Department of Engineering Science, University West, Trollh¨attan, Sweden
Received date: November 27, 2007; Revised date: March 13, 2008
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We introduce weak topological functors and show that they lift and preserve weak limits and weak colimits. We also show that if A ! B is a topological functor and J is a category, then the induced functor AJ ! BJ is topological. These results are applied to a generalization of Wyler’s top categories and in particular to functor categories of fuzzy maps, fuzzy relations, fuzzy topological spaces and fuzzy measurable spaces.
Almost forty years ago, Zadeh  introduced the category of fuzzy sets. The objects in this category are maps from ordinary sets to the unit interval and the morphisms are ordinary nondecreasing maps with respect to the fuzzy sets. Two years later, Goguen  replaced the unit interval by an arbitrary complete ordered lattice. Since then a lot of work has been devoted to proposing different versions of what fuzzy algebras of distinct types may be, e.g. fuzzy semigroups, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings (see  for an overview). Fuzziness has also been introduced in various topological settings, e.g. fuzzy topological spaces , fuzzy measurable spaces  and fuzzy topological groups . Categories of fuzzy structures are often topological over their corresponding base categories, that is, the category of modules in the case of fuzzy modules , the category of groups in the case with fuzzy topological groups  and so on. This is an important fact since it implies that many properties of the top category, such as completeness and cocompleteness, are automatically inherited from the ground category. However, the theory of topological functors (see e.g. ) does not cover the lifting of weak limits and colimits (that is, the uniqueness property of the mediating morphism is dropped). To cover these cases, we introduce weak topological functors (see Definition 2.1) and show a lifting result for such functors (see Proposition 2.2). To simultaneously treat all fuzzy algebraical constructions, we show a result (see Proposition 2.3) concerning the topologicality of functors between functor categories. In the end of the article (see Section 3), these results are applied to a generalization (see Proposition 2.5) of Wyler  to prove (weak) completeness and cocompleteness results for functor categories of categories of fuzzy maps, fuzzy relations, fuzzy continuous maps and fuzzy measurable maps.
Let A be a category. The family of objects and the family of morphisms in A is denoted ob(A) and mor(A) respectively. The domain and codomain of a morphism α in A is denoted d(α) and c(α) respectively. The composition of two morphisms α and β in A with d(α) = c(β) is denoted αβ. The identity morphism at a 2 ob(A) is denoted ida. We let homC(a, b) denote the collection of morphisms from a to b. Let Cat denote the category with small categories as objects and functors between such categories as morphisms. If B is another category, then let BA denote the category with functors from A to B as objects and natural transformations between such functors as morphisms. If then we let Ub denote the corresponding fibre category in the sense of Grothendieck , that is, the subcategory of A having as objects all and as morphisms all in mor(A) between such objects with A (simple) source in B is a pair consisting of an object b in B and a family of morphisms in B indexed by some class I (with cardinality one). If for some objects then S is called U-structured. A U-lift of such a U-structured source S is a source in A where satisfy We say that such a U-lift of S is weakly initial if for any U-lift of a U-structured source equipped with a in B with the property that there is a in A with and A weak initial U-lift is called initial if the morphism is unique. Concepts dual to ”initial” are called ”final”.
Proposition 2.1. Let U : A → B be a functor. (a) If each U-structured source (sink) in B has a weak initial (final) U-lift, then U is faithful. (b) Each U-structured source in B has a weak initial U-lift if and only if each U-structured sink in B has a weak final U-lift. (c) Each U-structured source in B has a unique initial U-lift if and only if each U-structured sink in B has a unique final U-lift.
Proof. Adapt the proofs of Theorems 21.3 and 21.9 in  to the weak situation.
Definition 2.1. We call a functor satisfying either of the equivalent criteria in Proposition 2.1(c) (or (b)) (weakly) topological. Note that our definition of topological functor coincides with the one given by Herrlich .
