

where Q denotes the usual composition of maps. In order to give an explicit description of the multiplication  we proceed as follows. Because of Proposition 5.9(a)  / we first observe that the distributor Rid determines a Q-functor P (X) P (P2 (X)) which is obviously P2 (X)



given by (a, f ) = (a, A(a, f ) ) with A(a, f ) (, b) = (a, f ), (a, f ) ∈ P (X). Then  : T 2 → T has the form: X = idT (X) ◦ idT 2 (X) , X (, b) = ( , b), (, b) ∈ Pr (P (P2 (X))) where  (a, f ) = (a, A(a, f ) ), (a, f ) ∈ P (X).

(5.17)

Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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The triple T = (T, , ) is also called the double presheaf monad on Cat(Q). Theorem 5.10. The double presheaf monad is the composition of the monad of contravariant presheaves with the monad of covariant presheaves — i.e. T = TPr · TP . Proof. First we compute the composition of the endofunctors P and Pr . For this purpose we fix a Q-functor  / Y and consider a covariant presheaf  on P (X) of type b and a contravariant presheaf g on Y of X type c. Then we obtain  (a, f ) · d ((a,  (a, f )), (c, g)) (a, f )∈P (X)

=



⎛ (a, f ) · ⎝

(a, f )∈P (X)

=



(a, f ) ·

(a, f )∈P (X)

=





⎛ f (x)  ⎝

x∈X





⎞⎞ dY (y, (x))  g(y)⎠⎠

y∈Y



f (x)  g((x))

x∈X

(a, f ) · d ((a, f ), (c, g · ))

(a, f )∈P (X)

= (c, gQ). Hence T () = Pr (P ()) holds (cf. (5.16)). Further, we fix x ∈ X and (a, f ) ∈ P (X) and observe: (e X (x),  x )r (a, f ) = d ((e X (x),  x ), (a, f )) = f (x) = x (a, f ). Hence X = rP (X) · X also holds.  Finally, we have to construct a natural transformation (so-called swapper map)  : P · Pr → Pr · P such that the following relation is valid: X = Pr (X ) · rP (P (X)) · Pr (P (X) ). It is well known (cf. [22]) that it is sufficient to verify the commutativity of the following diagrams: Pr (Pr (P (Pr (P (X)))))     Pr (X )     Pr (Pr (P (X))) Pr (P (Pr (P (P (X)))))     P (X)        Pr (P (P (X)))

rP

 (Pr (P (X)))

rP

 (X)

/ Pr (P (Pr (P (X))))     X    / Pr (P (X))

Pr (P (Pr ((X) )))

Pr (X )

/ Pr (P (Pr (P (X))))       X     / Pr (P (X))

(J1)

(J2)

Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Let  be a covariant presheaf on Pr (P (Pr (P (X)))) and f be a contravariant presheaf on X of type a. Then the commutativity of the diagram (J1) follows from the following relation:  (, b) · dr (X (, b), (, c)) · (a, f ) (,b)∈Pr (P (Pr (P (X)))), (,c)∈Pr (P (X))



=

(, b) · X (, b)(a, f )

(,b)∈Pr (P (Pr (P (X))))



=

(, b) · (a, A(a, f ) ))

(,b)∈Pr (P (Pr (P (X))))

where we have used the notation applied in (5.17). In order to verify the commutativity of the diagram (J2) we proceed as follows. Let  be a covariant presheaf on P (Pr (P (P (X)))) of type c. Because of T = Pr · P we first refer to (5.16) and obtain: Pr (P (Pr (X )))(, c) = ( , c) where  (b, ) = (b, QPr (X )), (b, ) ∈ P (Pr (P (X))). Further for every (a, F) ∈ P (P (X)) we define a contravariant presheaf A(a,F) on Pr (P (P (X))) of type a by A(a,F) (, c) = (a, F),

(, c) ∈ Pr (P (P (X))).

Then for (a, f ) ∈ P (X) and (, c) ∈ Pr (P (P (X))) we derive the following relation: A

P (X) (a, f )

(, c)

= (P (X) (a, f )) =



⎛

(a  , F) · ⎝

(a  ,F)∈P (P (X))

=



(a  ,F)∈P (P (X))

⎞



F(b, g)  d ((b, g), (a, f ))⎠

(b,g)∈P (X)

(a  , F) · d (X (a  , F), (a, f )).

Hence for all (a, f ) ∈ P (X) the relation A 

P (X) (a, f )

= A(a, f ) QPr (X )

holds where we recall (a, f ) = (a, A(a, f ) ). Finally, we fix again a covariant presheaf  on P (Pr (P (P (X)))). Because of (5.17) the expression P (X) (, c) has the form: P (X) (, c) = ( , c),  (b, F) = (b, A(b,F) ), (b, F) ∈ P (P (X)). Then for every contravariant presheaf f on X of type a we obtain:  (b, A(b,F) ) · d (X (b, F), (a, f )) (b,F)∈P (P (X))

=



⎛ (b, A(b,F) ) · ⎝

(b,F)∈P (P (X))

= (a, A

P (X) (a, f )



⎞ F(c, g)  d ((c, g), (a, f ))⎠

(c,g)∈P (X)

)

= (a, A(a, f ) QPr (X )). Hence the commutativity of the diagram (J2) is established.  Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Addition. It follows from the swap construction that the corresponding distributive law (so-called swapper transformation)  : P · Pr → Pr · P is determined by (cf. [4,22]): X = X · rP (Pr (P (X))) · P (Pr (X )).

(5.18)

In order to give an explicit description of X sending a contravariant presheaf on Pr (X) to a covariant presheaf on P (X) we first choose (g, b) ∈ Pr (X), (G, c) ∈ Pr (P (X)) and (a, F) ∈ P (Pr (X)). Because of

  G(a, f )  g(x) · f (x) (a, f )∈P (X)

=

 x∈X

=



⎛ ⎝

x∈X



⎞

(G(a, f )  f (x))  g(x)⎠

(a, f )∈P (X)

G(X (x))  g(x),

x∈X

the contravariant presheaf (Pr (X )) (F) on Pr (P (X)) of tye a has the form:  dr ((G, c), Pr (X )(g, b)) · F(g, b) = F(GQX , c). (Pr (X )) (F)(G, c) = (g,b)∈Pr (X)

Hence we conclude from (5.17) and (5.18) that the covariant presheaf H satisfying X (a, F) = (H, a) is explicitly given by  F(GQX , c)  G(a, f ), (a, f ) ∈ P (X).  H (a, f ) = (G,c)∈Pr (P (X))

The importance of Theorem 5.10 emerges among other things, if we recall the debate on the powerset operator foundations for Poslat fuzzy set theories. Remark 5.11. It is Rodabaugh who emphasizes the importance of Zadeh’s powerset operators for “many-valued mathematics” (cf. [33]). In fact Remarks 5.4 and 5.7 can be seen as a partial confirmation of this general insight, because in both cases the endofunctor of the respective presheaf monad is based on Zadeh’s forward powerset operator. In contrast to that Theorem 5.10 assigns an inferior role to Zadeh’s backward powerset operator. We will show how this operator can be constructed from the composition of the endofunctors P and Pr .  / Y be a Q-functor. Then the backward operator attains the form: Let X P (Y)

