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**Patrik LUNDSTRÖM ^{*}**

Department of Engineering Science, University West, Trollh¨attan, Sweden

- *Corresponding Author:
- Patrik LUNDSTRÖM

Department of Engineering Science

University West, Trollh¨attan, Sweden[email protected]

E-mail:

**Received date:** November 27, 2007; **Revised date:** March 13, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 [13] 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 [5] 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 [8] for an overview). Fuzziness has also been introduced in various topological settings, e.g. fuzzy topological spaces [4], fuzzy measurable spaces [9] and fuzzy topological groups [2]. Categories of fuzzy structures are often topological over their corresponding base categories, that is, the category of modules in the case of fuzzy modules [10], the category of groups in the case with fuzzy topological groups [3] 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. [1]) 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 [12] 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 id_{a}. We let *hom _{C}*(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 U

**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 [1] 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 [7].

**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 [1].

**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 s_{1} = s_{2} 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 U_{b} . 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 [1]). 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 a^{j }be a weak U-initial lift of b^{j}
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 [12] 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
[5] 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 Set^{J} 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, LT^{J} is (co)complete whenever Set^{J} is (co)complete, for any category J. The category LTop of L-topological spaces (Chang [4]) 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 [9]) 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, *LTop ^{J}* and
LMeas

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- ChangCL (1968) Fuzzy topological spaces. J Math Anal Appl24: 182–190.
- JA Goguen (1967) L-fuzzy sets. J Math Anal Appl18: 145–174.
- GrothendieckA (1961) Cat ́egoriesfibr ́eesetdescente. S ́em.g ́eom. alg ́ebrique I.H.E.S. Paris.
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- KandasamyWBV (2003) Smarandache Fuzzy Algebra.American Research Press.
- KlementEP (1980) Fuzzy sigma-algebras and fuzzy measurable functions. Fuzzy Sets and Systems 4: 83-93.
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