Proposition 2.2. Let U : A → B be a weakly topological functor. (a) If F : J → A is a functor with weak (co)limit L, then U(L) is a weak (co)limit of UF; (b) If L is a weak (co)limit of UF, then there is a weak (co)limit of F such that is weakly (co)complete if and only if B is weakly (co)complete; (d) If U is topological, then (a) and (b) hold with weakness removed. Furthermore, in that case, the (co)limit is unique subject to the condition
Proof. We only show the ”limit” part of the result. The proof of the ”colimit” part is dual and is therefore left to the reader. (a) Suppose that L is a limit of F : J → A with morphisms such that in J. Suppose that there are with the property that By weak topologicality of U, there is a U-lift with By Proposition 2.1(a) U is faithful. Therefore the equality implies that Then there is with This implies that is the desired map. Hence U(L) is a weak limit of UF. Now we show (b). Suppose that L is a weak limit of UF with morphisms such that for all α : i → j in J. Let be a weak initial U-lift of the U-structured source . By U-faithfullness, for all morphisms α : i → j in J. Now suppose that there is an object X in A and morphisms such that for all morphisms α : i → j. Since L is a weak limit of UF, there is a morphism By weak initiality there is a morphism in A such that which implies that is a weak limit of F. (c) follows directly from (a) and (b). (d) is Proposition 21.5 in .
Proposition 2.3. If J is a category and U : A → B is a topological functor, then the induced functor is topological.
Proof. Let structured source in BJ . Associated to this source, we define a functor F : J → A in the following way. For each be the unique initial lift of the U-structured source Take a morphism in J. Since, there is, by initiality of the lift, a unique with the property that Take another morphism in J. By uniqueness and initiality of and the fact that the following chain of equalities holds we get that Also note that since U is topological, the functor F is unique subject to the condition that Define a collection of natural transformations by the morphisms By the construction of the we get that they are indeed natural transformations. Suppose that and are natural transformations satisfying Suppose that there are natural transformations with Now we define a natural transformation with in the following way. Since U is topological there is for each a unique morphism in A subject to the conditions that and Define p by the morphisms Take a morphism in J. By the definition of F, we get that Therefore, Since each is a natural transformation, we get that and hence that Since is the unique initial lift of is a lift of and the morphisms satisfy we get that s1 = s2 and hence that p is natural transformation.
Let U : A → B be a functor. If U is weakly topological then each fibre of U has weak (co)products. In fact, every simple U-structured source in B has a weak U-initial lift. Take and suppose that we have a set of objects in Ub . Then the identity morphisms considered as a U-structured source, has a weak initial U-lift The weak initiality of this lift is equivalent to the condition that is a weak product of In the same way one can show that if U is topological, then the weak (co)product above makes each fibre of U a complete ordered lattice (see Proposition 21.11 in ). If we make two additional assumptions, then this condition is sufficient for U to be weakly topological:
Proposition 2.4. (a) A functor U : A → B is weakly topological if the following three properties hold: (i) Each simple U-structured source in B has a weak initial lift; (ii) Each fibre of U has weak products; (iii) The weak products in the fibres of U are weak products in A. (b) A functor U : A → B is topological if the following three properties hold: (i) Each simple U-structured source in B has a unique initial lift; (ii) Each fibre of U is a complete ordered lattice; (iii) The infima in the fibres of U are products in A.
Proof. Suppose that (i), (ii) and (iii) in (a) hold. Suppose that Ustructured source with For each let aj be a weak U-initial lift of bj with morphisms Let a be a weak product in with morphisms Put Suppose that satisfy Suppose that is a U-lift of By simple U-initiality, there are morphisms By (iii) there is a morphism in A such that Therefore Therefore (a) holds. (b) is a modification of (a).
Now we recall a folkloristic generalization of Wyler’s  construction of top categories. Let denote a contravariant functor. The category has as objects all pairs and as morphisms all pairs where in A and The composition of and is defined by The identity morphisms are defined by In the sequel, the forgetful functor will be denoted U. By the discussion preceding Proposition 2.4, we get that if U is weakly topological, then each has weak (co)products and that if U is topological, then each is a complete ordered lattice. If we make an additional assumption, then this condition is sufficient for U to be weakly topological:
Proposition 2.5. (a) If each has weak products and each respects weak products, then U is weakly topological.