←

/ P (X),

← (a, g) = (a, gQ), (a, g) ∈ P (Y)

where Q denotes the usual composition of maps. Because of Proposition 5.9(a) ← can be identified with the distributor R← : P (Y)X defined by R← ((a, g), x) = g((x)), (a, g) ∈ P (Y), x ∈ X. Because of Proposition 5.9(b) the distributor R← can again be identified with a Q-functor X which has the following form:



/ Pr (P (X))

 = T () · X = Y · . Since Theorem 5.10 implies T = Pr · P and  = r ·  , the backward operator is obviously uniquely determined by the forward operator — a fact which is also confirmed by the subsequent adjoint situation: d ([P ()](a, f ), (b, g)) = d ((a, f ), ← (b, g)), (a, f ) ∈ P (X), (b, g) ∈ P (Y). In this sense the backward operator is not a primitive concept of the “set-theoretical foundations” of “many-valued mathematics”. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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In the following considerations we view Cat(Q) as an (epi, extremal mono)-category and refer to the characterization of extremal monomorphisms we give above (see also the condition (EM)). Further, we consider always the opposite partial ordering op of the intrinsic partial ordering on the carrier of the Q-category P2 (X) := Pr (P (X)). Then op can be internalized as a partial ordering ≤P2 (X )

P2 (X )

P2 (X )

//

P2 (X )

on P2 (X) in the sense of Cat(Q) such that op is the carrier of the Q-category ≤P2 (X) . Since Q-functors are not only isotone w.r.t. the respective intrinsic partial orderings , but trivially also w.r.t. the corresponding opposite partial orderings op , the endofunctor T of the double presheaf monad factors through the category of partially ordered objects of Cat(Q) where T (X) = P2 (X) is provided with the partial ordering ( 2 , ). P (X) P2 (X) The proof of the isotonicity of the Kleisli composition is based on the following characterization of covariant presheaves on the Q-category of contravariant presheaves. Lemma 5.12. Let b be an object of Q and X be a Q-category. Further, let S b be the carrier of Sb (cf. Example 5.1)  / S b be a map. Then the following assertions are equivalent: and P (X) (i)  is a covariant presheaf on P (X) of type b. (ii)  satisfies the subsequent conditions: (F1)  is isotone w.r.t. the respective intrinsic partial orderings. (F2) For all objects a, c of Q, for all contravariant presheaves f of type a and for all  ∈ Q(c, a) the relation (a, f ) ·  ≤ (c, f · ) holds. Proof. The equivalence of (i) and (ii) follows immediately from the relations:  ≤ d ((a, f ), (c, f · )),  ∈ Q(c, a), f (x) · d ((a, f ), (c, g)) ≤ g(x), x ∈ X.



Theorem 5.13. The double presheaf monad T on Cat(Q) is an ordered monad satisfying the condition (R). Proof. A repeated application of Lemma 5.12 shows that the Kleisli composition is isotone in both variables separately. Hence T is an ordered monad on Cat(Q). In order to verify (R) it is sufficient to prove that for all Q-functors  the Q-functor T () and all components X of the multiplication have left adjoints.  / Y there exists a Q-functor P2 (Y) T ()∗ / P2 (X) determined by (a) In fact, for every X T ()∗ (, b) = (−1 (), b), (, b) ∈ Pr (P (Y)) where −1 ()(a, f ) =



{(a, g)|gQ ≤ f }, (a, f ) ∈ P (X)

(5.19)

and Q denotes the usual composition of maps. Then for all (, b) ∈ Pr (P (X)) and (, b) ∈ Pr (P (Y)) the relations (, b) op T ()(T ()∗ (, b)) and T ()∗ (T ()(, b)) op (, b) hold. (b) Let P (X)



/ P (P2 (X)) be the Q-functor corresponding to the distributor defined in (5.14). Then every 

covariant presheaf  on P (X) of type b induces a covariant presheaf  on P (P2 (X)) of type b as follows:   (a, T ) = {(c, g) · | ∈ Q(a, c), (a, A(c,g) · ) (a, T )}

(5.20)

Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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where (c, g) = (c, A(c,g) ), (a, T ) ∈ P (P2 (X)) and the intrinsic partial ordering on P (P2 (X)) is determined accordX / P2 (P2 (X)). Further, we ing to (5.2). The correspondence (, b)  ( , b) is evidently a Q-functor P2 (X) conclude from (5.17), (5.20), (F1) and (F2) that for all (a, f ) ∈ P (X), (, b) ∈ Pr (P (X)), (a, T ) ∈ P (P2 (X)) and (, b) ∈ Pr (P (P2 (X))) the following relations hold: [X · X (, b)](a, f ) =  ((a, f )) ≥ (a, f ) and [X · X (, b)](a, T ) =

 {((c, g)) · | ∈ Q(a, c), (c, g) ·  (a, T )}

≤ (a, T ). Hence X is left adjoint to X .  Comment. Since in the proof of Theorem 5.13 we have used the opposite partial ordering of the intrinsic partial ordering of P2 (X), we have shown that for all (, c) ∈ Pr (P (Y)), (, b) ∈ Pr (P (X)) and (, c) ∈ Pr (P (P2 (X))) the Q-functors T ()∗ and X satisfy the following conditions: dr (T ()(, b), (, c)) = dr ((, b), T ()∗ (, c)), dr (X (, c), (, b)) = dr ((, c), X (, b)). Hence T () and X have right adjoints in the sense of Q-categories; i.e. T () and X are cocontinuous where we have used the fact that the underlying Q-categories are always skeletal and cocomplete (cf. Remark 5.4, Remark 5.7).  Because of Theorem 5.13 there exist neighborhood systems on Q-categories in the sense of the double presheaf monad. Moreover, an application of Proposition 5.9 will show that these neighborhood systems permit an important characterization by certain extremal subobjects of the Q-category of contravariant presheaves. For this purpose we first explain the role played by coclosures on cocomplete and skeletal Q-categories. Definition 5.14. Let X be a skeletal Q-category and be its intrinsic partial ordering. A Q-functor X called a coclosure on X iff j satisfies the following conditions:

j

/ X is

(cc1) j(x) x f or all x ∈ X , i.e. j id X , (cc2) j = j · j. It follows immediately from (cc1) and (cc2) that coclosures j1 and j2 on a skeletal Q-category X coincide iff j1 (X ) = j2 (X ). Theorem 5.15. Let X be a cocomplete, skeletal Q-category and  be the formation of arbitrary internal joins in X. m / Further, let U / X be an extremal monomorphism. Then the following assertions are equivalent: (i) U is cocomplete and m is cocontinuous. (ii) There exists a coclosure on X with m(U ) = { j(x) | x ∈ X }. Proof. The first observation is that there exists always a Q-functor X  j(x) = (e X (x), f x ) with f x (z) = d X (z, m(u)) · d X (m(u), x)

j

/ X determined by (5.21)

u∈U

where x, z ∈ X . By definition j fulfills the following property: m(u) = j(m(u)), u ∈ U.

(5.22)

x that j satisfies always the axiom (cc1). Since  is necessarily isotone w.r.t. the intrinsic preorder, we infer from f x ≤   / U denotes the formation of arbitrary internal joins (a) We assume that the assertion (i) holds. Then P (U) in U. Since m is an extremal monomorphism, the map u  d X (m(u), x) is a contravariant presheaf on U of type e X (x) Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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for all x ∈ X . Now we make use of the cocontinuity of m and conclude from the commutativity of the diagram P (U)         U

P (m)

m

/ P (X)        /X

that for all x ∈ X the following relation holds: j(x) =  · P (m)(e X (x), d X (m( ), x)) = m · (e X (x), d X (m( ), x)).

(5.23)

Then the idempotence of j follows from (5.22) and (5.23). Hence j is a coclosure on X, and the assertion (ii) is evident. (b) Now we assume the validity of assertion (ii). Then we choose a contravariant presheaf f on U of type a and consider a coclosure j on X with m(U ) = { j(x) | x ∈ X }. Then for u ∈ U we conclude from f (u) = m  ( f )(m(u)) = d (X (m(u)), P (m)(a, f )) ≤ d X (m(u),  · P (m)(a, f )) and the idempotence of j that f (u) ≤ d X (m(u), j ·  · P (m)(a, f )) holds. Hence we obtain  d X (x, m(u)) · f (u) ≤ d X (x, j ·  · P (a, f )), x ∈ X. u∈U

Now we apply the isotonicity of  w.r.t. the intrinsic partial ordering:  · P (m)(a, f ) j ·  · P (m)(a, f ). Hence the relation  · P (m)(a, f ) = j ·  · P (m)(a, f )

(5.24)

follows from (cc1) and the antisymmetry of . Because of (5.24) the Q-functor  · P (m) factors through m — this means that U is cocomplete and m is cocontinuous.  After this digression we return to neighborhood systems on Q-categories. Theorem 5.16. Let X be a Q-category. Then there exists a bijective map between the set of all neighborhood systems on X and the set of all coclosures on P (X). Proof. It follows indeed from Proposition 5.9 that there exists a bijective map between Cat(Q)(X, Pr (P (X))) and Cat(Q)(P (X), P (X)). Since we use the opposite partial ordering w.r.t the intrinsic partial ordering of Pr (P (X)), it is easily seen that (N1) is equivalent (cc1). Referring to the construction of the Kleisli composition of the double presheaf monad the equivalence between the idempotency of neighborhood systems and coclosure operators is also clear.  As an immediate corollary from Theorems 5.15 and 5.16 we obtain: Proposition 5.17. There exists a bijective map between neighborhood systems on X in the sense of the double presheaf m / P (X) satisfying the monad and extremal subobjects of P (X) represented by an extremal monomorphism U / following condition: (O1) U is cocomplete and the embedding m is cocontinuous. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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The axiom (O1) plays evidently the role that U is internally closed under arbitrary joins in the sense of P (X). In this context we recall that the formation of arbitrary internal joins in P (X) is given by the X-component of the multiplication of the monad of contravariant presheaves (cf. Remark 5.4). Before we open the discussion on topological axioms, we try to simplify (O1). Because of (EM) every extremal m / P (X) where the carrier of U is a subobject of P (X) can be represented by an extremal monomorphism U / subset of P (X) and m is the inclusion map. Therewith the axiom (O1) attains the following form:  (O1) For every object b of Q and for every contravariant presheaf F on U of type b the contravariant presheaf F of type b   F(x) = f (x) · F(a, f ), x ∈ X, (a, f )∈U

is an element of U. Referring to Remark 5.6(d) an external characterization of (O1) can be given as follows. Proposition 5.18. Let U be a subset of the carrier of P (X). Then the following assertions are equivalent: (i) U satisfies (O1) . (ii) U fulfills the subsequent properties: (O1) U is closed under fiberwise joins — i.e. {(a, gi ) | i ∈ I } ⊆ U ⇒ (a, (t)  ∈ Q(c, a) and (a, f ) ∈ U ⇒ (c, f · ) ∈ U . (Tensor Condition)



gi ) ∈ U ,

i∈I

Remark 5.19. (a) Proposition 5.18 shows that (O1) describes externally arbitrary joins w.r.t. the intrinsic partial ordering of P (X), while (t) is related to the right action of P (X) (cf. Remark 5.6(a)) and forces the internalization of arbitrary joins. Hence neighborhood systems on a Q-category X in the sense of the double presheaf monad are equivalent to right Q-submodules of the right Q-module of contravariant presheaves on X (cf. Remark 5.4, Definition 5.5(b) and Remark 5.6(a)). (b) Special cases of the tensor 1 condition (t) appear already in the literature. Given the category of sheaves on a locale, the axiom (t) is called the truncation condition and is involved in the internalization of arbitrary unions (cf. [41]); in fuzzy topology (t) plays the role of the so-called constant condition (cf. [29]). m / P (X) Finally, the relationship between neighborhood systems  on X and extremal monomorphisms U / satisfying (O1) can be expressed as follows. For all x ∈ X we put (x , e X (x)) = (x). Then we obtain (a, f ) ∈ m(U )

⇐⇒

∀x ∈ X : f (x) ≤ x (a, f ).

(5.25)

6. Submonads of the double presheaf monad Even though the machinery developed in the previous section works very well — neighborhood systems are identified with so-called “open sets” and vice versa (cf. Proposition 5.17), it is not clear whether this approach covers all traditional axioms of general topology — for example: • The “non-emptiness” of neighborhood systems. • The intersection axiom. Before we solve these problems by certain submonads of the double presheaf monad, we need some further notation. The greatest contravariant presheaf on X of type a is always denoted by a where  a (x) = Q(a, e X (x)), x ∈ X. Analogously, the smallest contravariant presheaf ⊥a of type a evaluated at x is the universal lower bound of Q(a, e X (x)) — i.e. 1 For the definition of tensors in quantaloid enriched categories the reader is referred to [43].

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⊥a (x) =



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Q(a, e X (x)), x ∈ X.

Moreover, the universal bounds of contravariant presheaves on X give rise to Q-functors 1 1

⊥X

X

/ P (X) and

/ P (X) from the terminal object of Cat(Q) to P (X) as follows:

X (a) = (a, a ), ⊥X (a) = (a, ⊥a ), a ∈ obj Q. After these preliminaries we solve first the non-emptiness problem of neighborhood systems. Let  be a covariant presheaf on P (X) of type c. Since covariant presheaves are isotone w.r.t. the intrinsic partial ordering of P (X), the “non-emptiness” of  can be expressed by the property  (F0) (a, a ) = Q(a, c), a ∈ obj(Q). Then the set of all “non-empty” covariant presheaves on P (X) — this means all covariant presheaves satisfying (F0) — form the carrier of an extremal subobject of P2 (X) and leads to a new object function TS . It is not difficult to show that TS determines a submonad of the double presheaf monad which we denote by T S . This submonad is again ordered and satisfies the condition (R) where we have used the following relations: • a = a ·  for all Q-functors , • (a, a ) = Q(a, b) = a (, b) for all (, b) satisfying (F0). Hence there exist neighborhood systems in the sense of T S . A characterization of this type of neighborhoods can be given as follows. Proposition 6.1. There exists a bijective map between neighborhood systems on X in the sense of T S and extremal m / P (X) satisfying the following conditions: subobjects of P (X) represented by an extremal monomorphism U / (O1) U is cocomplete and the embedding m is cocontinuous. X / P (X) factors through m. (O2) The Q-functor 1 Proof. The condition (O2) is an immediate corollary from (5.25) and (F0). All other arguments are evident.  Extremal subobjects of P (X) satisfying the axioms (O1) and (O2) are called generalized topologies. If U is a generalized topology on X, then the pair (X, U) is said to be a generalized topological space. If the underlying quantaloid is given by the Boolean algebra 2, then generalized topologies on sets provided with the discrete partial ordering have been introduced by Sierpi´nski 1928 and were rediscovered by Appert under the name transitive topologies 1934 (cf. [2,40]). The characteristic feature of generalized topologies is the missing intersection axiom. If we impose such an axiom — and this is now our intention, it is necessary to enrich the underlying quantaloid by an extra local binary operation. Definition 6.2. A premultiplication on a quantaloid is a local binary operation  (i.e. a binary operation on each hom-set Q(a, b)) satisfying the following conditions: (pm1)  is distributive over non-empty joins in both variables, (pm2)  is subdistributive over the composition in both variables — i.e. for all a, b, c ∈ obj(Q) and ,  ∈ Q(a, b) the subsequent relations are valid:  · (  ) ≤ ( · )  ( · ),  ∈ Q(b, c) (  ) ·  ≤ ( · )  ( · ),  ∈ Q(c, a). A quantaloid with a premultiplication is also called premultiplicative. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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If Q is a premultiplicative quantaloid, then it is easily seen that also the quantaloid of distributors (cf. [42]) is premultiplicative where the premultiplication is defined pointwisely. Because of Proposition 5.9(a) the premultiplication ⱓ / P (X) with determines trivially a Q-functor P (X) × P (X) (a, f )ⱓ(a, g) := (a, f  g), ( f  g)(x) = f (x)  g(x), x ∈ X. Further, it is not difficult to see that the class of premultiplicative quantaloids is fairly large. With regard to many-valued and non-commutative topologies we select only two examples. Example 6.3. Let us consider the unital quantale I given by the canonical MV-algebra — this is the real unit interval provided with Łukasiewicz arithmetic conjunction ∗ — i.e.

∗  = max( +  − 1, 0), ,  ∈ [0, 1]. Further, let I be the quantaloid with one object induced by I — i.e. obj(I) = {a0 } and I(a0 , a0 ) = I . In this context the binary minimum, but also every binary convex combination defines a premultiplication on I. Example 6.4. Let (L ,  ) be a complete De Morgan algebra — this means that L is a complete (not necessarily distributive) lattice provided with an order reversing involution  . In particular, the universal upper (resp. lower) bound in L is denoted by  (resp. ⊥). Then we construct a quantaloid Q as follows. The set of objects of Q is given by L enlarged by a further element  ∈ / L — i.e. obj(Q) = L ∪ {}. The hom-sets of Q with their respective partial orderings are given by • • • •

Q(a, a) is the two-point lattice for all a ∈ L ∪ {}. Q(a, b) is a singleton, if a, b ∈ L with a  b. Q(, b) = { ∈ L | ≤ b} with the ordering from L, if   b. Q(a, ) = { ∈ L | a  ≤ } with the ordering from L op , if a  .

Then there exists a unique composition law satisfying the following properties: • The composition preserves arbitrary joins in each variable separately. • On Q(a, a) the composition is the meet of the two-point lattice. • If a, b, c ∈ L ∪ {} with a  b and b  c, then the composition attaches the universal lower bound of Q(a, c) to all ( 1 , 2 ) ∈ Q(a, b) × Q(b, c). Finally, the premultiplication on Q is determined as follows: On Q(a, a) we use again the meet, while on hom-sets consisting of a unique morphism the binary operation is evident. In order to complete the situation we have only to define binary operations on Q(, b) and Q(a, ) with a, b ∈ L:   1 , 2  ⊥, 2 , 1  , 1  2 = 1  2 = ⊥, 2 = ⊥.  , 1 = . All this shows that Q is a premultiplicative quantaloid. In the following considerations we always assume that the underlying quantaloid Q is provided with a premultiplication . Further, the universal lower bound in any hom-set of Q is denoted by ⊥. Then the concept of Q-filters on Q-categories can be introduced as follows. Definition 6.5. A covariant presheaf  on P (X) of type b is called a Q-filter on X iff  satisfies the following axioms:  (F0) (a, a ) = Q(a, b), (F3) (a, f )  (a, g) ≤ (a, f  g), (F4) (a, ⊥a ) = ⊥. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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The set F(X) of all Q-filters on X is the carrier of an extremal subobject F(X) of P2 (X) — the Q-category of Q-filters. Then the Q-enriched filter monad TF is a submonad of the double presheaf monad on Cat(Q) determined by the object function X  F(X). Obviously, TF is again an ordered monad. The data for an explicit description of the endofunctor TF and the respective components of the multiplication (cf. (5.16)–(5.17)) can easily be transferred from the double presheaf monad to the Q-enriched filter monad. As an  / F(Y) in the sense of the Q-enriched filter monad. example let us compute the extension  of a Q-functor X First we recall  =  ◦ idF(X) and put (x) = (x , e X (x)) for all x ∈ X . With regard to (5.13) every contravariant presheaf g on Y of type a induces a contravariant presheaf g on X of type a by g(x) = x (a, g), x ∈ X.

(6.1)

Hence for every Q-filter (, b) on X we obtain  (, b) = ( , b) where  (a, g) = (a, g), (a, g) ∈ P (Y).

(6.2)

Since the Q-enriched filter monad is a submonad of the double presheaf monad,  is evidently a Q-filter on Y. For the convenience of the reader we would like to add some comments related to the verification of the Q-filter axiom (F3). For this purpose we choose two contravariant presheaves g1 and g2 on Y of type a. Because of (F3) the relation g1 (x) g2 (x) ≤ g1  g2 (x) holds for all x ∈ X . Now we use (F3) again, but also the isotonicity of covariant presheaves and obtain:  (a, g1 )   (a, g2 ) ≤ (a, g1  g2 ) ≤ (a, g1  g2 ) =  (a, g1  g2 ). With regard to condition (R) we have the following result. Theorem 6.6. Let X a left adjoint F(Y)

 

/ F(Y) be a Q-functor. If the extension  of  is an epimorphism in Cat(Q), then  has

/ F(X).

Proof. For every object a of Q we introduce the set Pa (X) of all contravariant presheaves on X of type a. Obviously, (Pa (X), ) is a groupoid. Further, let (Ia , ) be the free groupoid generated by Pa (X). Then there exists a unique groupoid homomorphism Ia

Ha

/ P a (X) determined by the property: 

Ha ( f ) = f for all f ∈ Pa (X).

(6.3)

Step 1. We fix a Q-filter  on Y of type b. Since  is an epimorphism in Cat(Q), there exists a Q-filter  on X of type b provided with the property (, b)  (, b).