(b) If each is a complete ordered lattice and each respects infima, then U is topological.
Proof. (a) Conditions (ii) and (iii) of Proposition 2.4(a) are immediate by the assumptions. Now we show condition (i) of Proposition 2.4(a). Suppose that is a morphism in A and Clearly is a U-lift of α. Suppose now that are morphisms in A satisfying Suppose that are lifts of respectively, satisfying Since the left hand side of this equation simplifies to we have that This calculation shows that is a U-initial lift of α. (b) is a modification of the proof of (a) using Proposition 2.4(b).
The category CfzSet. Let X be a set and C a category. As a generalization of Goguen’s  definition of fuzzy sets, we say that a C-fuzzy set on X is a function Let be another C-fuzzy set. We say that a fuzzy function from μ to v is a pair (f, α) where f : X → Y is a function and This is indicated by writing be a third fuzzy set and a second fuzzy function. Let the composition of be defined by where It is easy to check that the collection of C-fuzzy sets and C-fuzzy functions form a category which we denote denote the forgetful functor. Now we wish to present as a top category. For each set X, let Note that the objects in are the C-fuzzy sets on X and the morphisms in are all ordered triples where μ and v are C-fuzzy sets on X and is a function, let be defined by and where The correspondence is a contravariant functor from Set to Cat and there is an isomorphism of categories. Hence, by Proposition 2.5 and a straightforward argument, the forgetful functor is (weakly) topological if C is (has weak products) a complete ordered lattice. Therefore, by Proposition 2.2, is (weakly) complete and cocomplete if C is a complete ordered lattice (has products or coproducts respectively). Furthermore, if we assume that C is a complete ordered lattice and J is a category, then, by Proposition 2.3, the category is (co)complete whenever SetJ is (co)complete.
The category LfzRel. For the rest of the article, let L be a complete ordered lattice. Suppose that are L-fuzzy sets. We say that a relation is that are L-fuzzy sets. We say that a relation is L-fuzzy if The collection of L-fuzzy sets and L-fuzzy relations form a category LfzRel. Define a functor on objects in the same way as for LfzSet and on relations where the infimum is taken over all with The correspondence is a contravariant functor and there is an isomorphism of categories. Therefore, by Proposition 2.5(b), the forgetful functor U : is topological. Hence, by Proposition 2.2(c), LfzRel is weakly complete and cocomplete.
The categories LT, LTop and LMeas. We define the category LT as the category with objects where X is a set and μ is a subset of the set of maps from X to L. A morphism in LT is a function f : X → Y satisfying Now we present LT as a top category. For each set X, let be the collection of subsets of ordered under reversed inclusion. If is a function and v is a subset of It is easy to see that there is an isomorphism of categories. Therefore, by Proposition 2.5(b), the forgetful functor U : LT → Set is topological. Therefore, by Propositions 2.2 and 2.3, LTJ is (co)complete whenever SetJ is (co)complete, for any category J. The category LTop of L-topological spaces (Chang ) is the full subcategory of LT with objects (X, μ) where μ is an L-fuzzy topology on X, that is, subsets of LX closed under finite pointwise infima and arbitrary pointwise suprema. If L is a complemented lattice, then the full subcategory LMeas of L-measurable spaces (Klement ) is defined as the full subcategory of LT with objects (X, μ) where μ is an L-sigma algebra, that is, a subset of LX closed under countable pointwise suprema and pointwise complements. By following the argument above, it is a straightforward task to show that the forgetful functors U : LTop → Set and U : LMeas → Set are topological. Therefore, by Propositions 2.2 and 2.3, LTopJ and LMeasJ are (co)complete whenever SetJ is (co)complete, for any category J.