(6.4)

In particular, the relation (c, g) ≤ (c, g), (c, g) ∈ P (Y)

(6.5)

follows from (6.2) and (6.4) where g is defined by (6.1). Moreover, the Q-filter  on Y of type b induces a covariant presheaf 1 on P (X) of type b by  1 (a, f ) = {(c, g) · | ∈ Q(a, c), g ·  ≤ f }, (a, f ) ∈ P (X). If a is the greatest contravariant presheaf on Y of type a, then we infer from (6.1) and (F0) that the relation  a (x) = Q(a, e X (x)), x ∈ X, holds. Hence 1 satisfies (F0), too. Because of (F1), (F2) and (6.5) we obtain for  ∈ Q(a, c) and g ·  ≤ f : (c, g) ·  ≤ (a, g) ·  ≤ (a, g · ) ≤ (a, f ). Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Hence 1 satisfies also (F4). Unfortunately, it is not clear whether 1 satisfies (F3). We will return to this point later. Here we just notice the following properties of 1 : (, b)  (, b)

⇐⇒

( 1 , b) (, b)

(6.6)

and dr ((2 , b2 ), (1 , b1 )) · 11 (a, f ) ≤ 12 (a, f )

(6.7)

for all (a, f ) ∈ P (X) and for all (1 , b1 ), (2 , b2 ) ∈ F(Y). Now we force the Q-filter axiom (F3). For this purpose we fix again an object a of Q and a Q-filter  on Y of type b. Since (Q(a, b), ) is a groupoid, there exists a unique groupoid homomorphism (1,a)

Ia

h (,b)

/ Q(a, b)

determined by (1,a)

h (,b) ( f ) = 1 (a, f ),

f ∈ Pa (X).

Since the premultiplication  is subdistributive over the composition, we obtain from (F3), (6.6) and (6.7) by induction over the length of terms that for all T ∈ Ia and for all (, b), (1 , b1 ), (2 , b2 ) ∈ F(Y) the following relations hold: (1,a)

(1,a)

dr ((2 , b2 ), (1 , b1 )) · h (1 ,b1 ) (T ) ≤ h (2 ,b2 ) (T ), (, b)  (, b) ⇒ h (,b) (T ) ≤ (Ha (T )). (1,a)

(6.8)

Now we introduce a map 1(,b) from P (X) to S b (cf. Example 5.1) as follows. Because of (6.3) we first observe that for all contravariant presheaves f on X of type a the set (1,a)

{h (,b) (T )|T ∈ Ia , Ha (T ) ≤ f } is always non-empty. Then we define 1(,b) by 1(,b) (a, f ) =

 (1,a) {h (,b) (T )|T ∈ Ia , Ha (T ) ≤ f }.

(6.9)

In particular, the relation 1 (a, f ) ≤ 1(,b) (a, f ) holds. Because of (6.8) for every Q-filter  on X of type b we have the following equivalence: (, b)  (, b) ⇐⇒ ∀(a, f ) ∈ P (X) : 1(,b) (a, f ) ≤ (a, f ).

(6.10)

Further, the map 1(,b) satisfies obviously the axiom (F1) and the condition 1(,b) (a, a ) =



Q(a, b).

(6.11)

Since the premultiplication  is distributive over non-empty joins, we conclude from (6.9) that 1(,b) also fulfills axioms (F3). The axiom (F4) follows from (6.10), because  is an epimorphism. Unfortunately, it is not clear whether 1(,b) satisfies (F2). Finally, the property dr ((2 , b2 ), (1 , b1 )) · 1(1 ,b1 ) (a, f ) ≤ 1(2 ,b2 ) (a, f )

(6.12)

follows immediately from (6.8) and (6.9) for all (a, f ) ∈ P (X). Step 2. In order to force the axioms (F2) and (F3) we iterate the construction in the previous step. Based on 1(,b) we define again the following data. We begin with the covariant presheaf 2 on P (X) of type b:  2 (a, f ) = {1(,b) (c, g) · |(c, g) ∈ P (X),  ∈ Q(a, c), g ·  ≤ f }. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Then 2 determines a unique groupoid homomorphism Ia (2,a)

h (,b) ( f ) = 2 (a, f ),

(2,a) h (,b)

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/ Q(a, b) by

f ∈ Pa (X).

Because of (6.10) and (6.12) the covariant presheaf 2 satisfies the conditions (6.6) and (6.7). Hence for all objects a (2,a) of Q the groupoid homomorphism h (,b) fulfills the property (6.8). Again we define a map 2(,b) from P (X) to S b as follows:  (2,a) 2(,b) (a, f ) = {h (,b) (T )|T ∈ Ia , Ha (T ) ≤ f }. (6.13) Then 2(,b) satisfies also (6.10), (6.11), (6.12), (F1), (F3) and (F4). Moreover, we conclude from the previous construction that the additional condition 1 (c, g) ·  ≤ 2 (a, g · ) ≤ 2(,b) (a, g · ),  ∈ Q(a, c)

(6.14)

is valid. Hence, by recursion there exists a nondecreasing sequence (n(,b) )n∈N of maps n(,b) from P (X) to S b satisfying the following properties:  (a) n(,b) (a, a ) = Q(a, b), (b) all maps n(,b) fulfill the axioms (F1), (F3) and (F4), c), n ∈ N, (c) n(,b) (c, g) ·  ≤ n+1 (,b) (a, g · ),  ∈ Q(a,  (d) (, b)  (, b) ⇐⇒ ∀(a, f ) ∈ P (X) : n∈N n(,b) (a, f ) ≤ (a, f ), (e) dr ((2 , b2 ), (1 , b1 )) · n(1 ,b1 ) (a, f ) ≤ n(2 ,b2 ) (a, f ), (a, f ) ∈ P (X). Step 3. Since the sequence (n(,b) )n∈N is nondecreasing and the premultiplication is distributive over non-empty joins, we conclude from (a) to (c) that (n(,b) )n∈N induces a Q-filter  on X of type b as follows:  n(,b) (a, f ), (a, f ) ∈ P (X).  (a, f ) = n∈N

Because of (e) the correspondence (, b)  ( , b) is a Q-functor from F(Y) to F(X) which is left adjoint to  because of (d).  As an immediate corollary from Theorem 6.6 we obtain the fact that the Q-enriched filter monad TF satisfies (R). Hence the regularity axiom can be expressed in the categorical framework determined by TF . The next proposition explains a characterization of neighborhood systems in the sense of TF . Proposition 6.7. There exists a bijective map between neighborhood systems on X in the sense of TF and extremal m / P (X) satisfying the following conditions: subobjects of P (X) represented by an extremal monomorphism U / (O1) U is cocomplete and the embedding m is cocontinuous. X / P (X) factors through m. (O2) The Q-functor 1 (O3) The Q-functor ⱓ · (m × m) factors through m. The previous proposition suggests to introduce the following terminology. Definition 6.8. Let X be a Q-category and P (X) be the Q-category of contravariant presheaves on X. A Q-enriched topology (Q-topology for short) on X is an extremal subobject on P (X) represented by an extremal monomorphism m / P (X) satisfying the axioms (O1), (O2) and (O3). U/ Comment. Obviously, the axiom (O3) plays the role of the intersection axiom for Q-topologies. In this context we emphasize that (O3) depends essentially on the chosen premultiplication! The next remark gives an external description of Q-topologies and explains the existence of initial and final structures. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Remark 6.9. Because of Proposition 5.18 and Remark 5.19(a) Q-topologies on X are evidently equivalent to right / obj(Q) of the right Q-module of all contravariant presheaves on X satisfying the additional Q-submodules U / axioms: (O2) (a, a ) ∈ U, a ∈ obj(Q), (O3) (a, f ), (a, g) ∈ U ⇒ (a, f  g) ∈ U . Since the axioms (O1) –(O3) are preserved under arbitrary non-empty intersections, it follows immediately that the category of topological TF -spaces is topological over Cat(Q). In particular, every subset S of P (X) generates a Q-topology on X as follows:  {V |S ⊆ V ⊆ P (X), V satisfies (O1)−(O3) } (6.15) US = The Q-topology corresponding to the right Q-submodule U S / by S.

/ obj(Q) is called the Q-topology on X generated

Let us now consider some special cases. If the quantaloid is determined by the Boolean algebra 2, then 2-topologies on sets provided with the discrete partial ordering are topologies in the traditional sense. If the quantaloid is given by Example 6.3, then I-topologies on sets provided with the crisp equality are just many valued topologies (see also [20]). If the quantaloid is given by Example 6.4, then the premultiplication is associative and idempotent, but not commutative. In this situation Q-topologies on Q-categories are called non-commutative topologies. Finally, we translate the separation axiom and the density of extremal subobjects into terms related directly to the corresponding Q-enriched topologies. m / P (X) be an extremal monomorphism repreProposition 6.10. Let (X, ) be a topological TF -space and U /  / X is an extremal monomorphism, then the following assertions senting the corresponding Q-topology. If V /

are equivalent:  / X is dense in X w.r.t. . (i) The subobject represented by V / (ii) For all (a, f ) ∈ m(U ) the following implication holds:

∀v ∈ V : f ((v)) = ⊥ ⇒ f = ⊥a . Proof. In order to verify (i) ⇒ (ii) we choose (a, f ) ∈ m(U ) satisfying the property f Q(v) = ⊥ for all v ∈ V  / X is dense, there where  denotes the usual composition of maps. Since the subobject of X represented by V /  / F(V) with the property  TF () · . For all x ∈ X we put (x) = (x , e X (x)) and exists a Q-functor X (x) = (x , e X (x)). Because of (5.25) the relation f (x) ≤ x (a, f ) ≤ [TF ()((x))](a, f ) = x (a, f Q). holds for all x ∈ X . Now we invoke the Q-filter axiom (F4) and obtain x (a, f Q) = ⊥, x ∈ X . Hence f coincides with ⊥a .  / On the other hand, if we assume (ii), then we are able to construct a Q-functor F(X) F(V) by (, b) = ( , b) with  (a, g) =



{(a, f )| f ·  ≤ g}, (a, g) ∈ P (V).

 / X is dense.  Then we observe:  TF () ·  · . Hence the subobject of X represented by V /

The separation axioms of a topological TF -space (X, ) can entirely be expressed by terms of the corresponding m / P (X) is an extremal monomorphism representing the Q-topology of (X, ), then the Q-topology. If U / Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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separation axioms attain the following form: • (X, ) is Kolmogoroff separated iff two distinct elements x and y of X with the same type are separated by some contravariant presheaf contained in the carrier of the image of m — i.e. ∃(a, g) ∈ m(U ) with g(x)  g(y). • (X, ) is Fréchet’s separated iff two distinct elements x and y of X with the same type are separated by a pair of contravariant presheaves (a, g1 ), (b, g2 ) ∈ m(U ) in the following sense: g1 (x) ⱕ g1 (y) and g2 (y) ⱕ g2 (x). The translation of Hausdorff’s separation axiom depends on the premultiplication of the underlying quantaloid. Since this is a more complicated topic, we omit here the details (see Proposition 7.1.3) and turn to a brief historical remark related to many-valued topologies. Historical remark. The characterization of density by assertion (ii) in Proposition 6.10 is equivalent to 0-density introduced by Rodabaugh 1995 (cf. [32, Definition 8.6]). Further, up to our knowledge the formulation of Kolmogoroff’s separation axiom goes also back to Rodabaugh (see his preprint on Poslat Topology 1986 and [31]) and is independently rediscovered by Šostak 1987 (cf. [37,38]). The formulation of Fréchet’s separation axiom is due to Kubiak 1995 (cf. [27]). 7. Examples of Q-enriched topologies 7.1. -Smooth Borel probability measures Let (X, T ) be an ordinary topological space satisfying the T3 -axiom — i.e. (X, O) is separated and regular. A Borel probability measure on X is -smooth iff for every family F of closed sets filtering to the left the following relation holds (cf. P13 in [44]):

 = inf (F).  F∈F

F∈F

Usually the set of -smooth Borel probability measures are topologized by the weak topology (cf. [44]). Here we explain an alternative approach based on many valued topologies (cf. [20]). Therefore we use the canonical MV-algebra I as unital quantale and consider the quantaloid I constructed in Example 6.3. Since I has only one object, we just suppress the type in the study of I-enriched categories. In particular, I-categories are simply fuzzy preordered sets (X, d) — d / i.e. X is a set and X × X [0, 1] is a map subjected to the following axioms: 1 = d(x, x), d(x, y) ∗ d(y, z) ≤ d(x, z) where ∗ denotes the Łukasiewicz arithmetic conjunction (cf. Example 6.3). Further, on I we consider the binary arithmetic mean as premultiplication. In order to translate the Hausdorff separation axiom in terms of I-enriched topologies, we first recall the concept of I-filters. Let X be a I-category. An I-filter on X is a covariant presheaf  on P (X) satisfying the following axioms: (F0) (1) = 1, (F3) (( f ) + (g))/2 ≤ (( f + g)/2), (F4) (0) = 0 where denotes the constant map determined by ∈ [0, 1]. Since in any MV-algebra the law of double negation holds, we conclude from (F0), (F1), (F2) and (F4) that every I-filter  fulfills the so-called constants condition ( ) = , ∈ [0, 1].

(7.1)

Further special properties of I-filters are explained in the next lemmas which are minor modifications of Corollary 4.3.5 and Proposition 4.3.6 in [17]. Lemma 7.1.1. Every I-filter  on X satisfies the following important property: f ∗ g = 0 ⇒ ( f ) ∗ (g) = 0. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Proof. Because of the special relationship between Łukasiewicz arithmetic conjunction and the binary mean   ( ∗ ) + 1

+ 1 = max , 2 2 2 the assertion follows immediately from (7.1) and (F3).



Lemma 7.1.2. Let 1 and 2 be I-filters on X. Then the following assertions are equivalent: (i) There exists an I-filter 3 on X s.t. 1 3 and 2 3 . (ii) For all f, g ∈ P (X) the following implication holds: f ∗ g = 0 ⇒ 1 ( f ) ∗ 2 (g) = 0. Proof. The implication (i) ⇒ (ii) follows from Lemma 7.1.1. If (ii) holds, then the covariant presheaf 3 defined by  {1 (g1 ) ∗ 2 (g2 )|g1 ∗ g2 ≤ f }, f ∈ P (X), 3 ( f ) = is a I-filter on X. Because of (F0) the property 1 3 and 2 3 holds. Hence the implication (ii) ⇒ (i) is verified.  m / P (X) be an extremal monomorphism Proposition 7.1.3. Let  be a neighborhood system on X and U / representing the corresponding I-topology on X. Then the following assertions are equivalent:

(i) The topological TF -space (X, ) is Hausdorff separated. (ii) For all x, y ∈ X with x  y there exist contravariant presheaves g1 , g2 ∈ m(U ) provided with the following property: g1 (x) ∗ g2 (y)  0 and g1 ∗ g2 = 0. Proof. For all x ∈ X we put (x) = x . Then the carrier of the limit relation associated with  has the form lim = {(, x) ∈ F(X) × X | x }. Then we conclude from Lemma 7.1.2 that the restriction of the projection onto the first component of lim is a monomorphism in Cat(I) (i.e. is injective) iff for all x, y ∈ X with x  y the following relation holds: ∃ f, g ∈ P (X) s.t. x ( f ) ∗  y (g)  0 and f ∗ g = 0. Hence the equivalence (i) ⇐⇒ (ii) follows from the neighborhood axioms (N1) and (N2) and from the relation (5.25).  Comment. It should be noted that the Hausdorff separation axiom defined in Definition 4.1 is stronger than that introduced by Kubiak in [27]. After these theoretical preparations we return to our example. The set of all -smooth Borel probability measures on f / [0, 1]. a T3 -space X is denoted by M1 . Further, let G(X ) be the set of all lower semicontinuous functions X Then every function f ∈ G(X ) induces a map M1  A f () = f d,  ∈ M1 .

Af

/ [0, 1] by (7.2)

X

Now we introduce a fuzzy preorder d on M1 : d(1 , 2 ) =

inf A f (2 ) → A f (1 )

f ∈G (X )

(7.3)

where → denotes Łukasiewicz’ implication — i.e. →  = min(1 − + , 1) for ,  ∈ [0, 1]. Then M = (M1 , d) is a I-category and every map A f is a contravariant presheaf on M. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Now, let U be the subset of all contravariant presheaves T on M which have the following representation: T () = sup(Agi ()) ∗ i ,  ∈ M1 .

(7.4)

i∈I

It is not difficult to show that U is a I-submodule of the I-module P (M) and satisfies the axioms (O2) and (O3) .  / Further, let M F(M) be the neighborhood system corresponding to U (cf. Proposition 6.7 and Remark 6.9). Then (M, ) is a topological TF -space and the corresponding I-topology is represented by U /  is the inclusion map from U to P (M). Since every ordinary filter F on M1 can be identified with an I-filter F determined by   F (T ) = sup inf T () , T ∈ P (M), F∈F



/ P (M) , where

∈F

we can characterize weak convergence as follows: An ordinary filter F is weakly convergent to a Borel probability measure  (see [44]) iff the pair (F , ) is an element of the carrier of the limit relation corresponding to  (cf. the pullback square (3.6)).  / P (M) is also called the I-enriched topology of weak converTherefore the I-topology represented by U / gence. As an immediate corollary from (7.2) and Proposition 6.10 we obtain the following result. Lemma 7.1.4. The extremal subobject determined by the set of all Dirac measures on X is dense in M w.r.t. . Proposition 7.1.5. The topological TF -space (M, ) is Hausdorff separated. Proof. It is sufficient to prove assertion (ii) in Proposition 7.1.3 which has already been verified in [18]. For the convenience of the reader we recall here the most important arguments. Let 1 and 2 be distinct -smooth Borel probability measures. Then there exists an open subset G of X with 1 (G)  2 (G) e.g. 1 (G) < 2 (G). Since (X, T ) is regular and all Borel probability measures are -smooth, we now choose disjoint open subsets G 1 and G 2 of X provided with the following properties: X ∩ G ⊆ G 1 , G 2 ⊆ G, 1 (G) < 2 (G 2 ). Then we obtain from the additivity and isotonicity of probability measures: (G 1 ) ∗ (G 2 ) = 0,  ∈ M1 , 1 (G 1 ) ∗ 2 (G 2 ) ≥ 1 (X ∩ G) + 2 (G 2 ) − 1 = 2 (G 2 ) − 1 (G) > 0. Finally, we identify open subsets with their characteristic functions and conclude from the previous observations that AG and AG satisfy the assertion (ii) in Proposition 7.13.  1

2

Since X is separated, the restriction of the fuzzy preorder d to the set of Dirac measures coincides with the discrete (pre)order. Hence we can summarize Lemma 7.1.4 and Proposition 7.1.5 as follows. The I-enriched topology of weak convergence fulfills two important properties: (1) Dirac measures form a dense subobject of the I-category of -smooth Borel probability measure, and (2) the I-enriched topology of weak convergence is Hausdorff separated. As a matter of fact such a situation is impossible on the basis of traditional, non-enriched topologies! For instance it is well known that the traditional weak topology is Hausdorff separated, but only Borel probability measures with finite support are dense (cf. [44]). Let us illustrate this situation by the following example. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Example 7.1.6. It is well known that the real unit interval [0, 1] can be regarded as the set of all Borel probability measures on {0, 1} provided with the discrete topology. Then the situation is as follows: • The traditional non-enriched topology of weak convergence on [0, 1] coincides with the usual topology on [0, 1]. • Every Borel probability measure has finite support. • The fuzzy preorder on [0, 1] is symmetric and has the form (cf. (7.3)): d(x1 , x2 ) = 1 − |x2 − x1 |, x1 , x2 ∈ [0, 1]. • The set of functions [0, 1]

g 

/ [0, 1]

determined by

  g  (x) = max ( − )x +  +  − 1, 0 , x, , ,  ∈ [0, 1] form a “basis” for the I-enriched topology of weak convergence on [0, 1] (cf. (7.4)). • {0, 1} is dense in [0, 1] w.r.t. the I-enriched topology of weak convergence. 7.2. The spectrum of C ∗ -algebras In this subsection we show that the spectrum of a non-commutative C ∗ -algebra A induces a Q-enriched topology on the Q-category of all locally irreducible representations of A in the sense of an appropriately chosen quantaloid Q. The spectrum (A) of a unital C ∗ -algebra A is always understood as the idempotent quantale of all closed rightideals (cf. [34]). Further, we agree with the conception that irreducible representations play the role of “points” for non-commutative C ∗ -algebras (cf. [1, p. 14]). By means of the GNS-construction every pure state of a C ∗ -algebra A induces an irreducible representation (H ,  ) of A (cf. [24]). In contrast to the commutative setting it is interesting to note that in the case of non-commutative C ∗ -algebras the Hilbert space dimension of the underlying Hilbert space H depends on the pure state and might possibly vary. In order to overcome this obstacle and to choose an underlying Hilbert space which is independent from the respective pure states, we recall some terminology introduced in [21,19]. Remark 7.2.1. Let P(A) be the set of all pure states of a unital C ∗ -algebra A. (a) A Hilbert space H is admissible for A iff for every pure state ∈ P(A) there exists an isometry H C ∗ -algebra



/ H.

Of course every has an admissible Hilbert space. (b) A representation (H, ) of A is called locally irreducible iff H is admissible for A and there exists a pure state of  / H s.t. for all a ∈ A the following relation holds: A and an isometry H (a) =  ◦  (a) ◦ ∗ , a ∈ A where H

∗

/ H

denotes the adjoint operator corresponding to  .

Now we fix a unital C ∗ -algebra A and choose an admissible Hilbert space H for A. Further, let L(H) be the C ∗ algebra of all bounded linear operators, L be the complete lattice of all closed linear subspaces of H provided with the orthogonal complement as order reversing involution, and let Q be the quantaloid induced by the complete De Morgan algebra (L , ⊥ ) according to Example 6.4 (see also the quantaloid constructed in [19, Section 6]).  / Finally, let X be the set of all ∗-homomorphisms A L(H) s.t. (H, ) is a locally irreducible representation of A. First, we construct the Q-category X = (X, e X , d X ) of locally irreducible representations. We observe that every fI / L mor(Q) as follows: closed right ideal I of A induces a map X f I () = {x ∈ H | ∀b ∈ I ∗ : (b)(x) = 0}⊥ where I ∗ = {a ∗ | a ∈ I } is the closed left-ideal being adjoint to I. The next theorem is a version of Proposition 4 in [8] which we provide with a direct proof. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Theorem 7.2.2. The family { f I | I ∈ (A)} satisfies the following properties: (C1) If  f I2 ()  ⊥, then f I1 () = f I1 ·I2 (). (C2) j∈J f I j () = f  j∈J I j (). (C3) The correspondence I  f I is injective. Proof. Let (H, ) be a locally irreducible representation of A. Then there exists a pure state () and an isometry  () / H s.t. the relation H (a) =  () ◦  () (a) ◦ ∗ () holds for all a ∈ A. In order to verify (C1) we proceed as follows. If f I2 () is not the trivial subspace, then there exist v ∈ H and c ∈ I2 s.t. (c∗ )(v)  0. Now we put z =  () (∗ () (v)). Then (c∗ )(z) = (c∗ )(v)  0 holds. It is sufficient to show: f I1 ·I2 ()⊥ = f I1 ()⊥ . The set-inclusion f I1 ()⊥ ⊆ f I1 ·I2 ()⊥ is obvious. Now we choose x ∈ / f I1 ()⊥ — this means that there exists b0 ∈ I1∗ s.t. (b0 )(x)  0. Since  () is irreducible, we obtain  () (∗ () (H)) = {(a)((b0 )(x)) | a ∈ A}.   Hence there exists an a0 ∈ A with z = (a0 ) (b0 )(x) . Since I1 is right-sided, a0 · b0 is an element of I1∗ . Now we use the fact that  is an algebra homomorphism and obtain: (c∗ · a0 · b0 )(x)  0. Since b0∗ · a0∗ · c is an element of I1 · I2 , we conclude from the previous relation that x is not contained in f I1 ·I2 ()⊥ . Hence we have proved: f I1 ·I2 ()⊥ ⊆ f I1 ()⊥ . The property (C2) follows immediately from  ( f I j ())⊥ = ( f  j∈J I j ())⊥ . j∈J

Finally, we verify (C3). For this purpose we choose I1 , I2 ∈ (A) with I1  I2 — e.g. I1 I2 . Since every proper closed left-ideal is an intersection of maximal left-ideals (cf. Theorem 10.2.10(iii) in [25]), there exists a maximal left-ideal Im provided with the following properties: I2∗ ⊆ Im ,

I1∗ Im .

(7.5)

Now we refer to Theorem 10.2.10(ii) in [25] and choose a unique pure state s.t. its left-kernel coincides with Im . Let (H ,  ) be the irreducible representation of A induced by according to the GNS-construction. Since H is admissible  / H. In particular, (H, ) determined by for A, there exists an isometry H (a) =  ◦  (a) ◦ ∗ , a ∈ A, is a locally irreducible representation of A. Further, let e be the unit of A. Then it follows again from the GNS-construction that [e] is a unit vector of H . Now we put x =  ([e]) and conclude from (7.5) that the following relations hold: ∀ b ∈ I2∗ : (b)(x ) = 0, ∃ a ∈ I1∗ : (a)(x )  0. Hence f I1 ()⊥  f I2 ()⊥ .  Now we are in a position to carry out the construction of the Q-enriched topologization of the spectrum (A) which we have announced at the beginning of this subsection. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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Construction. Define X

eX

/ obj(Q) and X × X

e X () = f A (), d X (1 , 2 ) =

 

dX

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43

/ mor(Q) by

 f I (1 )  f I (2 ) where f I () ∈ Q(, f A ()).

I ∈(A)

Then X = (X, e X , d X ) is the Q-category of all locally irreducible representations of A, and all maps f I turn out to be contravariant presheaves on X of type  where  is the extra element added to L according to Example 6.4. In particular, the relation f I1 ·I2 = f I1  f I2 ,

I1 , I2 ∈ (A),

follows from (C1). Hence we conclude from (C1) and (C2) that the set S = {(a, a ), (a, ⊥a ) | a ∈ L} ∪ {(, f I ) | I ∈ (A)} satisfies the topological axioms (O1) , (O2) and (O3) , but not necessarily the tensor condition (t) in Proposition 5.18 which is responsible for the internalization of the formation of arbitrary joins (cf. Remark 5.19(a)). Because of (C3) every closed right-ideal I can be identified with the “open” contravariant presheaf f I . Thus the Q-topology generated by S (cf. Remark 6.9) is the non-commutative topology associated with the spectrum of A. The    A / P (X). right Q-submodule corresponding to the non-commutative topology of A is always denoted by U  We finish this section with a description of “open” contravariant presheaves in the finite dimensional case. Remark 7.2.3. Let H be a finite dimensional Hilbert space and A = L(H) be the C ∗ -algebra of all linear operators on H. Then the Hilbert space occurring in the GNS-construction does not depend on the respective pure state and is isomorphic  / to H. In this situation the carrier X of the Q-category X consists of all ∗-homomorphisms L(H) L(H) s.t. (, H) is an irreducible representation of L(H), and the map e X has the form e X () = H for all  ∈ X . Obviously, / {, H}. Further, the greatest contravariant presheaf H on X of type H a = ⊥a for all objects a of Q with a ∈ coincides with the constant function 1H determined by the unit of Q(H, H). Because of (O2) the tensor condition (t) implies that for all  ∈ Q(, H) the constant contravariant presheaf H ·  =  on X of type  is contained in the  A / P (X) — i.e. (, ) ∈ U . Since L(H) does not contain any non-trivial, proper two-sided ideal, the carrier of U  relation f I ()  0,  ∈ X holds for every non-trivial right ideal. Now we refer to the premultiplication on Q(, H) (cf. Example 6.4) and obtain for   0 and I  0: f I   = f I and   f I = . Then the carrier U of the right Q-submodule U

   A / P (X) is given by 

U = {(, f I ∨ ) | I ∈ (L(H)),  ∈ Q(, H)} ∪ {(H, H )} ∪ {(a, ⊥a ) | a  }. Hence contravariant presheaves determined by right ideals and constant contravariant presheaves of type  form together a “basis” of the non-commutative topology of L(H). Acknowledgments I am very grateful for the great support I received by my friend T. Kubiak. The exchange of ideas on various topics of this paper was very successful. Also the comments by I. Stubbe at the Linz Seminar 2012 and during the preparation of the paper have been very helpful and are gratefully acknowledged. Please cite this article as: U. Höhle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.010

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